Convex hull

From Rosetta Code
Task
Convex hull
You are encouraged to solve this task according to the task description, using any language you may know.

Find the points which form a convex hull from a set of arbitrary two dimensional points.

For example, given the points (16,3), (12,17), (0,6), (-4,-6), (16,6), (16,-7), (16,-3), (17,-4), (5,19), (19,-8), (3,16), (12,13), (3,-4), (17,5), (-3,15), (-3,-9), (0,11), (-9,-3), (-4,-2) and (12,10) the convex hull would be (-9,-3), (-3,-9), (19,-8), (17,5), (12,17), (5,19) and (-3,15).

See also



11l[edit]

Translation of: Nim
F orientation(p, q, r)
   V val = (q.y - p.y) * (r.x - q.x) - (q.x - p.x) * (r.y - q.y)
   I val == 0
      R 0
   R I val > 0 {1} E 2

F calculateConvexHull(points)
   [(Int, Int)] result

   I points.len < 3
      R points

   V indexMinX = 0
   L(p) points
      V i = L.index
      I p.x < points[indexMinX].x
         indexMinX = i

   V p = indexMinX
   V q = 0

   L
      result.append(points[p])

      q = (p + 1) % points.len

      L(i) 0 .< points.len
         I orientation(points[p], points[i], points[q]) == 2
            q = i

      p = q
      I p == indexMinX
         L.break

   R result

V points = [(16, 3), 
            (12, 17), 
            (0, 6), 
            (-4, -6), 
            (16, 6), 
            (16, -7), 
            (17, -4), 
            (5, 19), 
            (19, -8), 
            (3, 16), 
            (12, 13), 
            (3, -4), 
            (17, 5), 
            (-3, 15), 
            (-3, -9), 
            (0, 11), 
            (-9, -3), 
            (-4, -2), 
            (12, 10)]

V hull = calculateConvexHull(points)
L(i) hull
   print(i)
Output:
(-9, -3)
(-3, -9)
(19, -8)
(17, 5)
(12, 17)
(5, 19)
(-3, 15)

Action![edit]

DEFINE POINTSIZE="4"
TYPE PointI=[INT x,y]

CARD FUNC GetAddr(INT ARRAY points BYTE index)
RETURN (points+POINTSIZE*index)

BYTE FUNC GetMinXIndex(INT ARRAY points BYTE pLen)
  PointI POINTER p
  BYTE i,index
  INT minX

  p=GetAddr(points,0)
  minX=p.x
  index=0
  FOR i=1 TO pLen-1
  DO
    p=GetAddr(points,i)
    IF p.x<minX THEN
      minX=p.x
      index=i
    FI
  OD  
RETURN (index)

BYTE FUNC Orientation(PointI POINTER p,q,r)
  INT v
  v=(q.y-p.y)*(r.x-q.x)
  v==-(q.x-p.x)*(r.y-q.y)

  IF v=0 THEN
    RETURN (0)
  ELSEIF v>0 THEN
    RETURN (1)
  FI
RETURN (2)

PROC ConvexHull(INT ARRAY points BYTE pLen
  INT ARRAY res BYTE POINTER resLen)
  PointI POINTER pSrc,pDst,p1,p2,p3
  BYTE minIndex,i,p,q

  IF pLen<3 THEN
    resLen^=pLen
    MoveBlock(res,points,pLen*POINTSIZE)
    RETURN
  FI

  resLen^=0
  minIndex=GetMinXIndex(points,pLen)
  p=minIndex q=0
  DO
    pSrc=GetAddr(points,p)
    pDst=GetAddr(res,resLen^)
    pDst.x=pSrc.x
    pDst.y=pSrc.y
    resLen^==+1

    q=(p+1) MOD pLen
    FOR i=0 TO pLen-1
    DO
      p1=GetAddr(points,p)
      p2=GetAddr(points,i)
      p3=GetAddr(points,q)
      IF Orientation(p1,p2,p3)=2 THEN
        q=i
      FI
    OD
    p=q
  UNTIL p=minIndex
  OD
RETURN

PROC PrintPoints(INT ARRAY points BYTE len)
  PointI POINTER p
  BYTE i

  FOR i=0 TO len-1
  DO
    p=GetAddr(points,i)
    PrintF("(%I,%I) ",p.x,p.y)
  OD
RETURN

PROC Main()
  INT ARRAY points=[
    16 3 12 17 0 6 65532 65530
    16 6 16 65529 17 65532 5 19
    19 65528 3 16 12 13 3 65532
    17 5 65533 15 65533 65527
    0 11 65527 65533 65532 65534
    12 10]
  INT ARRAY result(38)
  BYTE pLen=[19],rlen

  ConvexHull(points,pLen,result,@rlen)

  PrintE("Points:")
  PrintPoints(points,pLen)
  PutE() PutE()
  PrintE("Convex hull:")
  PrintPoints(result,rLen)
RETURN
Output:

Screenshot from Atari 8-bit computer

Points:
(16,3) (12,17) (0,6) (-4,-6) (16,6) (16,-7) (17,-4) (5,19) (19,-8) (3,16) (12,13) (3,-4) (17,5) (-3,15) (-3,-9) (0,11) (-9,-3) (-4,-2) (12,10)

Convex hull:
(-9,-3) (-3,-9) (19,-8) (17,5) (12,17) (5,19) (-3,15)

Ada[edit]

Translation of: D
with Ada.Text_IO;
with Ada.Containers.Vectors;

procedure Convex_Hull is

   type Point is record
      X, Y : Integer;
   end record;

   package Point_Vectors is
      new Ada.Containers.Vectors (Index_Type   => Positive,
                                  Element_Type => Point);
   use Point_Vectors;

   function Find_Convex_Hull (Vec : in Vector) return Vector is

      function Counter_Clock_Wise (A, B, C : in Point) return Boolean is
         ((B.X - A.X) * (C.Y - A.Y) > (B.Y - A.Y) * (C.X - A.X));

      function Less_Than (Left, Right : Point) return Boolean is
         (Left.X < Right.X);

      package Sorter is
         new Point_Vectors.Generic_Sorting (Less_Than);

      Sorted : Vector := Vec;
      Result : Vector := Empty_vector;
      use type Ada.Containers.Count_Type;
   begin
      if Vec = Empty_Vector then
         return Empty_Vector;
      end if;

      Sorter.Sort (Sorted);

      --  Lower hull
      for Index in Sorted.First_Index .. Sorted.Last_Index loop
         while
           Result.Length >= 2 and then
           not Counter_Clock_Wise (Result (Result.Last_Index - 1),
                                   Result (Result.Last_Index),
                                   Sorted (Index))
         loop
            Result.Delete_Last;
         end loop;
         Result.Append (Sorted (Index));
      end loop;

      --  Upper hull
      declare
         T : constant Ada.Containers.Count_Type := Result.Length + 1;
      begin
         for Index in reverse Sorted.First_Index .. Sorted.Last_Index loop
            while
              Result.Length >= T and then
              not Counter_Clock_Wise (Result (Result.Last_Index - 1),
                                      Result (Result.Last_Index),
                                      Sorted (Index))
            loop
               Result.Delete_Last;
            end loop;
            Result.Append (Sorted (Index));
         end loop;
      end;

      Result.Delete_Last;
      return Result;
   end Find_Convex_Hull;

   procedure Show (Vec : in Vector) is
      use Ada.Text_IO;
   begin
      Put ("[ ");
      for Point of Vec loop
         Put ("(" & Point.X'Image & "," & Point.Y'Image & ")");
      end loop;
      Put (" ]");
   end Show;

   Vec : constant Vector :=
     (16, 3) & (12,17) & ( 0, 6) & (-4,-6) & (16, 6) &
     (16,-7) & (16,-3) & (17,-4) & ( 5,19) & (19,-8) &
     ( 3,16) & (12,13) & ( 3,-4) & (17, 5) & (-3,15) &
     (-3,-9) & ( 0,11) & (-9,-3) & (-4,-2) & (12,10);
begin
   Show (Find_Convex_Hull (Vec));
   Ada.Text_IO.New_Line;
end Convex_Hull;
Output:
[ (-9,-3)(-3,-9)( 19,-8)( 17, 5)( 12, 17)( 5, 19)(-3, 15) ]

Arturo[edit]

Translation of: Rust
define :point [x y][]

orientation: function [P Q R][
    val: sub (Q\y - P\y)*(R\x - Q\x) (Q\x - P\x)*(R\y - Q\y)
    if val=0 -> return 0
    if val>0 -> return 1
    return 2
]

calculateConvexHull: function [points][
    result: new []

    if 3 > size points [
        loop points 'p -> 'result ++ p
    ]

    indexMinX: 0
    loop.with:'i points 'p [
        if p\x < get points\[indexMinX] 'x ->
            indexMinX: i
    ]

    p: indexMinX
    q: 0

    while [true][
        'result ++ points\[p]

        q: (p + 1) % size points

        loop 0..dec size points 'i [
            if 2 = orientation points\[p] points\[i] points\[q] ->
                q: i
        ]

        p: q

        if p=indexMinX -> break
    ]

    return result
]
 
points: @[
    to :point [16 3],
    to :point [12 17],
    to :point [0 6],
    to :point @[neg 4 neg 6],
    to :point [16 6],
    to :point @[16 neg 7],
    to :point @[17 neg 4],
    to :point [5 19],
    to :point @[19 neg 8],
    to :point [3 16],
    to :point [12 13],
    to :point @[3 neg 4],
    to :point [17 5],
    to :point @[neg 3 15],
    to :point @[neg 3 neg 9],
    to :point [0 11],
    to :point @[neg 9 neg 3],
    to :point @[neg 4 neg 2],
    to :point [12 10]
]

hull: calculateConvexHull points
loop hull 'i ->
    print i
Output:
[x:-9 y:-3]
[x:-3 y:-9]
[x:19 y:-8]
[x:17 y:5]
[x:12 y:17]
[x:5 y:19]
[x:-3 y:15]

ATS[edit]

Translation of: Standard ML


I purposely demonstrate various features, although I personally do not favor the syntax of the .x and .y overloads. (I have sometimes had this syntax get confused with record field syntax.)


//
// Convex hulls by Andrew's monotone chain algorithm.
//
// For a description of the algorithm, see
// https://en.wikibooks.org/w/index.php?title=Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain&stableid=40169
//
//
// The implementation is designed for use with a garbage collector.
// In other words, some of the memory is allowed to leak.
//
// Sometimes, where it is slightly easier if one does so, we do try to
// free memory. Boehm GC sometimes cannot free allocated memory
// itself, so perhaps it is just as well if an ATS program explicitly
// frees some of its memory.
//

#include "share/atspre_staload.hats"

#define NIL list_nil ()
#define ::  list_cons

(* Make the floating point type big, so use of a boxed representation
   for points makes some sense. *)
typedef floatpt = ldouble

extern castfn int2floatpt : int -<> floatpt
overload i2fp with int2floatpt

(*------------------------------------------------------------------*)
//
// Enough plane geometry for our purpose.
//

(* Let us make the point type boxed. Here is one way to do that. The
   name of the type will be "plane_point_t". The constructor will be
   named "plane_point". *)
datatype plane_point_t =
| plane_point of (floatpt, floatpt)

fn {}
plane_point_x (p : plane_point_t) :<> floatpt =
  let
    val+ plane_point (x, _) = p
  in
    x
  end

fn {}
plane_point_y (p : plane_point_t) :<> floatpt =
  let
    val+ plane_point (_, y) = p
  in
    y
  end

fn {}
plane_point_eq (p : plane_point_t,
                q : plane_point_t) :<> bool =
  let
    val+ plane_point (xp, yp) = p
    val+ plane_point (xq, yq) = q
  in
    xp = xq && yp = yq
  end

(* Impose a total order on points, making it one that will work for
   Andrew's monotone chain algorithm. *)
fn {}
plane_point_order (p : plane_point_t,
                   q : plane_point_t) :<> bool =
  let
    val+ plane_point (xp, yp) = p
    val+ plane_point (xq, yq) = q
  in
    xp < xq || (xp = xq && yp < yq)
  end

(* Subtraction is really a vector or multivector operation. *)
fn {}
plane_point_sub (p : plane_point_t,
                 q : plane_point_t) :<> plane_point_t =
  let
    val+ plane_point (xp, yp) = p
    val+ plane_point (xq, yq) = q
  in
    plane_point (xp - xq, yp - yq)
  end

(* Cross product is really a multivector operation. *)
fn {}
plane_point_cross (p : plane_point_t,
                   q : plane_point_t) :<> floatpt =
  let
    val+ plane_point (xp, yp) = p
    val+ plane_point (xq, yq) = q
  in
    (xp * yq) - (yp * xq)
  end  

overload .x with plane_point_x
overload .y with plane_point_y
overload = with plane_point_eq
overload order with plane_point_order
overload < with plane_point_order
overload - with plane_point_sub

(* Make "cross" a left-associative infix operator with the same
   precedence as "*". *)
infixl ( * ) cross
overload cross with plane_point_cross

(*------------------------------------------------------------------*)
//
// Sorting an array of points.
//

fn
plane_point_array_sort
          {n   : int}
          (arr : &(@[plane_point_t][n]) >> _,
           n   : size_t n) :<!wrt> void =
  let
    (* The comparison will be inlined by template expansion. *)
    implement
    array_quicksort$cmp<plane_point_t> (p, q) =
      if order (p, q) then    (* An overload for plane_point_order. *)
        ~1
      else if q < p then (* Another overload for plane_point_order. *)
        1
      else
        0
  in
    (* The following sort routine is in the ATS2 prelude. *)
    array_quicksort<plane_point_t> (arr, n)
  end

(*------------------------------------------------------------------*)
//
// Removing duplicates from a sorted array. Returns a new array.
//

extern fun {a : t@ype}
array_delete_neighbor_dups$eq (p : a, q : a) :<> bool

fn {a : t@ype}
array_delete_neighbor_dups
          {n   : int}
          (arr : &(@[a][n]),
           n   : size_t n)
     :<!wrt> [m       : nat | m <= n]
             [parr1   : addr]
             @(@[a][m] @ parr1,
               mfree_gc_v parr1 |
               ptr parr1,
               size_t m) =
  let
    macdef eq = array_delete_neighbor_dups$eq<a>

    fn
    nondups_list {n   : int}
                 (arr : &(@[a][n]),
                  n   : size_t n)
        :<> [m : nat | m <= n] list (a, m) =
      let
        fun
        loop {i : nat | i < n}
             {k : nat | k < n - i}
             .<i>.              (* <-- proof of termination. *)
             (arr : &(@[a][n]),
              i   : size_t i,
              lst : list (a, k))
            :<> [m : nat | m <= n] list (a, m) =
          (* Cons a list of non-duplicates, going backwards through
             the array so the list will be in forwards order. (The
             order does not really matter in ATS, though, because in
             the prelude there are both array_initize_list *and*
             array_initize_rlist. *)
          if i = i2sz 0 then
            arr[i] :: lst
          (* The "\" in the following line makes eq temporarily
             infix. *)
          else if arr[i - 1] \eq arr[i] then
            loop (arr, pred i, lst)
          else
            loop (arr, pred i, arr[i] :: lst)

        prval () = lemma_array_param arr
      in
        if n = i2sz 0 then
          NIL
        else
          loop (arr, pred n, NIL)
      end

    val lst = nondups_list (arr, n)
    prval () = lemma_list_param lst
    val m = i2sz (length lst)

    val @(pfarr1, pfgc1 | parr1) = array_ptr_alloc<a> (m)
    val () = array_initize_list<a> (!parr1, sz2i m, lst)
  in
    @(pfarr1, pfgc1 | parr1, m)
  end

(*------------------------------------------------------------------*)
//
// Removing duplicates from a sorted plane_point_t array. Returns a
// new array.
//

fn
plane_point_array_delete_neighbor_dups
          {n  : int}
          (arr : &(@[plane_point_t][n]),
           n   : size_t n)
     :<!wrt> [m       : nat | m <= n]
             [parr1   : addr]
             @(@[plane_point_t][m] @ parr1,
               mfree_gc_v parr1 |
               ptr parr1,
               size_t m) =
  let
    implement
    array_delete_neighbor_dups$eq<plane_point_t> (p, q) =
      p = q            (* Here = is an overload for plane_point_eq. *)
  in
    array_delete_neighbor_dups<plane_point_t> (arr, n)
  end

(*------------------------------------------------------------------*)
//
// The convex hull algorithm.
//

fn
cross_test {m, k : int | 1 <= k; k < m}
           (pt_i : plane_point_t,
            hull : &(@[plane_point_t][m]),
            k    : size_t k) :<> bool =
  let
    val hull_k = hull[k]
    and hull_k1 = hull[k - 1]
  in
    i2fp 0 < (hull_k - hull_k1) cross (pt_i - hull_k1)
  end

fn
construct_lower_hull {n  : int | 2 <= n}
                     (pt : &(@[plane_point_t][n]),
                      n  : size_t n)
    :<!wrt> [m     : int | 2 <= m; m <= n]
            [phull : addr]
            @(@[plane_point_t][n] @ phull,
              mfree_gc_v phull |
              ptr phull,
              size_t m) =
  let
    val @(pfhull, pfgc | phull) = array_ptr_alloc<plane_point_t> n

    (* It is easier to work with an array if it is fully
       initialized. (Yes, there are also ways to cheat and so merely
       pretend the array has been initialized.)  *)
    val arbitrary_point = plane_point (i2fp 0, i2fp 0)
    val () = array_initize_elt<plane_point_t> (!phull, n,
                                               arbitrary_point)

    (* The macro "hull" can be used with index notation "[]". *)
    macdef hull = !phull

    val () = hull[0] := pt[0]
    val () = hull[1] := pt[1]

    fun
    outer_loop {i    : int | 0 <= i; i <= n}
               {j    : int | 1 <= j; j < n}
               .<n - i>.        (* <-- proof of termination. *)
               (pt   : &(@[plane_point_t][n]),
                hull : &(@[plane_point_t][n]),
                i    : size_t i,
                j    : size_t j)
        :<!wrt> [m : int | 2 <= m; m <= n] size_t m =
      if i = n then
        succ j
      else
        let
          val pt_i = pt[i]

          fun
          inner_loop {k : int | 0 <= k; k < n - 1}
                     .<k>.      (* <-- proof of termination. *)
                     (hull : &(@[plane_point_t][n]),
                      k    : size_t k)
              :<!wrt> [j : int | 1 <= j; j < n] size_t j =
            if k = i2sz 0 then
              begin
                hull[succ k] := pt_i;
                succ k
              end
            else if cross_test (pt_i, hull, k) then
              begin
                hull[succ k] := pt_i;
                succ k
              end
            else
              inner_loop (hull, pred k)

          (* I do not know how to write a proof of the following, so
             let us test it at runtime. *)
          val () = $effmask_exn assertloc (j < n - 1)
        in
          outer_loop (pt, hull, succ i, inner_loop (hull, j))
        end

    val hull_size = outer_loop (pt, hull, i2sz 2, i2sz 1)
  in
    @(pfhull, pfgc | phull, hull_size)
  end

fn
construct_upper_hull {n  : int | 2 <= n}
                     (pt : &(@[plane_point_t][n]),
                      n  : size_t n)
    :<!wrt> [m     : int | 2 <= m; m <= n]
            [phull : addr]
            @(@[plane_point_t][n] @ phull,
              mfree_gc_v phull |
              ptr phull,
              size_t m) =
  let
    val @(pfhull, pfgc | phull) = array_ptr_alloc<plane_point_t> n

    (* It is easier to work with an array if it is fully
       initialized. (Yes, there are also ways to cheat and so merely
       pretend the array has been initialized.)  *)
    val arbitrary_point = plane_point (i2fp 0, i2fp 0)
    val () = array_initize_elt<plane_point_t> (!phull, n,
                                               arbitrary_point)

    (* The macro "hull" can be used with index notation "[]". *)
    macdef hull = !phull

    val () = hull[0] := pt[n - 1]
    val () = hull[1] := pt[n - 2]

    fun
    outer_loop {i1   : int | 0 <= i1; i1 <= n - 2}
               {j    : int | 1 <= j; j < n}
               .<i1>.           (* <-- proof of termination. *)
               (pt   : &(@[plane_point_t][n]),
                hull : &(@[plane_point_t][n]),
                i1   : size_t i1,
                j    : size_t j)
        :<!wrt> [m : int | 2 <= m; m <= n] size_t m =
      if i1 = i2sz 0 then
        succ j
      else
        let
          val i = pred i1
          val pt_i = pt[i]

          fun
          inner_loop {k : int | 0 <= k; k < n - 1}
                     .<k>.      (* <-- proof of termination. *)
                     (hull : &(@[plane_point_t][n]),
                      k    : size_t k)
              :<!wrt> [j : int | 1 <= j; j < n] size_t j =
            if k = i2sz 0 then
              begin
                hull[succ k] := pt_i;
                succ k
              end
            else if cross_test (pt_i, hull, k) then
              begin
                hull[succ k] := pt_i;
                succ k
              end
            else
              inner_loop (hull, pred k)

          (* I do not know how to write a proof of the following, so
             let us test it at runtime. *)
          val () = $effmask_exn assertloc (j < n - 1)
        in
          outer_loop (pt, hull, pred i1, inner_loop (hull, j))
        end

    val hull_size = outer_loop (pt, hull, n - i2sz 2, i2sz 1)
  in
    @(pfhull, pfgc | phull, hull_size)
  end

fn
construct_hull {n  : int | 2 <= n}
               (pt : &(@[plane_point_t][n]),
                n  : size_t n)
    :<!wrt> [hull_size : int | 2 <= hull_size]
            [phull     : addr]
            @(@[plane_point_t][hull_size] @ phull,
              mfree_gc_v phull |
              ptr phull,
              size_t hull_size) =

  (* The following implementation demonstrates slightly complicated
     "safe" initialization. By "safe" I mean so one never *reads* from
     an uninitialized entry. Elsewhere in the program I did this more
     simply, with prelude routines.

     I also demonstrate freeing of a linear object. If you remove the
     calls to array_ptr_free, you cannot compile the program. *)

  let
    (* Side note: Construction of the lower and upper hulls can be
       done in parallel. *)
    val [lower_hull_size : int]
        [p_lower_hull : addr]
        @(pf_lower_hull, pfgc_lower | p_lower_hull, lower_hull_size) =
      construct_lower_hull (pt, n)
    and [upper_hull_size : int]
        [p_upper_hull : addr]
        @(pf_upper_hull, pfgc_upper | p_upper_hull, upper_hull_size) =
      construct_upper_hull (pt, n)

    stadef hull_size = lower_hull_size + upper_hull_size - 2
    val hull_size : size_t hull_size =
      lower_hull_size + upper_hull_size - i2sz 2

    val [phull : addr] @(pfhull, pfgc_hull | phull) =
      array_ptr_alloc<plane_point_t> hull_size

    (* Split off the part of each partial hull's view that we actually
       will use. *)
    prval @(pf_lower, pf_lower_rest) =
      array_v_split {plane_point_t} {p_lower_hull} {n}
                    {lower_hull_size - 1}
                    pf_lower_hull
    prval @(pf_upper, pf_upper_rest) =
      array_v_split {plane_point_t} {p_upper_hull} {n}
                    {upper_hull_size - 1}
                    pf_upper_hull

    (* Split the new array's view in two. The question mark means
       that the array elements are uninitialized. *)
    prval @(pfleft, pfright) =
      array_v_split {plane_point_t?} {phull} {hull_size}
                    {lower_hull_size - 1}
                    pfhull

    (* Copy the lower hull, thus initializing the left side of the new
       array. *)
    val () = array_copy<plane_point_t> (!phull, !p_lower_hull,
                                        pred lower_hull_size)

    (* Copy the upper hull, thus initializing the left side of the new
       array. *)
    val phull_right =
      ptr_add<plane_point_t> (phull, pred lower_hull_size)
    val () = array_copy<plane_point_t> (!phull_right, !p_upper_hull,
                                        pred upper_hull_size)

    (* Join the views of the initialized halves. *)
    prval pfhull = array_v_unsplit (pfleft, pfright)

    (* Restore the views of the partial hulls. *)
    prval () = pf_lower_hull :=
      array_v_unsplit (pf_lower, pf_lower_rest)
    prval () = pf_upper_hull :=
      array_v_unsplit (pf_upper, pf_upper_rest)

    (* We do not need the lower and upper hulls anymore, and so can
       free them. (Of course, if there is a garbage collector, you
       could make freeing be a no-operation.) *)
    val () = array_ptr_free (pf_lower_hull, pfgc_lower | p_lower_hull)
    val () = array_ptr_free (pf_upper_hull, pfgc_upper | p_upper_hull)
  in
    @(pfhull, pfgc_hull | phull, hull_size)
  end

fn
plane_convex_hull {num_points : int}
                  (points_lst : list (plane_point_t, num_points))
    :<!wrt> [hull_size : int | 0 <= hull_size]
            [phull     : addr]
            @(@[plane_point_t][hull_size] @ phull,
              mfree_gc_v phull |
              ptr phull,
              size_t hull_size) =
  (* Takes an arbitrary list of points, which may be in any order and
     may contain duplicates. Returns an ordered array of points that
     make up the convex hull. If the initial list of points is empty,
     the returned array is empty. *)
  let
    prval () = lemma_list_param points_lst
    val num_points = i2sz (length points_lst)

    (* Copy the list to an array. *)
    val @(pf_points, pfgc_points | p_points) =
      array_ptr_alloc<plane_point_t> num_points
    val () =
      array_initize_list<plane_point_t> (!p_points, sz2i num_points,
                                         points_lst)

    (* Sort the array. *)
    val () = plane_point_array_sort (!p_points, num_points)

    (* Create a new sorted array that has the duplicates removed. *)
    val @(pf_pt, pfgc_pt | p_pt, n) =
      plane_point_array_delete_neighbor_dups (!p_points, num_points)

    (* The original array no longer is needed. *)
    val () = array_ptr_free (pf_points, pfgc_points | p_points)
  in
    if n <= 2 then
      @(pf_pt, pfgc_pt | p_pt, n)
    else
      let
        val @(pfhull, pfgc_hull | phull, hull_size) =
          construct_hull (!p_pt, n)
        val () = array_ptr_free (pf_pt, pfgc_pt | p_pt)
      in
        @(pfhull, pfgc_hull | phull, hull_size)
      end
  end

(*------------------------------------------------------------------*)

implement
main0 () =
  {
    val example_points =
      $list (plane_point (i2fp 16, i2fp 3),
             plane_point (i2fp 12, i2fp 17),
             plane_point (i2fp 0, i2fp 6),
             plane_point (i2fp ~4, i2fp ~6),
             plane_point (i2fp 16, i2fp 6),
             plane_point (i2fp 16, i2fp ~7),
             plane_point (i2fp 16, i2fp ~3),
             plane_point (i2fp 17, i2fp ~4),
             plane_point (i2fp 5, i2fp 19),
             plane_point (i2fp 19, i2fp ~8),
             plane_point (i2fp 3, i2fp 16),
             plane_point (i2fp 12, i2fp 13),
             plane_point (i2fp 3, i2fp ~4),
             plane_point (i2fp 17, i2fp 5),
             plane_point (i2fp ~3, i2fp 15),
             plane_point (i2fp ~3, i2fp ~9),
             plane_point (i2fp 0, i2fp 11),
             plane_point (i2fp ~9, i2fp ~3),
             plane_point (i2fp ~4, i2fp ~2),
             plane_point (i2fp 12, i2fp 10))

    val (pf_hull, pfgc_hull | p_hull, hull_size) =
      plane_convex_hull example_points

    macdef hull = !p_hull

    val () =
      let
        var i : [i : nat] size_t i
      in
        for (i := i2sz 0; i < hull_size; i := succ i)
          println! ("(", hull[i].x(), " ", hull[i].y(), ")")
      end

    val () = array_ptr_free (pf_hull, pfgc_hull | p_hull)
  }

(*------------------------------------------------------------------*)
Output:
$ patscc -O3 -DATS_MEMALLOC_GCBDW convex_hull_task.dats -lgc && ./a.out
(-9.000000 -3.000000)
(-3.000000 -9.000000)
(19.000000 -8.000000)
(17.000000 5.000000)
(12.000000 17.000000)
(5.000000 19.000000)
(-3.000000 15.000000)

C[edit]

Translation of: C++
#include <assert.h>
#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>
 
typedef struct tPoint {
    int x, y;
} Point;
 
bool ccw(const Point *a, const Point *b, const Point *c) {
    return (b->x - a->x) * (c->y - a->y)
         > (b->y - a->y) * (c->x - a->x);
}
 
int comparePoints(const void *lhs, const void *rhs) {
    const Point* lp = lhs;
    const Point* rp = rhs;
    if (lp->x < rp->x)
        return -1;
    if (rp->x < lp->x)
        return 1;
    if (lp->y < rp->y)
        return -1;
    if (rp->y < lp->y)
        return 1;
    return 0;
}

void fatal(const char* message) {
    fprintf(stderr, "%s\n", message);
    exit(1);
}

void* xmalloc(size_t n) {
    void* ptr = malloc(n);
    if (ptr == NULL)
        fatal("Out of memory");
    return ptr;
}

void* xrealloc(void* p, size_t n) {
    void* ptr = realloc(p, n);
    if (ptr == NULL)
        fatal("Out of memory");
    return ptr;
}

void printPoints(const Point* points, int len) {
    printf("[");
    if (len > 0) {
        const Point* ptr = points;
        const Point* end = points + len;
        printf("(%d, %d)", ptr->x, ptr->y);
        ++ptr;
        for (; ptr < end; ++ptr)
            printf(", (%d, %d)", ptr->x, ptr->y);
    }
    printf("]");
}
 
Point* convexHull(Point p[], int len, int* hsize) {
    if (len == 0) {
        *hsize = 0;
        return NULL;
    }

    int i, size = 0, capacity = 4;
    Point* hull = xmalloc(capacity * sizeof(Point));

    qsort(p, len, sizeof(Point), comparePoints);
 
    /* lower hull */
    for (i = 0; i < len; ++i) {
        while (size >= 2 && !ccw(&hull[size - 2], &hull[size - 1], &p[i]))
            --size;
        if (size == capacity) {
            capacity *= 2;
            hull = xrealloc(hull, capacity * sizeof(Point));
        }
        assert(size >= 0 && size < capacity);
        hull[size++] = p[i];
    }
 
    /* upper hull */
    int t = size + 1;
    for (i = len - 1; i >= 0; i--) {
        while (size >= t && !ccw(&hull[size - 2], &hull[size - 1], &p[i]))
            --size;
        if (size == capacity) {
            capacity *= 2;
            hull = xrealloc(hull, capacity * sizeof(Point));
        }
        assert(size >= 0 && size < capacity);
        hull[size++] = p[i];
    }
    --size;
    assert(size >= 0);
    hull = xrealloc(hull, size * sizeof(Point));
    *hsize = size;
    return hull;
}
 
int main() {
    Point points[] = {
        {16,  3}, {12, 17}, { 0,  6}, {-4, -6}, {16,  6},
        {16, -7}, {16, -3}, {17, -4}, { 5, 19}, {19, -8},
        { 3, 16}, {12, 13}, { 3, -4}, {17,  5}, {-3, 15},
        {-3, -9}, { 0, 11}, {-9, -3}, {-4, -2}, {12, 10}
    };
    int hsize;
    Point* hull = convexHull(points, sizeof(points)/sizeof(Point), &hsize);
    printf("Convex Hull: ");
    printPoints(hull, hsize);
    printf("\n");
    free(hull);
 
    return 0;
}
Output:
Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]

C#[edit]

using System;
using System.Collections.Generic;

namespace ConvexHull {
    class Point : IComparable<Point> {
        private int x, y;

        public Point(int x, int y) {
            this.x = x;
            this.y = y;
        }

        public int X { get => x; set => x = value; }
        public int Y { get => y; set => y = value; }

        public int CompareTo(Point other) {
            return x.CompareTo(other.x);
        }

        public override string ToString() {
            return string.Format("({0}, {1})", x, y);
        }
    }

    class Program {
        private static List<Point> ConvexHull(List<Point> p) {
            if (p.Count == 0) return new List<Point>();
            p.Sort();
            List<Point> h = new List<Point>();

            // lower hull
            foreach (var pt in p) {
                while (h.Count >= 2 && !Ccw(h[h.Count - 2], h[h.Count - 1], pt)) {
                    h.RemoveAt(h.Count - 1);
                }
                h.Add(pt);
            }

            // upper hull
            int t = h.Count + 1;
            for (int i = p.Count - 1; i >= 0; i--) {
                Point pt = p[i];
                while (h.Count >= t && !Ccw(h[h.Count - 2], h[h.Count - 1], pt)) {
                    h.RemoveAt(h.Count - 1);
                }
                h.Add(pt);
            }

            h.RemoveAt(h.Count - 1);
            return h;
        }

        private static bool Ccw(Point a, Point b, Point c) {
            return ((b.X - a.X) * (c.Y - a.Y)) > ((b.Y - a.Y) * (c.X - a.X));
        }

        static void Main(string[] args) {
            List<Point> points = new List<Point>() {
                new Point(16, 3),
                new Point(12, 17),
                new Point(0, 6),
                new Point(-4, -6),
                new Point(16, 6),

                new Point(16, -7),
                new Point(16, -3),
                new Point(17, -4),
                new Point(5, 19),
                new Point(19, -8),

                new Point(3, 16),
                new Point(12, 13),
                new Point(3, -4),
                new Point(17, 5),
                new Point(-3, 15),

                new Point(-3, -9),
                new Point(0, 11),
                new Point(-9, -3),
                new Point(-4, -2),
                new Point(12, 10)
            };

            List<Point> hull = ConvexHull(points);
            Console.Write("Convex Hull: [");
            for (int i = 0; i < hull.Count; i++) {
                if (i > 0) {
                    Console.Write(", ");
                }
                Point pt = hull[i];
                Console.Write(pt);
            }
            Console.WriteLine("]");
        }
    }
}
Output:
Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]

C++[edit]

Translation of: D
#include <algorithm>
#include <iostream>
#include <ostream>
#include <vector>
#include <tuple>

typedef std::tuple<int, int> point;

std::ostream& print(std::ostream& os, const point& p) {
    return os << "(" << std::get<0>(p) << ", " << std::get<1>(p) << ")";
}

std::ostream& print(std::ostream& os, const std::vector<point>& v) {
    auto it = v.cbegin();
    auto end = v.cend();

    os << "[";

    if (it != end) {
        print(os, *it);
        it = std::next(it);
    }
    while (it != end) {
        os << ", ";
        print(os, *it);
        it = std::next(it);
    }

    return os << "]";
}

// returns true if the three points make a counter-clockwise turn
bool ccw(const point& a, const point& b, const point& c) {
    return ((std::get<0>(b) - std::get<0>(a)) * (std::get<1>(c) - std::get<1>(a)))
         > ((std::get<1>(b) - std::get<1>(a)) * (std::get<0>(c) - std::get<0>(a)));
}

std::vector<point> convexHull(std::vector<point> p) {
    if (p.size() == 0) return std::vector<point>();
    std::sort(p.begin(), p.end(), [](point& a, point& b){
        if (std::get<0>(a) < std::get<0>(b)) return true;
        return false;
    });

    std::vector<point> h;

    // lower hull
    for (const auto& pt : p) {
        while (h.size() >= 2 && !ccw(h.at(h.size() - 2), h.at(h.size() - 1), pt)) {
            h.pop_back();
        }
        h.push_back(pt);
    }

    // upper hull
    auto t = h.size() + 1;
    for (auto it = p.crbegin(); it != p.crend(); it = std::next(it)) {
        auto pt = *it;
        while (h.size() >= t && !ccw(h.at(h.size() - 2), h.at(h.size() - 1), pt)) {
            h.pop_back();
        }
        h.push_back(pt);
    }

    h.pop_back();
    return h;
}

int main() {
    using namespace std;

    vector<point> points = {
        make_pair(16, 3),  make_pair(12, 17), make_pair(0,  6),  make_pair(-4, -6), make_pair(16,  6),
        make_pair(16, -7), make_pair(16, -3), make_pair(17, -4), make_pair(5, 19),  make_pair(19, -8),
        make_pair(3, 16),  make_pair(12, 13), make_pair(3, -4),  make_pair(17,  5), make_pair(-3, 15),
        make_pair(-3, -9), make_pair(0, 11),  make_pair(-9, -3), make_pair(-4, -2), make_pair(12, 10)
    };

    auto hull = convexHull(points);
    auto it = hull.cbegin();
    auto end = hull.cend();

    cout << "Convex Hull: ";
    print(cout, hull);
    cout << endl;

    return 0;
}
Output:
Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]

Common Lisp[edit]

Translation of: Scheme


The program is wrapped in a Roswell script, which is one of the popular ways to get around differences between Common Lisp implementations.

(Side note: It occurs to me that the "complete with tail recursions" comment in the code is not meant to include the Scheme do constructs, which are secretly really tail recursions. How Common Lisps implement the loop macro I do not know; it might be tail recursion, or it might not. Note also how the Scheme uses "named let" syntactic sugar, where the Common Lisp has defun fully written out.)


#!/bin/sh
#|-*- mode:lisp -*-|#
#|
exec ros -Q -- $0 "$@"
|#
(progn ;;init forms
  (ros:ensure-asdf)
  #+quicklisp(ql:quickload '() :silent t)
  )

(defpackage :ros.script.convex-hull-task.3861520611
  (:use :cl))
(in-package :ros.script.convex-hull-task.3861520611)

;;;
;;; Convex hulls by Andrew's monotone chain algorithm.
;;;
;;; For a description of the algorithm, see
;;; https://en.wikibooks.org/w/index.php?title=Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain&stableid=40169
;;;
;;; This program is translated rather faithfully from the Scheme,
;;; complete with tail recursions.
;;;

;; x and y coordinates of a "point". A "point" is represented by a
;; list of length 2.
(defun x@ (u) (car u))
(defun y@ (u) (cadr u))

(defun cross (u v)
  ;; Cross product (as a signed scalar).
  (- (* (x@ u) (y@ v)) (* (y@ u) (x@ v))))

(defun point- (u v)
  (list (- (x@ u) (x@ v)) (- (y@ u) (y@ v))))

(defun sort-points-vector (points-vector)
  ;; Ascending sort on x-coordinates, followed by ascending sort
  ;; on y-coordinates.
  (sort points-vector #'(lambda (u v)
                          (or (< (x@ u) (x@ v))
                              (and (= (x@ u) (x@ v))
                                   (< (y@ u) (y@ v)))))))

(defun construct-lower-hull (sorted-points-vector)
  (let* ((pt sorted-points-vector)
         (n (length pt))
         (hull (make-array n))
         (j 1))
    (setf (aref hull 0) (aref pt 0))
    (setf (aref hull 1) (aref pt 1))
    (loop for i from 2 to (1- n)
          do (progn
               (defun inner-loop ()
                 (if (or (zerop j)
                         (plusp
                          (cross (point- (aref hull j)
                                         (aref hull (1- j)))
                                 (point- (aref pt i)
                                         (aref hull (1- j))))))
                     (progn
                       (setf j (1+ j))
                       (setf (aref hull j) (aref pt i)))
                     (progn
                       (setf j (1- j))
                       (inner-loop))))
               (inner-loop)))
    (values (+ j 1) hull)))             ; Hull size, hull points.

(defun construct-upper-hull (sorted-points-vector)
  (let* ((pt sorted-points-vector)
         (n (length pt))
         (hull (make-array n))
         (j 1))
    (setf (aref hull 0) (aref pt (- n 1)))
    (setf (aref hull 1) (aref pt (- n 2)))
    (loop for i from (- n 3) downto 0
          do (progn
               (defun inner-loop ()
                 (if (or (zerop j)
                         (plusp
                          (cross (point- (aref hull j)
                                         (aref hull (1- j)))
                                 (point- (aref pt i)
                                         (aref hull (1- j))))))
                     (progn
                       (setf j (1+ j))
                       (setf (aref hull j) (aref pt i)))
                     (progn
                       (setf j (1- j))
                       (inner-loop))))
               (inner-loop)))
    (values (+ j 1) hull)))             ; Hull size, hull points.

(defun construct-hull (sorted-points-vector)
  ;; Notice that the lower and upper hulls could be constructed in
  ;; parallel. (The Scheme "let-values" macro made this apparent,
  ;; despite not actually doing the computation in parallel. The
  ;; coding here makes it less obvious.)
  (multiple-value-bind (lower-hull-size lower-hull)
      (construct-lower-hull sorted-points-vector)
    (multiple-value-bind (upper-hull-size upper-hull)
        (construct-upper-hull sorted-points-vector)
      (let* ((hull-size (+ lower-hull-size upper-hull-size -2))
             (hull (make-array hull-size)))
        (loop for i from 0 to (- lower-hull-size 2)
              do (setf (aref hull i) (aref lower-hull i)))
        (loop for i from 0 to (- upper-hull-size 2)
              do (setf (aref hull (+ i (1- lower-hull-size)))
                       (aref upper-hull i)))
        hull))))

(defun vector-delete-neighbor-dups (elt= v)
  ;; A partial clone of the SRFI-132 procedure of the same name. This
  ;; implementation is similar to the reference implementation for
  ;; SRFI-132, and may use a bunch of stack space.  That reference
  ;; implementation is by Olin Shivers and rests here:
  ;; https://github.com/scheme-requests-for-implementation/srfi-132/blob/master/sorting/delndups.scm
  ;; The license is:
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; This code is
;;;     Copyright (c) 1998 by Olin Shivers.
;;; The terms are: You may do as you please with this code, as long as
;;; you do not delete this notice or hold me responsible for any outcome
;;; related to its use.
;;;
;;; Blah blah blah. Don't you think source files should contain more lines
;;; of code than copyright notice?
;;;
  (let ((start 0)
        (end (length v)))
    (let ((x (aref v start)))
      (defun recur (x i j)
        (if (< i end)
            (let ((y (aref v i))
                  (nexti (1+ i)))
              (if (funcall elt= x y)
                  (recur x nexti j)
                  (let ((ansvec (recur y nexti (1+ j))))
                    (setf (aref ansvec j) y)
                    ansvec)))
            (make-array j)))
      (let ((ans (recur x start 1)))
        (setf (aref ans 0) x)
        ans))))

(defun vector-convex-hull (points)
  (let* ((points-vector (coerce points 'vector))
         (sorted-points-vector
           (vector-delete-neighbor-dups
            #'equalp
            (sort-points-vector points-vector))))
    (if (<= (length sorted-points-vector) 2)
        sorted-points-vector
        (construct-hull sorted-points-vector))))

(defun list-convex-hull (points)
  (coerce (vector-convex-hull points) 'list))

(defconstant example-points
  '((16 3) (12 17) (0 6) (-4 -6) (16 6)
    (16 -7) (16 -3) (17 -4) (5 19) (19 -8)
    (3 16) (12 13) (3 -4) (17 5) (-3 15)
    (-3 -9) (0 11) (-9 -3) (-4 -2) (12 10)))

(defun main (&rest argv)
  (declare (ignorable argv))
  (write (list-convex-hull example-points))
  (terpri))

;;; vim: set ft=lisp lisp:
Output:
$ ./convex-hull-task.ros
((-9 -3) (-3 -9) (19 -8) (17 5) (12 17) (5 19) (-3 15))

D[edit]

Translation of: Kotlin
import std.algorithm.sorting;
import std.stdio;

struct Point {
    int x;
    int y;

    int opCmp(Point rhs) {
        if (x < rhs.x) return -1;
        if (rhs.x < x) return 1;
        return 0;
    }

    void toString(scope void delegate(const(char)[]) sink) const {
        import std.format;
        sink("(");
        formattedWrite(sink, "%d", x);
        sink(",");
        formattedWrite(sink, "%d", y);
        sink(")");
    }
}

Point[] convexHull(Point[] p) {
    if (p.length == 0) return [];
    p.sort;
    Point[] h;

    // lower hull
    foreach (pt; p) {
        while (h.length >= 2 && !ccw(h[$-2], h[$-1], pt)) {
            h.length--;
        }
        h ~= pt;
    }

    // upper hull
    auto t = h.length + 1;
    foreach_reverse (i; 0..(p.length - 1)) {
        auto pt = p[i];
        while (h.length >= t && !ccw(h[$-2], h[$-1], pt)) {
            h.length--;
        }
        h ~= pt;
    }

    h.length--;
    return h;
}

/* ccw returns true if the three points make a counter-clockwise turn */
auto ccw(Point a, Point b, Point c) {
    return ((b.x - a.x) * (c.y - a.y)) > ((b.y - a.y) * (c.x - a.x));
}

void main() {
    auto points = [
        Point(16,  3), Point(12, 17), Point( 0,  6), Point(-4, -6), Point(16,  6),
        Point(16, -7), Point(16, -3), Point(17, -4), Point( 5, 19), Point(19, -8),
        Point( 3, 16), Point(12, 13), Point( 3, -4), Point(17,  5), Point(-3, 15),
        Point(-3, -9), Point( 0, 11), Point(-9, -3), Point(-4, -2), Point(12, 10)
    ];
    auto hull = convexHull(points);
    writeln("Convex Hull: ", hull);
}
Output:
Convex Hull: [(-9,-3), (-3,-9), (19,-8), (17,5), (12,17), (5,19), (-3,15)]

Delphi[edit]

Translation of: C#
Works with: Delphi version XE10
program ConvexHulls;

{$APPTYPE CONSOLE}

{$R *.res}

uses
  System.Types,
  System.SysUtils,
  System.Generics.Defaults,
  System.Generics.Collections;

  function Ccw(const a, b, c: TPoint): Boolean;
  begin
    Result := ((b.X - a.X) * (c.Y - a.Y)) > ((b.Y - a.Y) * (c.X - a.X));
  end;

  function ConvexHull(const p: TList<TPoint>): TList<TPoint>;
  var
    pt: TPoint;
    i, t: Integer;
  begin
    Result := TList<TPoint>.Create;

    if (p.Count = 0) then Exit;

    p.Sort(TComparer<TPoint>.Construct(
          function(const Left, Right: TPoint): Integer
          begin
            Result := Left.X - Right.X;
          end
    ));

    // lower hull
    for i := 0 to p.Count-1 do
    begin
      pt := p[i];
      while ((Result.Count >= 2) and (not Ccw(Result[Result.Count - 2], Result[Result.Count - 1], pt))) do
      begin
        Result.Delete(Result.Count - 1);
      end;
      Result.Add(pt);
    end;

    // upper hull
    t := Result.Count + 1;
    for i := p.Count-1 downto 0 do
    begin
      pt := p[i];
      while ((Result.Count >= t) and (not Ccw(Result[Result.Count - 2], Result[Result.Count - 1], pt))) do
      begin
        Result.Delete(Result.Count - 1);
      end;
      Result.Add(pt);
    end;

    Result.Delete(Result.Count - 1);
  end;

var
  points: TList<TPoint>;
  hull: TList<TPoint>;
  i: Integer;
begin

  hull := nil;
  points := TList<TPoint>.Create;
  try

    points.AddRange([
        Point(16, 3),
        Point(12, 17),
        Point(0, 6),
        Point(-4, -6),
        Point(16, 6),
        Point(16, -7),
        Point(16, -3),
        Point(17, -4),
        Point(5, 19),
        Point(19, -8),
        Point(3, 16),
        Point(12, 13),
        Point(3, -4),
        Point(17, 5),
        Point(-3, 15),
        Point(-3, -9),
        Point(0, 11),
        Point(-9, -3),
        Point(-4, -2),
        Point(12, 10)
        ]);

    hull := ConvexHull(points);

    // Output the result
    Write('Convex Hull: [');
    for i := 0 to hull.Count-1 do
    begin
      if (i > 0) then Write(', ');
      Write(Format('(%d, %d)', [hull[i].X, hull[i].Y]));
    end;
    WriteLn(']');

  finally
    hull.Free;
    points.Free;
  end;

end.
Output:
Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]

F#[edit]

Translation of: C
open System

type Point =
    struct
        val X : int
        val Y : int
        new (x : int, y : int ) = {X = x; Y = y}
    end

let (poly : Point list) = [ Point(16,  3); Point(12, 17); Point( 0,  6); Point(-4, -6); Point(16,  6);
                            Point(16, -7); Point(16, -3); Point(17, -4); Point( 5, 19); Point(19, -8);
                            Point( 3, 16); Point(12, 13); Point( 3, -4); Point(17,  5); Point(-3, 15);
                            Point(-3, -9); Point( 0, 11); Point(-9, -3); Point(-4, -2); Point(12, 10)]


let affiche (lst : Point list) = 
    let mutable (str : string) = List.fold (fun  acc (p : Point) -> acc + sprintf "(%d, %d) " p.X p.Y) "Convex Hull: [" lst
    printfn "%s" (str.[0.. str.Length - 2] + "]")

let ccw (p1 : Point) (p2 : Point) (p3 : Point) =
    (p2.X - p1.X) * (p3.Y - p1.Y) > (p2.Y - p1.Y) * (p3.X - p1.X)

let convexHull (poly : Point list) =
    let mutable (outHull : Point list) = List.Empty
    let mutable (k : int) = 0

    for p in poly do
        while k >= 2 && not (ccw outHull.[k-2] outHull.[k-1] p) do
            k <- k - 1
        if k >= outHull.Length
        then outHull <- outHull @ [p]
        else outHull <- outHull.[0..k - 1] @ [p]
        k <- k + 1

    let (t : int) = k + 1
    for p in List.rev poly do
        while k >= t && not (ccw outHull.[k-2] outHull.[k-1] p) do
            k <- k - 1
        if k >= outHull.Length
        then outHull <- outHull @ [p]
        else outHull <- outHull.[0..k - 1] @ [p]
        k <- k + 1

    outHull.[0 .. k - 2]

affiche (convexHull (List.sortBy (fun (x : Point) -> x.X, x.Y) poly))
Output:
Convex Hull: [(-9,-3), (-3,-9), (19,-8), (17,5), (12,17), (5,19), (-3,15)]

Fortran[edit]

Translation of: Scheme
Works with: gfortran version 11.3.0


module convex_hulls
  !
  ! Convex hulls by Andrew's monotone chain algorithm.
  !
  ! For a description of the algorithm, see
  ! https://en.wikibooks.org/w/index.php?title=Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain&stableid=40169
  !
  ! For brevity in the task, I shall use the built-in "complex" type
  ! to represent objects in the plane. One could have fun rewriting
  ! this implementation in terms of geometric algebra.
  !

  implicit none
  private

  public :: find_convex_hull

contains

  elemental function x (u)
    complex, intent(in) :: u
    real :: x

    x = real (u)
  end function x

  elemental function y (u)
    complex, intent(in) :: u
    real :: y

    y = aimag (u)
  end function y

  elemental function cross (u, v) result (p)
    complex, intent(in) :: u, v
    real :: p

    ! The cross product as a signed scalar.
    p = (x (u) * y (v)) - (y (u) * x (v))
  end function cross

  subroutine sort_points (num_points, points)
    integer, intent(in) :: num_points
    complex, intent(inout) :: points(0:*)

    ! Sort first in ascending order by x-coordinates, then in
    ! ascending order by y-coordinates. Any decent sort algorithm will
    ! suffice; for the sake of interest, here is the Shell sort of
    ! https://en.wikipedia.org/w/index.php?title=Shellsort&oldid=1084744510

    integer, parameter :: gaps(1:8) = (/ 701, 301, 132, 57, 23, 10, 4, 1 /)

    integer :: i, j, k, gap, offset
    complex :: temp
    logical :: done

    do k = 1, 8
       gap = gaps(k)
       do offset = 0, gap - 1
          do i = offset, num_points - 1, gap
             temp = points(i)
             j = i
             done = .false.
             do while (.not. done)
                if (j < gap) then
                   done = .true.
                else if (x (points(j - gap)) < x (temp)) then
                   done = .true.
                else if (x (points(j - gap)) == x (temp) .and. &
                     &    (y (points(j - gap)) <= y (temp))) then
                   done = .true.
                else
                   points(j) = points(j - gap)
                   j = j - gap
                end if
             end do
             points(j) = temp
          end do
       end do
    end do
  end subroutine sort_points

  subroutine delete_neighbor_duplicates (n, pt)
    integer, intent(inout) :: n
    complex, intent(inout) :: pt(0:*)

    call delete_trailing_duplicates
    call delete_nontrailing_duplicates

  contains

    subroutine delete_trailing_duplicates
      integer :: i
      logical :: done

      i = n - 1
      done = .false.
      do while (.not. done)
         if (i == 0) then
            n = 1
            done = .true.
         else if (pt(i - 1) /= pt(i)) then
            n = i + 1
            done = .true.
         else
            i = i - 1
         end if
      end do
    end subroutine delete_trailing_duplicates

    subroutine delete_nontrailing_duplicates
      integer :: i, j, num_deleted
      logical :: done

      i = 0
      do while (i < n - 1)
         j = i + 1
         done = .false.
         do while (.not. done)
            if (j == n) then
               done = .true.
            else if (pt(j) /= pt(i)) then
               done = .true.
            else
               j = j + 1
            end if
         end do
         if (j /= i + 1) then
            num_deleted = j - i - 1
            do while (j /= n)
               pt(j - num_deleted) = pt(j)
               j = j + 1
            end do
            n = n - num_deleted
         end if
         i = i + 1
      end do
    end subroutine delete_nontrailing_duplicates

  end subroutine delete_neighbor_duplicates

  subroutine construct_lower_hull (n, pt, hull_size, hull)
    integer, intent(in) :: n    ! Number of points.
    complex, intent(in) :: pt(0:*)
    integer, intent(inout) :: hull_size
    complex, intent(inout) :: hull(0:*)

    integer :: i, j
    logical :: done

    j = 1
    hull(0:1) = pt(0:1)
    do i = 2, n - 1
       done = .false.
       do while (.not. done)
          if (j == 0) then
             j = j + 1
             hull(j) = pt(i)
             done = .true.
          else if (0.0 < cross (hull(j) - hull(j - 1), &
               &                pt(i) - hull(j - 1))) then
             j = j + 1
             hull(j) = pt(i)
             done = .true.
          else
             j = j - 1
          end if
       end do
    end do
    hull_size = j + 1
  end subroutine construct_lower_hull

  subroutine construct_upper_hull (n, pt, hull_size, hull)
    integer, intent(in) :: n    ! Number of points.
    complex, intent(in) :: pt(0:*)
    integer, intent(inout) :: hull_size
    complex, intent(inout) :: hull(0:*)

    integer :: i, j
    logical :: done

    j = 1
    hull(0:1) = pt(n - 1 : n - 2 : -1)
    do i = n - 3, 0, -1
       done = .false.
       do while (.not. done)
          if (j == 0) then
             j = j + 1
             hull(j) = pt(i)
             done = .true.
          else if (0.0 < cross (hull(j) - hull(j - 1), &
               &                pt(i) - hull(j - 1))) then
             j = j + 1
             hull(j) = pt(i)
             done = .true.
          else
             j = j - 1
          end if
       end do
    end do
    hull_size = j + 1
  end subroutine construct_upper_hull

  subroutine contruct_hull (n, pt, hull_size, hull)
    integer, intent(in) :: n    ! Number of points.
    complex, intent(in) :: pt(0:*)
    integer, intent(inout) :: hull_size
    complex, intent(inout) :: hull(0:*)

    integer :: lower_hull_size, upper_hull_size
    complex :: lower_hull(0 : n - 1), upper_hull(0 : n - 1)
    integer :: ihull0

    ihull0 = lbound (hull, 1)

    ! A side note: the calls to construct_lower_hull and
    ! construct_upper_hull could be done in parallel.
    call construct_lower_hull (n, pt, lower_hull_size, lower_hull)
    call construct_upper_hull (n, pt, upper_hull_size, upper_hull)

    hull_size = lower_hull_size + upper_hull_size - 2

    hull(:ihull0 + lower_hull_size - 2) =                          &
         & lower_hull(:lower_hull_size - 2)
    hull(ihull0 + lower_hull_size - 1 : ihull0 + hull_size - 1) =  &
         & upper_hull(0 : upper_hull_size - 2)
  end subroutine contruct_hull

  subroutine find_convex_hull (n, points, hull_size, hull)
    integer, intent(in) :: n            ! Number of points.
    complex, intent(in) :: points(*)    ! Input points.
    integer, intent(inout) :: hull_size ! The size of the hull.
    complex, intent(inout) :: hull(*)   ! Points of the hull.

    !
    ! Yes, you can call this with something like
    !
    !    call find_convex_hull (n, points, n, points)
    !
    ! and in the program below I shall demonstrate that.
    !

    complex :: pt(0 : n - 1)
    integer :: ipoints0, ihull0, numpt

    ipoints0 = lbound (points, 1)
    ihull0 = lbound (hull, 1)

    pt = points(:ipoints0 + n - 1)
    numpt = n

    call sort_points (numpt, pt)
    call delete_neighbor_duplicates (numpt, pt)

    if (numpt == 0) then
       hull_size = 0
    else if (numpt <= 2) then
       hull_size = numpt
       hull(:ihull0 + numpt - 1) = pt(:numpt - 1)
    else
       call contruct_hull (numpt, pt, hull_size, hull)
    end if
  end subroutine find_convex_hull

end module convex_hulls

program convex_hull_task
  use, non_intrinsic :: convex_hulls
  implicit none

  complex, parameter :: example_points(20) =                   &
       & (/ (16, 3), (12, 17), (0, 6), (-4, -6), (16, 6),      &
       &    (16, -7), (16, -3), (17, -4), (5, 19), (19, -8),   &
       &    (3, 16), (12, 13), (3, -4), (17, 5), (-3, 15),     &
       &    (-3, -9), (0, 11), (-9, -3), (-4, -2), (12, 10) /)

  integer :: n, i
  complex :: points(0:100)
  character(len = 100) :: fmt

  n = 20
  points(1:n) = example_points
  call find_convex_hull (n, points(1:n), n, points(1:n))

  write (fmt, '("(", I20, ''("(", F3.0, 1X, F3.0, ") ")'', ")")') n
  write (*, fmt) (points(i), i = 1, n)

end program convex_hull_task
Output:

If you compile with -Wextra you may get warnings about the use of == on floating-point numbers. I hate those warnings and turn them off if I am using -Wextra. (Unfortunately, there is much incorrect information about == and floating point that is widely taught as inviolable doctrine.)

$ gfortran -fcheck=all -std=f2018 convex_hull_task.f90 && ./a.out
(-9. -3.) (-3. -9.) (19. -8.) (17.  5.) (12. 17.) ( 5. 19.) (-3. 15.)

FreeBASIC[edit]

#include "crt.bi"
Screen 20
Window (-20,-20)-(30,30)

Type Point
    As Single x,y
    As Long done
End Type

#macro rotate(pivot,p,a,scale)
Type<Point>(scale*(Cos(a*.0174533)*(p.x-pivot.x)-Sin(a*.0174533)*(p.y-pivot.y))+pivot.x, _
scale*(Sin(a*.0174533)*(p.x-pivot.x)+Cos(a*.0174533)*(p.y-pivot.y))+pivot.y)
#endmacro


Dim As Point p(1 To ...)={(16,3),(12,17),(0,6),(-4,-6),(16,6),(16,-7),(16,-3),(17,-4),(5,19), _
(19,-8),(3,16),(12,13),(3,-4),(17,5),(-3,15),(-3,-9),(0,11),(-9,-3),(-4,-2),(12,10)}


Function south(p() As Point,Byref idx As Long) As Point
    Dim As Point s=Type(0,100)
    For n As Long=Lbound(p) To Ubound(p)
        Circle(p(n).x,p(n).y),.2,7,,,,f 
        If s.y>p(n).y Then s=p(n):idx=n 
    Next n
    Return s
End Function

Function segment_distance(lx1 As Single, _
                          ly1 As Single, _
                          lx2 As Single, _
                          ly2 As Single, _
                          px As Single,_
                          py As Single, _
                          Byref ox As Single=0,_
                          Byref oy As Single=0) As Single
    Dim As Single M1,M2,C1,C2,B
    B=(Lx2-Lx1):If B=0 Then B=1e-20
    M2=(Ly2-Ly1)/B:If M2=0 Then M2=1e-20
    M1=-1/M2
    C1=py-M1*px
    C2=(Ly1*Lx2-Lx1*Ly2)/B
    Var L1=((px-lx1)*(px-lx1)+(py-ly1)*(py-ly1)),L2=((px-lx2)*(px-lx2)+(py-ly2)*(py-ly2))
    Var a=((lx1-lx2)*(lx1-lx2) + (ly1-ly2)*(ly1-ly2))
    Var a1=a+L1
    Var a2=a+L2
    Var f1=a1>L2,f2=a2>L1
    If f1 Xor f2 Then
        Var d1=((px-Lx1)*(px-Lx1)+(py-Ly1)*(py-Ly1))
        Var d2=((px-Lx2)*(px-Lx2)+(py-Ly2)*(py-Ly2))
        If d1<d2 Then Ox=Lx1:Oy=Ly1 : Return Sqr(d1) Else  Ox=Lx2:Oy=Ly2:Return Sqr(d2)
    End If
    Var M=M1-M2:If M=0 Then M=1e-20
    Ox=(C2-C1)/(M1-M2)
    Oy=(M1*C2-M2*C1)/M
    Return Sqr((px-Ox)*(px-Ox)+(py-Oy)*(py-Oy))
End Function


Dim As Long idx
Var s= south(p(),idx)
p(idx).done=1
Redim As Point ans(1 To 1)
ans(1)=s
Dim As Point e=s
e.x=1000
Dim As Long count=1
Dim As Single z
Circle(s.x,s.y),.4,5

Do
    z+=.05
    Var pt=rotate(s,e,z,1)
    For n As Long=Lbound(p) To Ubound(p)
        If segment_distance(s.x,s.y,pt.x,pt.y,p(n).x,p(n).y)<.05  Then
            s=p(n)
            If p(n).done=0 Then
                count+=1
                Redim Preserve ans(1 To count)
                ans(count)=p(n)
                p(n).done=1
            End If
        End If
        Circle(s.x,s.y),.4,5
    Next n
Loop Until z>360

For n As Long=Lbound(ans) To Ubound(ans)
    printf (!"(%2.0f , %2.0f )\n", ans(n).x, ans(n).y)
Next
Sleep
Output:
Graphical, plus console:
(-3 , -9 )
(19 , -8 )
(17 ,  5 )
(12 , 17 )
( 5 , 19 )
(-3 , 15 )
(-9 , -3 )

Go[edit]

package main

import (
	"fmt"
	"image"
	"sort"
)


// ConvexHull returns the set of points that define the
// convex hull of p in CCW order starting from the left most.
func (p points) ConvexHull() points {
	// From https://en.wikibooks.org/wiki/Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain
	// with only minor deviations.
	sort.Sort(p)
	var h points

	// Lower hull
	for _, pt := range p {
		for len(h) >= 2 && !ccw(h[len(h)-2], h[len(h)-1], pt) {
			h = h[:len(h)-1]
		}
		h = append(h, pt)
	}

	// Upper hull
	for i, t := len(p)-2, len(h)+1; i >= 0; i-- {
		pt := p[i]
		for len(h) >= t && !ccw(h[len(h)-2], h[len(h)-1], pt) {
			h = h[:len(h)-1]
		}
		h = append(h, pt)
	}

	return h[:len(h)-1]
}

// ccw returns true if the three points make a counter-clockwise turn
func ccw(a, b, c image.Point) bool {
	return ((b.X - a.X) * (c.Y - a.Y)) > ((b.Y - a.Y) * (c.X - a.X))
}

type points []image.Point

func (p points) Len() int      { return len(p) }
func (p points) Swap(i, j int) { p[i], p[j] = p[j], p[i] }
func (p points) Less(i, j int) bool {
	if p[i].X == p[j].X {
		return p[i].Y < p[i].Y
	}
	return p[i].X < p[j].X
}

func main() {
	pts := points{
		{16, 3}, {12, 17}, {0, 6}, {-4, -6}, {16, 6},
		{16, -7}, {16, -3}, {17, -4}, {5, 19}, {19, -8},
		{3, 16}, {12, 13}, {3, -4}, {17, 5}, {-3, 15},
		{-3, -9}, {0, 11}, {-9, -3}, {-4, -2}, {12, 10},
	}
	hull := pts.ConvexHull()
	fmt.Println("Convex Hull:", hull)
}
Output:
Convex Hull: [(-9,-3) (-3,-9) (19,-8) (17,5) (12,17) (5,19) (-3,15)]

Groovy[edit]

Translation of: Java
class ConvexHull {
    private static class Point implements Comparable<Point> {
        private int x, y

        Point(int x, int y) {
            this.x = x
            this.y = y
        }

        @Override
        int compareTo(Point o) {
            return Integer.compare(x, o.x)
        }

        @Override
        String toString() {
            return String.format("(%d, %d)", x, y)
        }
    }

    private static List<Point> convexHull(List<Point> p) {
        if (p.isEmpty()) return Collections.emptyList()
        p.sort(new Comparator<Point>() {
            @Override
            int compare(Point o1, Point o2) {
                return o1 <=> o2
            }
        })
        List<Point> h = new ArrayList<>()

        // lower hull
        for (Point pt : p) {
            while (h.size() >= 2 && !ccw(h.get(h.size() - 2), h.get(h.size() - 1), pt)) {
                h.remove(h.size() - 1)
            }
            h.add(pt)
        }

        // upper hull
        int t = h.size() + 1
        for (int i = p.size() - 1; i >= 0; i--) {
            Point pt = p.get(i)
            while (h.size() >= t && !ccw(h.get(h.size() - 2), h.get(h.size() - 1), pt)) {
                h.remove(h.size() - 1)
            }
            h.add(pt)
        }

        h.remove(h.size() - 1)
        return h
    }

    // ccw returns true if the three points make a counter-clockwise turn
    private static boolean ccw(Point a, Point b, Point c) {
        return ((b.x - a.x) * (c.y - a.y)) > ((b.y - a.y) * (c.x - a.x))
    }

    static void main(String[] args) {
        List<Point> points = Arrays.asList(new Point(16, 3),
                new Point(12, 17),
                new Point(0, 6),
                new Point(-4, -6),
                new Point(16, 6),

                new Point(16, -7),
                new Point(16, -3),
                new Point(17, -4),
                new Point(5, 19),
                new Point(19, -8),

                new Point(3, 16),
                new Point(12, 13),
                new Point(3, -4),
                new Point(17, 5),
                new Point(-3, 15),

                new Point(-3, -9),
                new Point(0, 11),
                new Point(-9, -3),
                new Point(-4, -2),
                new Point(12, 10))

        List<Point> hull = convexHull(points)
        println("Convex Hull: $hull")
    }
}
Output:
Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]

Haskell[edit]

import Data.List (sortBy, groupBy, maximumBy)
import Data.Ord (comparing)

(x, y) = ((!! 0), (!! 1))

compareFrom
  :: (Num a, Ord a)
  => [a] -> [a] -> [a] -> Ordering
compareFrom o l r =
  compare ((x l - x o) * (y r - y o)) ((y l - y o) * (x r - x o))

distanceFrom
  :: Floating a
  => [a] -> [a] -> a
distanceFrom from to = ((x to - x from) ** 2 + (y to - y from) ** 2) ** (1 / 2)

convexHull
  :: (Floating a, Ord a)
  => [[a]] -> [[a]]
convexHull points =
  let o = minimum points
      presorted = sortBy (compareFrom o) (filter (/= o) points)
      collinears = groupBy (((EQ ==) .) . compareFrom o) presorted
      outmost = maximumBy (comparing (distanceFrom o)) <$> collinears
  in dropConcavities [o] outmost

dropConcavities
  :: (Num a, Ord a)
  => [[a]] -> [[a]] -> [[a]]
dropConcavities (left:lefter) (right:righter:rightest) =
  case compareFrom left right righter of
    LT -> dropConcavities (right : left : lefter) (righter : rightest)
    EQ -> dropConcavities (left : lefter) (righter : rightest)
    GT -> dropConcavities lefter (left : righter : rightest)
dropConcavities output lastInput = lastInput ++ output

main :: IO ()
main =
  mapM_ print $
  convexHull
    [ [16, 3]
    , [12, 17]
    , [0, 6]
    , [-4, -6]
    , [16, 6]
    , [16, -7]
    , [16, -3]
    , [17, -4]
    , [5, 19]
    , [19, -8]
    , [3, 16]
    , [12, 13]
    , [3, -4]
    , [17, 5]
    , [-3, 15]
    , [-3, -9]
    , [0, 11]
    , [-9, -3]
    , [-4, -2]
    , [12, 10]
    ]
Output:
[-3.0,-9.0]
[19.0,-8.0]
[17.0,5.0]
[12.0,17.0]
[5.0,19.0]
[-3.0,15.0]
[-9.0,-3.0]

Haxe[edit]

Translation of: Nim
typedef Point = {x:Float, y:Float};

class Main {
    
    // Calculate orientation for 3 points
    // 0 -> Straight line
    // 1 -> Clockwise
    // 2 -> Counterclockwise
    static function orientation(pt1:Point, pt2:Point, pt3:Point): Int
    {
      var val = ((pt2.x - pt1.x) * (pt3.y - pt1.y)) - 
                ((pt2.y - pt1.y) * (pt3.x - pt1.x));
      if (val == 0)
        return 0;
      else if (val > 0)
        return 1;
      else return 2;
    }

    static function convexHull(pts:Array<Point>):Array<Point> {
      var result = new Array<Point>();

      // There must be at least 3 points
      if (pts.length < 3)
        for (i in pts) result.push(i);
      
      // Find the leftmost point
      var indexMinX = 0;
      for (i in 0...(pts.length - 1))
        if (pts[i].x < pts[indexMinX].x)
          indexMinX = i;
      
      var p = indexMinX;
      var q = 0;

      while (true) {
        // The leftmost point must be part of the hull.
        result.push(pts[p]);

        q = (p + 1) % pts.length;

        for (i in 0...(pts.length - 1))
          if (orientation(pts[p], pts[i], pts[q]) == 2) q = i;
        
        p = q;

        // Break from loop once we reach the first point again.
        if (p == indexMinX) 
          break;
      }
      return result;
    }

    static function main() {
      var pts = new Array<Point>();
      pts.push({x: 16, y: 3});
      pts.push({x: 12, y: 17});
      pts.push({x: 0, y: 6});
      pts.push({x: -4, y: -6});
      pts.push({x: 16, y: 6});

      pts.push({x: 16, y: -7});
      pts.push({x: 16, y: -3});
      pts.push({x: 17, y: -4});
      pts.push({x: 5, y: 19});
      pts.push({x: 19, y: -8});

      pts.push({x: 3, y: 16});
      pts.push({x: 12, y: 13});
      pts.push({x: 3, y: -4});
      pts.push({x: 17, y: 5});
      pts.push({x: -3, y: 15});

      pts.push({x: -3, y: -9});
      pts.push({x: 0, y: 11});
      pts.push({x: -9, y: -3});
      pts.push({x: -4, y: -2});
      pts.push({x: 12, y: 10});

      var hull = convexHull(pts);
      Sys.print('Convex Hull: [');
      var length = hull.length;
      var i = 0;
      while (length > 0) {
        if (i > 0)
          Sys.print(', ');
        Sys.print('(${hull[i].x}, ${hull[i].y})');
        length--;
        i++;
      }
      Sys.println(']');
    }
}
Output:
Convex Hull: [(-9, -3), (-3, 15), (5, 19), (12, 17), (17, 5), (19, -8), (-3, -9)]

Icon[edit]

Translation of: ObjectIcon


#
# Convex hulls by Andrew's monotone chain algorithm.
#
# For a description of the algorithm, see
# https://en.wikibooks.org/w/index.php?title=Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain&stableid=40169
#

record PlanePoint (x, y)

######################################################################
#
# Merge sort adapted from the Object Icon IPL (public domain code).
#

# A merge sort implementation. This returns a sorted copy, leaving the
# original unchanged.
#
# :Parameters :
# :  `l` - the list to sort
# :  `cmp` - a comparator function
#
procedure mergesort (l, cmp)
  return mergesort1 (l, cmp, 1, *l)
end

procedure mergesort1 (l, cmp, first, last)
  local l1, l2, l3, m, v1
  if last <= first then
      return l[first:last + 1]
  m := (first + last) / 2
  l1 := mergesort1 (l, cmp, first, m)
  l2 := mergesort1 (l, cmp, m + 1, last)
  l3 := []
  every v1 := !l1 do {
    while cmp (v1, l2[1]) > 0 do
        put (l3, get(l2))
    put (l3, v1)
  }
  every put(l3, !l2)
  return l3
end

######################################################################

procedure point_equals (p, q)
  if p.x = q.x & p.y = q.y then return else fail
end

# Impose a total order on points, making it one that will work for
# Andrew's monotone chain algorithm. *)
procedure point_comes_before (p, q)
  if (p.x < q.x) | (p.x = q.x & p.y < q.y) then return else fail
end

# Subtraction is really a vector or multivector operation.
procedure point_subtract (p, q)
  return PlanePoint (p.x - q.x, p.y - q.y)
end

# Cross product is really a multivector operation.
procedure point_cross (p, q)
  return (p.x * q.y) - (p.y * q.x)
end

procedure point_to_string (p)
  return "(" || string (p.x) || " " || string (p.y) || ")"
end

######################################################################

# Comparison like C's strcmp(3).
procedure compare_points (p, q)
  local cmp

  if point_comes_before (p, q) then
      cmp := -1
  else if point_comes_before (q, p) then
      cmp := 1
  else
      cmp := 0
  return cmp
end

procedure sort_points (points)
  # Non-destructive sort.
  return mergesort (points, compare_points)
end

procedure delete_neighbor_dups (arr, equals)
  local arr1, i

  if *arr = 0 then {
    arr1 := []
  } else {
    arr1 := [arr[1]]
    i := 2
    while i <= *arr do {
      if not (equals (arr[i], arr1[-1])) then
          put (arr1, arr[i])
      i +:= 1
    }
  }
  return arr1
end

procedure construct_lower_hull (pt)
  local hull, i, j

  hull := list (*pt)
  hull[1] := pt[1]
  hull[2] := pt[2]
  j := 2
  every i := 3 to *pt do {
    while (j ~= 1 &
           point_cross (point_subtract (hull[j], hull[j - 1]),
                        point_subtract (pt[i], hull[j - 1])) <= 0) do j -:= 1
    j +:= 1
    hull[j] := pt[i]
  }
  return hull[1 : j + 1]
end

procedure construct_upper_hull (pt)
  local hull, i, j

  hull := list (*pt)
  hull[1] := pt[-1]
  hull[2] := pt[-2]
  j := 2
  every i := 3 to *pt do {
    while (j ~= 1 &
           point_cross (point_subtract (hull[j], hull[j - 1]),
                        point_subtract (pt[-i], hull[j - 1])) <= 0) do j -:= 1
    j +:= 1
    hull[j] := pt[-i]
  }
  return hull[1 : j + 1]
end

procedure construct_hull (pt)
  local lower_hull, upper_hull

  lower_hull := construct_lower_hull (pt)
  upper_hull := construct_upper_hull (pt)
  return lower_hull[1 : -1] ||| upper_hull [1 : -1]
end

procedure find_convex_hull (points)
  local pt, hull

  if *points = 0 then {
    hull := []
  } else {
    pt := delete_neighbor_dups (sort_points (points), point_equals)
    if *pt <= 2 then {
      hull := pt
    } else {
      hull := construct_hull (pt)
    }
  }
  return hull
end

procedure main ()
  local example_points, hull

  example_points :=
      [PlanePoint (16.0, 3.0),
       PlanePoint (12.0, 17.0),
       PlanePoint (0.0, 6.0),
       PlanePoint (-4.0, -6.0),
       PlanePoint (16.0, 6.0),
       PlanePoint (16.0, -7.0),
       PlanePoint (16.0, -3.0),
       PlanePoint (17.0, -4.0),
       PlanePoint (5.0, 19.0),
       PlanePoint (19.0, -8.0),
       PlanePoint (3.0, 16.0),
       PlanePoint (12.0, 13.0),
       PlanePoint (3.0, -4.0),
       PlanePoint (17.0, 5.0),
       PlanePoint (-3.0, 15.0),
       PlanePoint (-3.0, -9.0),
       PlanePoint (0.0, 11.0),
       PlanePoint (-9.0, -3.0),
       PlanePoint (-4.0, -2.0),
       PlanePoint (12.0, 10.0)]

  hull := find_convex_hull (example_points)

  every write (point_to_string (!hull))
end

######################################################################
Output:
$ icon convex_hull_task-Icon.icn
(-9.0 -3.0)
(-3.0 -9.0)
(19.0 -8.0)
(17.0 5.0)
(12.0 17.0)
(5.0 19.0)
(-3.0 15.0)

J[edit]

Restated from the implementation at [1] which in turn is Kukuruku Hub's [2] translation of Dr. K. L. Metlov's original Russian article [3].

counterclockwise =: ({. , }. /: 12 o. }. - {.) @ /:~
crossproduct =: 11 o. [: (* +)/ }. - {.
removeinner =: #~ (1 , (0 > 3 crossproduct\ ]) , 1:)
hull =: [: removeinner^:_ counterclockwise

Example use:

   hull 16j3 12j17 0j6 _4j_6 16j6 16j_7 16j_3 17j_4 5j19 19j_8 3j16 12j13 3j_4 17j5 _3j15 _3j_9 0j11 _9j_3 _4j_2 12j10
_9j_3 _3j_9 19j_8 17j5 12j17 5j19 _3j15

   ] A =: ~. 20 2 ?.@$ 9
0 2
1 3
2 1
4 1
8 7
3 5
4 8
7 5
6 1
2 6
1 7
0 1
6 8
4 0
8 6
7 6
5 8
0 4
5 3
   hull&.:(+.inv"1) A
0 1
4 0
6 1
8 6
8 7
6 8
4 8
1 7
0 4

Java[edit]

Translation of: Kotlin
import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;
import java.util.stream.Collectors;

import static java.util.Collections.emptyList;

public class ConvexHull {
    private static class Point implements Comparable<Point> {
        private int x, y;

        public Point(int x, int y) {
            this.x = x;
            this.y = y;
        }

        @Override
        public int compareTo(Point o) {
            return Integer.compare(x, o.x);
        }

        @Override
        public String toString() {
            return String.format("(%d, %d)", x, y);
        }
    }

    private static List<Point> convexHull(List<Point> p) {
        if (p.isEmpty()) return emptyList();
        p.sort(Point::compareTo);
        List<Point> h = new ArrayList<>();

        // lower hull
        for (Point pt : p) {
            while (h.size() >= 2 && !ccw(h.get(h.size() - 2), h.get(h.size() - 1), pt)) {
                h.remove(h.size() - 1);
            }
            h.add(pt);
        }

        // upper hull
        int t = h.size() + 1;
        for (int i = p.size() - 1; i >= 0; i--) {
            Point pt = p.get(i);
            while (h.size() >= t && !ccw(h.get(h.size() - 2), h.get(h.size() - 1), pt)) {
                h.remove(h.size() - 1);
            }
            h.add(pt);
        }

        h.remove(h.size() - 1);
        return h;
    }

    // ccw returns true if the three points make a counter-clockwise turn
    private static boolean ccw(Point a, Point b, Point c) {
        return ((b.x - a.x) * (c.y - a.y)) > ((b.y - a.y) * (c.x - a.x));
    }

    public static void main(String[] args) {
        List<Point> points = Arrays.asList(new Point(16, 3),
                                           new Point(12, 17),
                                           new Point(0, 6),
                                           new Point(-4, -6),
                                           new Point(16, 6),

                                           new Point(16, -7),
                                           new Point(16, -3),
                                           new Point(17, -4),
                                           new Point(5, 19),
                                           new Point(19, -8),

                                           new Point(3, 16),
                                           new Point(12, 13),
                                           new Point(3, -4),
                                           new Point(17, 5),
                                           new Point(-3, 15),

                                           new Point(-3, -9),
                                           new Point(0, 11),
                                           new Point(-9, -3),
                                           new Point(-4, -2),
                                           new Point(12, 10));

        List<Point> hull = convexHull(points);
        System.out.printf("Convex Hull: %s\n", hull);
    }
}
Output:
Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]

JavaScript[edit]

function convexHull(points) {
    points.sort(comparison);
    var L = [];
    for (var i = 0; i < points.length; i++) {
        while (L.length >= 2 && cross(L[L.length - 2], L[L.length - 1], points[i]) <= 0) {
            L.pop();
        }
        L.push(points[i]);
    }
    var U = [];
    for (var i = points.length - 1; i >= 0; i--) {
        while (U.length >= 2 && cross(U[U.length - 2], U[U.length - 1], points[i]) <= 0) {
            U.pop();
        }
        U.push(points[i]);
    }
    L.pop();
    U.pop();
    return L.concat(U);
}

function comparison(a, b) {
    return a.x == b.x ? a.y - b.y : a.x - b.x;
}

function cross(a, b, o) {
    return (a.x - o.x) * (b.y - o.y) - (a.y - o.y) * (b.x - o.x);
}

For usage: convexhull.js

var points = [];
var hull = [];

function setup() {
    createCanvas(1132, 700);
    frameRate(10);

    strokeWeight(4);
    stroke(220);
}

function draw() {
    background(40);
    // draw points
    for (i = 0; i < points.length; i++) {
        point(points[i].x, points[i].y);
    };
    console.log(hull);
    // draw hull
    noFill();
    beginShape();
    for (i = 0; i < hull.length; i++) {
        vertex(hull[i].x, hull[i].y);
    };
    endShape(CLOSE);
}

function mouseClicked() {
    points.push(createVector(mouseX, mouseY));
    hull = convexHull(points);
    noFill();
    //console.log(hull);
    beginShape();
    for (var i = 0; i < hull.length; i++) {
        vertex(hull[i].x, hull[i].y);
    }
    endShape(CLOSE);
    return false;
}

// https://en.wikibooks.org/wiki/Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain
function convexHull(points) {
    points.sort(comparison);
    var L = [];
    for (var i = 0; i < points.length; i++) {
        while (L.length >= 2 && cross(L[L.length - 2], L[L.length - 1], points[i]) <= 0) {
            L.pop();
        }
        L.push(points[i]);
    }
    var U = [];
    for (var i = points.length - 1; i >= 0; i--) {
        while (U.length >= 2 && cross(U[U.length - 2], U[U.length - 1], points[i]) <= 0) {
            U.pop();
        }
        U.push(points[i]);
    }
    L.pop();
    U.pop();
    return L.concat(U);
}

function comparison(a, b) {
    return a.x == b.x ? a.y - b.y : a.x - b.x;
}

function cross(a, b, o) {
    return (a.x - o.x) * (b.y - o.y) - (a.y - o.y) * (b.x - o.x);
}

convexhull.html

<html>

<head>
    <script src="https://cdnjs.cloudflare.com/ajax/libs/p5.js/0.7.2/p5.js"></script>
    <script src="convexhull.js"></script>
</head>

<body>
    <table>
        <tr>
            <th><h1>Convex Hull</h4></th>
            <th><h4>Left mouse: Add points</h6></th>
        </tr>
    </table>

</body>

</html>

jq[edit]

Translation of: Wren
Works with: jq

Works with gojq, the Go implementation of jq

# ccw returns true if the three points make a counter-clockwise turn
def ccw(a; b; c):
  a as [$ax, $ay]
  | b as [$bx, $by]
  | c as [$cx, $cy]
  | (($bx - $ax) * ($cy - $ay)) > (($by - $ay) * ($cx - $ax)) ;

def convexHull:
  if . == [] then []
  else sort as $pts
  # lower hull:
  | reduce $pts[] as $pt ([];
      until (length < 2 or ccw(.[-2]; .[-1]; $pt); .[:-1] )
      | . + [$pt] )
  # upper hull
  | (length + 1) as $t
  | reduce range($pts|length-2; -1; -1) as $i (.;
      $pts[$i] as $pt
      | until (length < $t or ccw(.[-2]; .[-1]; $pt); .[:-1] )
      | . + [$pt])
  | .[:-1]
  end ;

The task

def pts: [
    [16,  3], [12, 17], [ 0,  6], [-4, -6], [16,  6],
    [16, -7], [16, -3], [17, -4], [ 5, 19], [19, -8],
    [ 3, 16], [12, 13], [ 3, -4], [17,  5], [-3, 15],
    [-3, -9], [ 0, 11], [-9, -3], [-4, -2], [12, 10]
];

"Convex Hull: \(pts|convexHull)"
Output:
Convex Hull: [[-9,-3],[-3,-9],[19,-8],[17,5],[12,17],[5,19],[-3,15]]


Julia[edit]

# v1.0.4
# https://github.com/JuliaPolyhedra/Polyhedra.jl/blob/master/examples/operations.ipynb
using Polyhedra, CDDLib

A = vrep([[16,3],  [12,17], [0,6], [-4,-6], [16,6], [16,-7], [16,-3], [17,-4], [5,19], [19,-8], [3,16], [12,13], [3,-4], [17,5], [-3,15], [-3,-9], [0,11], [-9,-3], [-4,-2], [12,10]])
P = polyhedron(A, CDDLib.Library())
Pch = convexhull(P, P)
removevredundancy!(Pch)
println("$Pch")
Output:
convexhull([5.0, 19.0], [19.0, -8.0], [17.0, 5.0], [-3.0, 15.0], [-9.0, -3.0], [12.0, 17.0], [-3.0, -9.0])

Kotlin[edit]

Translation of: Go
// version 1.1.3

class Point(val x: Int, val y: Int) : Comparable<Point> {

    override fun compareTo(other: Point) = this.x.compareTo(other.x)

    override fun toString() = "($x, $y)"
}

fun convexHull(p: Array<Point>): List<Point> {
    if (p.isEmpty()) return emptyList()
    p.sort()
    val h = mutableListOf<Point>()

    // lower hull
    for (pt in p) {
        while (h.size >= 2 && !ccw(h[h.size - 2], h.last(), pt)) {
            h.removeAt(h.lastIndex)
        }
        h.add(pt)
    }

    // upper hull
    val t = h.size + 1
    for (i in p.size - 2 downTo 0) {
        val pt = p[i]
        while (h.size >= t && !ccw(h[h.size - 2], h.last(), pt)) {
            h.removeAt(h.lastIndex)
        }
        h.add(pt)
    }

    h.removeAt(h.lastIndex)
    return h
}

/* ccw returns true if the three points make a counter-clockwise turn */
fun ccw(a: Point, b: Point, c: Point) =
    ((b.x - a.x) * (c.y - a.y)) > ((b.y - a.y) * (c.x - a.x))

fun main(args: Array<String>) {
    val points = arrayOf(
        Point(16,  3), Point(12, 17), Point( 0,  6), Point(-4, -6), Point(16,  6),
        Point(16, -7), Point(16, -3), Point(17, -4), Point( 5, 19), Point(19, -8),
        Point( 3, 16), Point(12, 13), Point( 3, -4), Point(17,  5), Point(-3, 15),
        Point(-3, -9), Point( 0, 11), Point(-9, -3), Point(-4, -2), Point(12, 10)
    )
    val hull = convexHull(points)
    println("Convex Hull: $hull")
}
Output:
Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]

WARRNING: Wrong results for case:

val points = arrayOf(
  Point(1387, 2842),
  Point(1247, 2842),   // A
  Point(1247, 2382),   // B (it should be in result)
  Point(1387, 2462),
  Point(1247, 2462),
  Point(1527, 2762),
  Point(1387, 2382),
)

Also if we swap points A and B then result will be different (!). Probably the problem is that when we use p.sort() we sort only by x - but for case when A.x == B.x we should also sort by y (like in e.g. JavaScript version). So below code in class Point method compareTo probably should be used

    override fun compareTo(other: Point): Int {
        val x = this.x.compareTo(other.x)
        
        if (x==0) {
            return this.y.compareTo(other.y)
        } else {
            return x
        }
    }

Lua[edit]

Translation of: C++
function print_point(p)
    io.write("("..p.x..", "..p.y..")")
    return nil
end

function print_points(pl)
    io.write("[")
    for i,p in pairs(pl) do
        if i>1 then
            io.write(", ")
        end
        print_point(p)
    end
    io.write("]")
    return nil
end

function ccw(a,b,c)
    return (b.x - a.x) * (c.y - a.y) > (b.y - a.y) * (c.x - a.x)
end

function pop_back(ta)
    table.remove(ta,#ta)
    return ta
end

function convexHull(pl)
    if #pl == 0 then
        return {}
    end
    table.sort(pl, function(left,right)
        return left.x < right.x
    end)

    local h = {}

    -- lower hull
    for i,pt in pairs(pl) do
        while #h >= 2 and not ccw(h[#h-1], h[#h], pt) do
            table.remove(h,#h)
        end
        table.insert(h,pt)
    end

    -- upper hull
    local t = #h + 1
    for i=#pl, 1, -1 do
        local pt = pl[i]
        while #h >= t and not ccw(h[#h-1], h[#h], pt) do
            table.remove(h,#h)
        end
        table.insert(h,pt)
    end

    table.remove(h,#h)
    return h
end

-- main
local points = {
    {x=16,y= 3},{x=12,y=17},{x= 0,y= 6},{x=-4,y=-6},{x=16,y= 6},
    {x=16,y=-7},{x=16,y=-3},{x=17,y=-4},{x= 5,y=19},{x=19,y=-8},
    {x= 3,y=16},{x=12,y=13},{x= 3,y=-4},{x=17,y= 5},{x=-3,y=15},
    {x=-3,y=-9},{x= 0,y=11},{x=-9,y=-3},{x=-4,y=-2},{x=12,y=10}
}
local hull = convexHull(points)

io.write("Convex Hull: ")
print_points(hull)
print()
Output:
Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]

Maple[edit]

Function are available in the geometry, ComputationalGeometry and simplex packages.

pts:=[[16,3],[12,17],[0,6],[-4,-6],[16,6],[16,-7],[16,-3],[17,-4],[5,19],[19,-8],
[3,16],[12,13],[3,-4],[17,5],[-3,15],[-3,-9],[0,11],[-9,-3],[-4,-2],[12,10]]:

with(geometry):
map(coordinates,convexhull([seq(point(P||i,pts[i]),i=1..nops(pts))]));
# [[-9, -3], [-3, -9], [19, -8], [17, 5], [12, 17], [5, 19], [-3, 15]]

with(ComputationalGeometry):
pts[ConvexHull(pts)];
# [[12, 17], [5, 19], [-3, 15], [-9, -3], [-3, -9], [19, -8], [17, 5]]

simplex:-convexhull(pts);
# [[-9, -3], [-3, -9], [19, -8], [17, 5], [12, 17], [5, 19], [-3, 15]]

Mathematica/Wolfram Language[edit]

hullPoints[data_] := MeshCoordinates[ConvexHullMesh[data]];
hullPoints[{{16, 3}, {12, 17}, {0, 6}, {-4, -6}, {16, 6}, {16, -7}, {16, -3}, {17, -4}, {5, 19}, {19, -8}, {3, 16}, {12, 13}, {3, -4}, {17, 5}, {-3, 15}, {-3, -9}, {0, 11}, {-9, -3}, {-4, -2}, {12, 10}}]
Output:
{{12., 17.}, {5., 19.}, {19., -8.}, {17., 5.}, {-3., 15.}, {-3., -9.}, {-9., -3.}}

Mercury[edit]

Translation of: Standard ML
Works with: Mercury version 20.06.1


:- module convex_hull_task.

:- interface.
:- import_module io.
:- pred main(io, io).
:- mode main(di, uo) is det.

:- implementation.
:- import_module exception.
:- import_module float.
:- import_module int.
:- import_module list.
:- import_module pair.
:- import_module string.
:- import_module version_array.

%%--------------------------------------------------------------------

%% fetch_items/3 for version_array, similar to the library function
%% for regular array.
:- func fetch_items(version_array(T), int, int) = list(T).
fetch_items(Arr, I, J) = fetch_items_(Arr, I, J, []).

:- func fetch_items_(version_array(T), int, int, list(T)) = list(T).
fetch_items_(Arr, I, J, Lst0) = Lst :-
  if (J < I) then (Lst = Lst0)
  else (J1 = J - 1,
        Lst = fetch_items_(Arr, I, J1, [Arr^elem(J) | Lst0])).

%%--------------------------------------------------------------------

:- type point == pair(float).
:- type point_list == list(point).
:- type point_array == version_array(point).

:- pred point_comes_before(point, point).
:- mode point_comes_before(in, in) is semidet.
point_comes_before(P, Q) :-
  (fst(P) < fst(Q); fst(P) - fst(Q) = (0.0),
                    snd(P) < snd(Q)).

:- pred point_compare(point, point, comparison_result).
:- mode point_compare(in, in, out) is det.
point_compare(P, Q, Cmp) :-
  if (point_comes_before(P, Q)) then (Cmp = (<))
  else if (point_comes_before(Q, P)) then (Cmp = (>))
  else (Cmp = (=)).

:- func point_subtract(point, point) = point.
point_subtract(P, Q) = pair(fst(P) - fst(Q),
                            snd(P) - snd(Q)).

:- func point_cross_product(point, point) = float.
point_cross_product(P, Q) = (fst(P) * snd(Q)) - (snd(P) * fst(Q)).

:- func point_to_string(point) = string.
point_to_string(P) = ("(" ++ from_float(fst(P)) ++
                      " " ++ from_float(snd(P)) ++ ")").

%%--------------------------------------------------------------------

:- func convex_hull(point_list) = point_list.
convex_hull(Pt) = Hull :-
  Pt1 = unique_points_sorted(Pt),
  (if (Pt1 = []; Pt1 = [_]; Pt1 = [_, _]) then (Hull = Pt1)
   else (Hull = construct_hull(Pt1))).

:- func unique_points_sorted(point_list) = point_list.
unique_points_sorted(Pt0) = Pt :-
  sort_and_remove_dups(point_compare, Pt0, Pt).

:- func construct_hull(point_list) = point_list.
construct_hull(Pt) = (construct_lower_hull(Pt) ++
                      construct_upper_hull(Pt)).

:- func construct_lower_hull(point_list) = point_list.
construct_lower_hull(Pt) = Hull :-
  if (Pt = [P0, P1 | Rest])
  then (N = length(Pt),
        Arr0 = (version_array.init(N, P0)),
        Arr1 = (Arr0^elem(1) := P1),
        hull_construction(Rest, Arr1, Arr2, 1, N_Hull),
        %% In the fetch_items/3 call, we leave out the last item. It
        %% is redundant with the first item of the upper hull.
        N_Hull2 = N_Hull - 2,
        Hull = fetch_items(Arr2, 0, N_Hull2))
  else throw("construct_lower_hull expects list of length >= 3").

:- func construct_upper_hull(point_list) = point_list.
%% An upper hull is merely a lower hull for points going the other
%% way.
construct_upper_hull(Pt) = construct_lower_hull(reverse(Pt)).

:- pred hull_construction(point_list, point_array, point_array,
                          int, int).
:- mode hull_construction(in, in, out, in, out) is det.
hull_construction([], Arr0, Arr, J, N_Hull) :-
  Arr = Arr0,
  N_Hull = J + 1.
hull_construction([P | Rest], Arr0, Arr, J, N_Hull) :-
  if cross_test(P, Arr0, J)
  then (J1 = J + 1,
        Arr1 = (Arr0^elem(J1) := P),
        hull_construction(Rest, Arr1, Arr, J1, N_Hull))
  else (J1 = J - 1,
        hull_construction([P | Rest], Arr0, Arr, J1, N_Hull)).

:- pred cross_test(point, point_array, int).
:- mode cross_test(in, in, in) is semidet.
cross_test(P, Arr, J) :-
  if (J = 0) then true
  else (Elem_J = Arr^elem(J),
        J1 = J - 1,
        Elem_J1 = Arr^elem(J1),
        0.0 < point_cross_product(point_subtract(Elem_J, Elem_J1),
                                  point_subtract(P, Elem_J1))).

%%--------------------------------------------------------------------

main(!IO) :-
  Example_points = [pair(16.0, 3.0),
                    pair(12.0, 17.0),
                    pair(0.0, 6.0),
                    pair(-4.0, -6.0),
                    pair(16.0, 6.0),
                    pair(16.0, -7.0),
                    pair(16.0, -3.0),
                    pair(17.0, -4.0),
                    pair(5.0, 19.0),
                    pair(19.0, -8.0),
                    pair(3.0, 16.0),
                    pair(12.0, 13.0),
                    pair(3.0, -4.0),
                    pair(17.0, 5.0),
                    pair(-3.0, 15.0),
                    pair(-3.0, -9.0),
                    pair(0.0, 11.0),
                    pair(-9.0, -3.0),
                    pair(-4.0, -2.0),
                    pair(12.0, 10.0)],
  Hull = convex_hull(Example_points),
  HullStr = join_list(" ", map(point_to_string, Hull)),
  write_string(HullStr, !IO),
  nl(!IO).

%%%-------------------------------------------------------------------
%%% local variables:
%%% mode: mercury
%%% prolog-indent-width: 2
%%% end:
Output:
$ mmc convex_hull_task.m && ./convex_hull_task
(-9.0 -3.0) (-3.0 -9.0) (19.0 -8.0) (17.0 5.0) (12.0 17.0) (5.0 19.0) (-3.0 15.0)

Modula-2[edit]

MODULE ConvexHull;
FROM FormatString IMPORT FormatString;
FROM Storage IMPORT ALLOCATE, DEALLOCATE;
FROM SYSTEM IMPORT TSIZE;
FROM Terminal IMPORT WriteString,WriteLn,ReadChar;

PROCEDURE WriteInt(n : INTEGER);
VAR buf : ARRAY[0..15] OF CHAR;
BEGIN
    FormatString("%i", buf, n);
    WriteString(buf);
END WriteInt;

TYPE
    Point = RECORD
        x, y : INTEGER;
    END;

PROCEDURE WritePoint(pt : Point);
BEGIN
    WriteString("(");
    WriteInt(pt.x);
    WriteString(", ");
    WriteInt(pt.y);
    WriteString(")");
END WritePoint;

TYPE
    NextNode = POINTER TO PNode;
    PNode = RECORD
        value : Point;
        next : NextNode;
    END;

PROCEDURE WriteNode(it : NextNode);
BEGIN
    IF it = NIL THEN
        RETURN
    END;
    WriteString("[");

    WritePoint(it^.value);
    it := it^.next;

    WHILE it # NIL DO
        WriteString(", ");
        WritePoint(it^.value);
        it := it^.next
    END;
    WriteString("]")
END WriteNode;

PROCEDURE AppendNode(pn : NextNode; p : Point) : NextNode;
VAR it,nx : NextNode;
BEGIN
    IF pn = NIL THEN
        ALLOCATE(it,TSIZE(PNode));
        it^.value := p;
        it^.next := NIL;
        RETURN it
    END;

    it := pn;
    WHILE it^.next # NIL DO
        it := it^.next
    END;

    ALLOCATE(nx,TSIZE(PNode));
    nx^.value := p;
    nx^.next := NIL;

    it^.next := nx;
    RETURN pn
END AppendNode;

PROCEDURE DeleteNode(VAR pn : NextNode);
BEGIN
    IF pn = NIL THEN RETURN END;
    DeleteNode(pn^.next);

    DEALLOCATE(pn,TSIZE(PNode));
    pn := NIL
END DeleteNode;

PROCEDURE SortNode(VAR pn : NextNode);
VAR
    it : NextNode;
    tmp : Point;
    done : BOOLEAN;
BEGIN
    REPEAT
        done := TRUE;
        it := pn;
        WHILE (it # NIL) AND (it^.next # NIL) DO
            IF it^.next^.value.x < it^.value.x THEN
                tmp := it^.value;
                it^.value := it^.next^.value;
                it^.next^.value := tmp;
                done := FALSE
            END;
            it := it^.next;
        END
    UNTIL done;
END SortNode;

PROCEDURE NodeLength(it : NextNode) : INTEGER;
VAR length : INTEGER;
BEGIN
    length := 0;
    WHILE it # NIL DO
        INC(length);
        it := it^.next;
    END;
    RETURN length
END NodeLength;

PROCEDURE ReverseNode(fp : NextNode) : NextNode;
VAR rp,tmp : NextNode;
BEGIN
    IF fp = NIL THEN RETURN NIL END;

    ALLOCATE(tmp,TSIZE(PNode));
    tmp^.value := fp^.value;
    tmp^.next := NIL;
    rp := tmp;
    fp := fp^.next;

    WHILE fp # NIL DO
        ALLOCATE(tmp,TSIZE(PNode));
        tmp^.value := fp^.value;
        tmp^.next := rp;
        rp := tmp;
        fp := fp^.next;
    END;

    RETURN rp
END ReverseNode;

(* ccw returns true if the three points make a counter-clockwise turn *)
PROCEDURE CCW(a,b,c : Point) : BOOLEAN;
BEGIN
    RETURN ((b.x - a.x) * (c.y - a.y)) > ((b.y - a.y) * (c.x - a.x))
END CCW;

PROCEDURE ConvexHull(p : NextNode) : NextNode;
VAR
    hull,it,h1,h2 : NextNode;
    t : INTEGER;
BEGIN
    IF p = NIL THEN RETURN NIL END;
    SortNode(p);
    hull := NIL;

    (* lower hull *)
    it := p;
    WHILE it # NIL DO
        IF hull # NIL THEN
            WHILE hull^.next # NIL DO
                (* At least two points in the list *)
                h2 := hull;
                h1 := hull^.next;
                WHILE h1^.next # NIL DO
                    h2 := h1;
                    h1 := h2^.next;
                END;

                IF CCW(h2^.value, h1^.value, it^.value) THEN
                    BREAK
                ELSE
                    h2^.next := NIL;
                    DeleteNode(h1);
                    h1 := NIL
                END
            END
        END;

        hull := AppendNode(hull, it^.value);
        it := it^.next;
    END;

    (* upper hull *)
    t := NodeLength(hull) + 1;
    p := ReverseNode(p);
    it := p;
    WHILE it # NIL DO
        WHILE NodeLength(hull) >= t DO
            h2 := hull;
            h1 := hull^.next;
            WHILE h1^.next # NIL DO
                h2 := h1;
                h1 := h2^.next;
            END;

            IF CCW(h2^.value, h1^.value, it^.value) THEN
                BREAK
            ELSE
                h2^.next := NIL;
                DeleteNode(h1);
                h1 := NIL
            END
        END;

        hull := AppendNode(hull, it^.value);
        it := it^.next;
    END;
    DeleteNode(p);

    h2 := hull;
    h1 := h2^.next;
    WHILE h1^.next # NIL DO
        h2 := h1;
        h1 := h1^.next;
    END;
    h2^.next := NIL;
    DeleteNode(h1);
    RETURN hull
END ConvexHull;

(* Main *)
VAR nodes,hull : NextNode;
BEGIN
    nodes := AppendNode(NIL, Point{16, 3});
    AppendNode(nodes, Point{12,17});
    AppendNode(nodes, Point{ 0, 6});
    AppendNode(nodes, Point{-4,-6});
    AppendNode(nodes, Point{16, 6});
    AppendNode(nodes, Point{16,-7});
    AppendNode(nodes, Point{16,-3});
    AppendNode(nodes, Point{17,-4});
    AppendNode(nodes, Point{ 5,19});
    AppendNode(nodes, Point{19,-8});
    AppendNode(nodes, Point{ 3,16});
    AppendNode(nodes, Point{12,13});
    AppendNode(nodes, Point{ 3,-4});
    AppendNode(nodes, Point{17, 5});
    AppendNode(nodes, Point{-3,15});
    AppendNode(nodes, Point{-3,-9});
    AppendNode(nodes, Point{ 0,11});
    AppendNode(nodes, Point{-9,-3});
    AppendNode(nodes, Point{-4,-2});
    AppendNode(nodes, Point{12,10});

    hull := ConvexHull(nodes);
    WriteNode(hull);
    DeleteNode(hull);

    DeleteNode(nodes);
    ReadChar
END ConvexHull.
Output:
[(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]

Nim[edit]

Translation of: Rust
type
  Point = object
    x: float
    y: float

# Calculate orientation for 3 points
# 0 -> Straight line
# 1 -> Clockwise
# 2 -> Counterclockwise
proc orientation(p, q, r: Point): int =
  let val = (q.y - p.y) * (r.x - q.x) -
    (q.x - p.x) * (r.y - q.y)
  
  if val == 0: 0
  elif val > 0: 1
  else: 2
  
proc calculateConvexHull(points: openArray[Point]): seq[Point] =
  result = newSeq[Point]()
  
  # There must be at least 3 points
  if len(points) < 3:
    for i in points: result.add(i)
  
  # Find the leftmost point
  var indexMinX = 0
  for i, _ in points:
    if points[i].x < points[indexMinX].x:
      indexMinX = i
  
  var p = indexMinX
  var q = 0
  
  while true:
    # The leftmost point must be part of the hull.
    result.add(points[p])
    
    q = (p + 1) mod len(points)
    
    for i in 0..<len(points):
      if orientation(points[p], points[i], points[q]) == 2:
        q = i
   
    p = q
    
    # Break from loop once we reach the first point again
    if p == indexMinX:
      break

var points = @[Point(x: 16, y: 3),
               Point(x: 12, y: 17),
               Point(x: 0, y: 6),
               Point(x: -4, y: -6),
               Point(x: 16, y: 6),
               Point(x: 16, y: -7),
               Point(x: 17, y: -4),
               Point(x: 5, y: 19),
               Point(x: 19, y: -8),
               Point(x: 3, y: 16),
               Point(x: 12, y: 13),
               Point(x: 3, y: -4),
               Point(x: 17, y: 5),
               Point(x: -3, y: 15),
               Point(x: -3, y: -9),
               Point(x: 0, y: 11),
               Point(x: -9, y: -3),
               Point(x: -4, y: -2),
               Point(x: 12, y: 10)]

let hull = calculateConvexHull(points)
for i in hull:
  echo i
Output:
(x: -9.0, y: -3.0)
(x: -3.0, y: -9.0)
(x: 19.0, y: -8.0)
(x: 17.0, y: 5.0)
(x: 12.0, y: 17.0)
(x: 5.0, y: 19.0)
(x: -3.0, y: 15.0)

ObjectIcon[edit]

Translation of: Fortran
Translation of: Standard ML


# -*- ObjectIcon -*-
#
# Convex hulls by Andrew's monotone chain algorithm.
#
# For a description of the algorithm, see
# https://en.wikibooks.org/w/index.php?title=Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain&stableid=40169
#

import io
import ipl.sort

class PlanePoint ()           # Enough plane geometry for our purpose.

  private readable x, y

  public new (x, y)
    self.x := x
    self.y := y
    return
  end

  public equals (other)
    if self.x = other.x & self.y = other.y then
      return
    else
      fail
  end

  # Impose a total order on points, making it one that will work for
  # Andrew's monotone chain algorithm. *)
  public comes_before (other)
    if (self.x < other.x) | (self.x = other.x & self.y < other.y) then
      return
    else
      fail
  end

  # Subtraction is really a vector or multivector operation.
  public minus (other)
    return PlanePoint (self.x - other.x, self.y - other.y)
  end

  # Cross product is really a multivector operation.
  public cross (other)
    return (self.x * other.y) - (self.y * other.x)
  end

  public to_string ()
    return "(" || string (self.x) || " " || string (self.y) || ")"
  end

end

# Comparison like C's strcmp(3).
procedure compare_points (p, q)
  local cmp

  if p.comes_before (q) then
    cmp := -1
  else if q.comes_before (p) then
    cmp := 1
  else
    cmp := 0
  return cmp
end

procedure sort_points (points)
  # Non-destructive sort.
  return mergesort (points, compare_points)
end

procedure delete_neighbor_dups (arr, equals)
  local arr1, i

  if *arr = 0 then {
    arr1 := []
  } else {
    arr1 := [arr[1]]
    i := 2
    while i <= *arr do {
      unless equals (arr[i], arr1[-1]) then
        put (arr1, arr[i])
      i +:= 1
    }
  }
  return arr1
end

procedure construct_lower_hull (pt)
  local hull, i, j

  hull := list (*pt)
  hull[1] := pt[1]
  hull[2] := pt[2]
  j := 2
  every i := 3 to *pt do {
    while (j ~= 1 &
           (hull[j].minus (hull[j - 1])).cross (pt[i].minus (hull[j - 1])) <= 0) do
      j -:= 1
    j +:= 1
    hull[j] := pt[i]
  }
  return hull[1 : j + 1]
end

procedure construct_upper_hull (pt)
  local hull, i, j

  hull := list (*pt)
  hull[1] := pt[-1]
  hull[2] := pt[-2]
  j := 2
  every i := 3 to *pt do {
    while (j ~= 1 &
           (hull[j].minus (hull[j - 1])).cross (pt[-i].minus (hull[j - 1])) <= 0) do
      j -:= 1
    j +:= 1
    hull[j] := pt[-i]
  }
  return hull[1 : j + 1]
end

procedure construct_hull (pt)
  local lower_hull, upper_hull

  lower_hull := construct_lower_hull (pt)
  upper_hull := construct_upper_hull (pt)
  return lower_hull[1 : -1] ||| upper_hull [1 : -1]
end

procedure points_equal (p, q)
  if p.equals (q) then
    return
  else
    fail
end

procedure find_convex_hull (points)
  local pt, hull

  if *points = 0 then {
    hull := []
  } else {
    pt := delete_neighbor_dups (sort_points (points), points_equal)
    if *pt <= 2 then
      hull := pt
    else
      hull := construct_hull (pt)
  }
  return hull
end

procedure main ()
  local example_points, hull

  example_points :=
    [PlanePoint (16.0, 3.0),
     PlanePoint (12.0, 17.0),
     PlanePoint (0.0, 6.0),
     PlanePoint (-4.0, -6.0),
     PlanePoint (16.0, 6.0),
     PlanePoint (16.0, -7.0),
     PlanePoint (16.0, -3.0),
     PlanePoint (17.0, -4.0),
     PlanePoint (5.0, 19.0),
     PlanePoint (19.0, -8.0),
     PlanePoint (3.0, 16.0),
     PlanePoint (12.0, 13.0),
     PlanePoint (3.0, -4.0),
     PlanePoint (17.0, 5.0),
     PlanePoint (-3.0, 15.0),
     PlanePoint (-3.0, -9.0),
     PlanePoint (0.0, 11.0),
     PlanePoint (-9.0, -3.0),
     PlanePoint (-4.0, -2.0),
     PlanePoint (12.0, 10.0)]

  hull := find_convex_hull (example_points)

  every write ((!hull).to_string ())
end
Output:
$ oiscript convex_hull_task-OI.icn
(-9.0 -3.0)
(-3.0 -9.0)
(19.0 -8.0)
(17.0 5.0)
(12.0 17.0)
(5.0 19.0)
(-3.0 15.0)

OCaml[edit]

Translation of: Standard ML
Works with: OCaml version 4.14.0


(*
 * Convex hulls by Andrew's monotone chain algorithm.
 *
 * For a description of the algorithm, see
 * https://en.wikibooks.org/w/index.php?title=Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain&stableid=40169
 *)

(*------------------------------------------------------------------*)
(* Just enough plane geometry for our purpose.  *)

module Plane_point =
  struct
    type t = float * float

    let make (xy : t) = xy
    let to_tuple ((x, y) : t) = (x, y)

    let x ((x, _) : t) = x
    let y ((_, y) : t) = y

    let equal (u : t) (v : t) = (x u = x v && y u = y v)

    (* Impose a total order on points, making it one that will work
       for Andrew's monotone chain algorithm. *)
    let order (u : t) (v : t) =
      (x u < x v) || (x u = x v && y u < y v)

    (* The Array module's sort routines expect a "cmp" function. *)
    let cmp u v =
      if order u v then
        (-1)
      else if order v u then
        (1)
      else
        (0)

    (* Subtraction is really a vector or multivector operation. *)
    let sub (u : t) (v : t) = make (x u -. x v, y u -. y v)

    (* Cross product is really a multivector operation. *)
    let cross (u : t) (v : t) = (x u *. y v) -. (y u *. x v)

    let to_string ((x, y) : t) =
      "(" ^ string_of_float x ^ " " ^ string_of_float y ^ ")"
  end
;;

(*------------------------------------------------------------------*)
(* We want something akin to array_delete_neighbor_dups of Scheme's
   SRFI-132. *)

let array_delete_neighbor_dups equal arr =
  (* Returns a Seq.t rather than an array. *)
  let rec loop i lst =
    (* Cons a list of non-duplicates, going backwards through the
       array so the list will be in forwards order. *)
    if i = 0 then
      arr.(i) :: lst
    else if equal arr.(i - 1) arr.(i) then
      loop (i - 1) lst
    else
      loop (i - 1) (arr.(i) :: lst)
  in
  let n = Array.length arr in
  List.to_seq (if n = 0 then [] else loop (n - 1) [])
;;

(*------------------------------------------------------------------*)
(* The convex hull algorithm. *)

let cross_test pt_i hull j =
  let hull_j = hull.(j)
  and hull_j1 = hull.(j - 1) in
  0.0 < Plane_point.(cross (sub hull_j hull_j1)
                       (sub pt_i hull_j1))

let construct_lower_hull n pt =
  let hull = Array.make n (Plane_point.make (0.0, 0.0)) in
  let () = hull.(0) <- pt.(0)
  and () = hull.(1) <- pt.(1) in
  let rec outer_loop i j =
    if i = n then
      j + 1
    else
      let pt_i = pt.(i) in
      let rec inner_loop j =
        if j = 0 || cross_test pt_i hull j then
          begin
            hull.(j + 1) <- pt_i;
            j + 1
          end
        else
          inner_loop (j - 1)
      in
      outer_loop (i + 1) (inner_loop j)
  in
  let hull_size = outer_loop 2 1 in
  (hull_size, hull)

let construct_upper_hull n pt =
  let hull = Array.make n (Plane_point.make (0.0, 0.0)) in
  let () = hull.(0) <- pt.(n - 1)
  and () = hull.(1) <- pt.(n - 2) in
  let rec outer_loop i j =
    if i = (-1) then
      j + 1
    else
      let pt_i = pt.(i) in
      let rec inner_loop j =
        if j = 0 || cross_test pt_i hull j then
          begin
            hull.(j + 1) <- pt_i;
            j + 1
          end
        else
          inner_loop (j - 1)
      in
      outer_loop (i - 1) (inner_loop j)
  in
  let hull_size = outer_loop (n - 3) 1 in
  (hull_size, hull)

let construct_hull n pt =

  (* Side note: Construction of the lower and upper hulls can be done
     in parallel. *)
  let (lower_hull_size, lower_hull) = construct_lower_hull n pt
  and (upper_hull_size, upper_hull) = construct_upper_hull n pt in

  let hull_size = lower_hull_size + upper_hull_size - 2 in
  let hull = Array.make hull_size (Plane_point.make (0.0, 0.0)) in

  begin
    Array.blit lower_hull 0 hull 0 (lower_hull_size - 1);
    Array.blit upper_hull 0 hull (lower_hull_size - 1)
      (upper_hull_size - 1);
    hull
  end

let plane_convex_hull points =
  (* Takes an arbitrary sequence of points, which may be in any order
     and may contain duplicates. Returns an ordered array of points
     that make up the convex hull. If the sequence of points is empty,
     the returned array is empty. *)
  let pt = Array.of_seq points in
  let () = Array.fast_sort Plane_point.cmp pt in
  let pt = Array.of_seq
             (array_delete_neighbor_dups Plane_point.equal pt) in
  let n = Array.length pt in
  if n <= 2 then
    pt
  else
    construct_hull n pt
;;

(*------------------------------------------------------------------*)

let example_points =
  [Plane_point.make (16.0, 3.0);
   Plane_point.make (12.0, 17.0);
   Plane_point.make (0.0, 6.0);
   Plane_point.make (-4.0, -6.0);
   Plane_point.make (16.0, 6.0);
   Plane_point.make (16.0, -7.0);
   Plane_point.make (16.0, -3.0);
   Plane_point.make (17.0, -4.0);
   Plane_point.make (5.0, 19.0);
   Plane_point.make (19.0, -8.0);
   Plane_point.make (3.0, 16.0);
   Plane_point.make (12.0, 13.0);
   Plane_point.make (3.0, -4.0);
   Plane_point.make (17.0, 5.0);
   Plane_point.make (-3.0, 15.0);
   Plane_point.make (-3.0, -9.0);
   Plane_point.make (0.0, 11.0);
   Plane_point.make (-9.0, -3.0);
   Plane_point.make (-4.0, -2.0);
   Plane_point.make (12.0, 10.0)]
;;

let hull = plane_convex_hull (List.to_seq example_points)
;;

Array.iter
  (fun p -> (print_string (Plane_point.to_string p);
             print_string " "))
  hull;
print_newline ()
;;

(*------------------------------------------------------------------*)
Output:
$ ocaml convex_hull_task.ml
(-9. -3.) (-3. -9.) (19. -8.) (17. 5.) (12. 17.) (5. 19.) (-3. 15.)

Pascal[edit]

Translation of: Fortran
{$mode ISO}

program convex_hull_task (output);

{ Convex hulls, by Andrew's monotone chain algorithm.
  
  For a description of the algorithm, see
  https://en.wikibooks.org/w/index.php?title=Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain&stableid=40169 }

const max_points = 1000;
type points_range = 0 .. max_points - 1;

type point =
  record
    x, y : real
  end;
type point_array = array [points_range] of point;

var ciura_gaps : array [1 .. 8] of integer;

var example_points : point_array;
var hull           : point_array;
var hull_size      : integer;
var index          : integer;

function make_point (x, y : real) : point;
begin
  make_point.x := x;
  make_point.y := y;
end;

{ The cross product as a signed scalar. }
function cross (u, v : point) : real;
begin
  cross := (u.x * v.y) - (u.y * v.x)
end;

function point_subtract (u, v : point) : point;
begin
  point_subtract := make_point (u.x - v.x, u.y - v.y)
end;

function point_equal (u, v : point) : boolean;
begin
  point_equal := (u.x = v.x) and (u.y = v.y)
end;

procedure sort_points (num_points : integer;
                       var points : point_array);
{ Sort first in ascending order by x-coordinates, then in
  ascending order by y-coordinates. Any decent sort algorithm will
  suffice; for the sake of interest, here is the Shell sort of
  https://en.wikipedia.org/w/index.php?title=Shellsort&oldid=1084744510 }
var
  i, j, k, gap, offset : integer;
  temp                 : point;
  done                 : boolean;
begin
  for k := 1 to 8 do
    begin
      gap := ciura_gaps[k];
      for offset := 0 to gap - 1 do
        begin
          i := offset;
          while i <= num_points - 1 do
            begin
              temp := points[i];
              j := i;
              done := false;
              while not done do
                begin
                  if j < gap then
                    done := true
                  else if points[j - gap].x < temp.x then
                    done := true
                  else if ((points[j - gap].x = temp.x)
                             and (points[j - gap].y < temp.y)) then
                    done := true
                  else
                    begin
                      points[j] := points[j - gap];
                      j := j - gap
                    end
                end;
              points[j] := temp;
              i := i + gap
            end
        end
    end
end; { sort_points }

procedure delete_neighbor_duplicates (var n  : integer;
                                      var pt : point_array);

  procedure delete_trailing_duplicates;
  var
    i    : integer;
    done : boolean;
  begin
    i := n - 1;
    done := false;
    while not done do
      begin
        if i = 0 then
          begin
            n := 1;
            done := true
          end
        else if not point_equal (pt[i - 1], pt[i]) then
          begin
            n := i + 1;
            done := true
          end
        else
          i := i + 1
      end
  end;

  procedure delete_nontrailing_duplicates;
  var
    i, j, num_deleted : integer;
    done              : boolean;
  begin
    i := 0;
    while i < n - 1 do
      begin
        j := i + 1;
        done := false;
        while not done do
          begin
            if j = n then
              done := true
            else if not point_equal (pt[j], pt[i]) then
              done := true
            else
              j := j + 1
          end;
        if j <> i + 1 then
          begin
            num_deleted := j - i - 1;
            while j <> n do
              begin
                pt[j - num_deleted] := pt[j];
                j := j + 1
              end;
            n := n - num_deleted
          end;
        i := i + 1
      end
  end;

begin
  delete_trailing_duplicates;
  delete_nontrailing_duplicates
end; { delete_neighbor_duplicates }

procedure construct_lower_hull (n             : integer;
                                pt            : point_array;
                                var hull_size : integer;
                                var hull      : point_array);
var
  i, j : integer;
  done : boolean;
begin
  j := 1;
  hull[0] := pt[0];
  hull[1] := pt[1];
  for i := 2 to n - 1 do
    begin
      done := false;
      while not done do
        begin
          if j = 0 then
            begin
              j := j + 1;
              hull[j] := pt[i];
              done := true
            end
          else if 0.0 < cross (point_subtract (hull[j],
                                               hull[j - 1]),
                               point_subtract (pt[i],
                                               hull[j - 1])) then
            begin
              j := j + 1;
              hull[j] := pt[i];
              done := true
            end
          else
            j := j - 1
        end
    end;
  hull_size := j + 1
end; { construct_lower_hull }

procedure construct_upper_hull (n             : integer;
                                pt            : point_array;
                                var hull_size : integer;
                                var hull      : point_array);
var
  i, j : integer;
  done : boolean;
begin
  j := 1;
  hull[0] := pt[n - 1];
  hull[1] := pt[n - 2];
  for i := n - 3 downto 0 do
    begin
      done := false;
      while not done do
        begin
          if j = 0 then
            begin
              j := j + 1;
              hull[j] := pt[i];
              done := true
            end
          else if 0.0 < cross (point_subtract (hull[j],
                                               hull[j - 1]),
                               point_subtract (pt[i],
                                               hull[j - 1])) then
            begin
              j := j + 1;
              hull[j] := pt[i];
              done := true
            end
          else
            j := j - 1
        end
    end;
  hull_size := j + 1
end; { construct_upper_hull }

procedure contruct_hull (n             : integer;
                         pt            : point_array;
                         var hull_size : integer;
                         var hull      : point_array);
var
  i                                : integer;
  lower_hull_size, upper_hull_size : integer;
  lower_hull, upper_hull           : point_array;
begin
  { A side note: the calls to construct_lower_hull and
    construct_upper_hull could be done in parallel. }
  construct_lower_hull (n, pt, lower_hull_size, lower_hull);
  construct_upper_hull (n, pt, upper_hull_size, upper_hull);

  hull_size := lower_hull_size + upper_hull_size - 2;

  for i := 0 to lower_hull_size - 2 do
    hull[i] := lower_hull[i];
  for i := 0 to upper_hull_size - 2 do
    hull[lower_hull_size - 1 + i] := upper_hull[i]
end; { contruct_hull }

procedure find_convex_hull (n             : integer;
                            points        : point_array;
                            var hull_size : integer;
                            var hull      : point_array);
var
  pt    : point_array;
  numpt : integer;
  i     : integer;
begin
  for i := 0 to n - 1 do
    pt[i] := points[i];
  numpt := n;

  sort_points (numpt, pt);
  delete_neighbor_duplicates (numpt, pt);

  if numpt = 0 then
    hull_size := 0
  else if numpt <= 2 then
    begin
      hull_size := numpt;
      for i := 0 to numpt - 1 do
        hull[i] := pt[i]
    end
  else
    contruct_hull (numpt, pt, hull_size, hull)
end; { find_convex_hull }

begin
  ciura_gaps[1] := 701;
  ciura_gaps[2] := 301;
  ciura_gaps[3] := 132;
  ciura_gaps[4] := 57;
  ciura_gaps[5] := 23;
  ciura_gaps[6] := 10;
  ciura_gaps[7] := 4;
  ciura_gaps[8] := 1;

  example_points[0] := make_point (16, 3);
  example_points[1] := make_point (12, 17);
  example_points[2] := make_point (0, 6);
  example_points[3] := make_point (-4, -6);
  example_points[4] := make_point (16, 6);
  example_points[5] := make_point (16, -7);
  example_points[6] := make_point (16, -3);
  example_points[7] := make_point (17, -4);
  example_points[8] := make_point (5, 19);
  example_points[9] := make_point (19, -8);
  example_points[10] := make_point (3, 16);
  example_points[11] := make_point (12, 13);
  example_points[12] := make_point (3, -4);
  example_points[13] := make_point (17, 5);
  example_points[14] := make_point (-3, 15);
  example_points[15] := make_point (-3, -9);
  example_points[16] := make_point (0, 11);
  example_points[17] := make_point (-9, -3);
  example_points[18] := make_point (-4, -2);
  example_points[19] := make_point (12, 10);

  find_convex_hull (19, example_points, hull_size, hull);

  for index := 0 to hull_size - 1 do
    writeln (hull[index].x, ' ', hull[index].y)
end.

{--------------------------------------------------------------------}
{ The Emacs Pascal mode is intolerable.
  Until I can find a substitute: }
{ local variables:  }
{ mode: fundamental }
{ end:              }
Output:
$ fpc convex_hull_task.pas && ./convex_hull_task
Free Pascal Compiler version 3.2.2 [2021/06/27] for x86_64
Copyright (c) 1993-2021 by Florian Klaempfl and others
Target OS: Linux for x86-64
Compiling convex_hull_task.pas
Linking convex_hull_task
324 lines compiled, 0.1 sec
-9.0000000000000000e+000 -3.0000000000000000e+000
-3.0000000000000000e+000 -9.0000000000000000e+000
 1.9000000000000000e+001 -8.0000000000000000e+000
 1.7000000000000000e+001  5.0000000000000000e+000
 1.2000000000000000e+001  1.7000000000000000e+001
 5.0000000000000000e+000  1.9000000000000000e+001
-3.0000000000000000e+000  1.5000000000000000e+001

Perl[edit]

Translation of: Raku
use strict;
use warnings;
use feature 'say';

{
package Point;
use Class::Struct;
struct( x => '$', y => '$',);
sub print { '(' . $_->x . ', ' . $_->y . ')' }
}

sub ccw {
    my($a, $b, $c) = @_;
    ($b->x - $a->x)*($c->y - $a->y) - ($b->y - $a->y)*($c->x - $a->x);
}

sub tangent {
    my($a, $b) = @_;
    my $opp = $b->x - $a->x;
    my $adj = $b->y - $a->y;
    $adj != 0 ? $opp / $adj : 1E99;
}

sub graham_scan {
    our @coords; local *coords = shift;
    my @sp = sort { $a->y <=> $b->y or $a->x <=> $b->x } map { Point->new( x => $_->[0], y => $_->[1] ) } @coords;

    # need at least 3 points to make a hull
    return @sp if @sp < 3;

    # first point on hull is minimum y point
    my @h  = shift @sp;

    # re-sort the points by angle, secondary on x (classic Schwartzian)
    @sp =
    map { $sp[$_->[0]] }
    sort { $b->[1] <=> $a->[1] or $a->[2] <=> $b->[2] }
    map { [$_, tangent($h[0], $sp[$_]), $sp[$_]->x] }
    0..$#sp;

    # first point of re-sorted list is guaranteed to be on hull
    push @h, shift @sp;

    # check through the remaining list making sure that there is always a positive angle
    for my $point (@sp) {
        if (ccw( @h[-2,-1], $point ) >= 0) {
            push @h, $point;
        } else {
            pop @h;
            redo;
        }
    }
    @h
}

my @hull_1 = graham_scan(
   [[16, 3], [12,17], [ 0, 6], [-4,-6], [16, 6], [16,-7], [16,-3],
    [17,-4], [ 5,19], [19,-8], [ 3,16], [12,13], [ 3,-4], [17, 5],
    [-3,15], [-3,-9], [ 0,11], [-9,-3], [-4,-2], [12,10]]
  );

my @hull_2 = graham_scan(
   [[16, 3], [12,17], [ 0, 6], [-4,-6], [16, 6], [16,-7], [16,-3],
    [17,-4], [ 5,19], [19,-8], [ 3,16], [12,13], [ 3,-4], [17, 5],
    [-3,15], [-3,-9], [ 0,11], [-9,-3], [-4,-2], [12,10], [14,-9], [1,-9]]
  );

my $list = join ' ', map { Point::print($_) } @hull_1;
say "Convex Hull (@{[scalar @hull_1]} points): [$list]";
   $list = join ' ', map { Point::print($_) } @hull_2;
say "Convex Hull (@{[scalar @hull_2]} points): [$list]";
Output:
Convex Hull (7 points): [(-3, -9) (19, -8) (17, 5) (12, 17) (5, 19) (-3, 15) (-9, -3)]
Convex Hull (9 points): [(-3, -9) (1, -9) (14, -9) (19, -8) (17, 5) (12, 17) (5, 19) (-3, 15) (-9, -3)]

Phix[edit]

Library: Phix/pGUI
Library: Phix/online

You can run this online here.

--
-- demo/rosetta/Convex_hull.exw
-- ============================
--
with javascript_semantics
constant tests = {{{16,  3}, {12, 17}, { 0,  6}, {-4, -6}, {16,  6},
                   {16, -7}, {16, -3}, {17, -4}, { 5, 19}, {19, -8},
                   { 3, 16}, {12, 13}, { 3, -4}, {17,  5}, {-3, 15},
                   {-3, -9}, { 0, 11}, {-9, -3}, {-4, -2}, {12, 10}},
                  {{16,  3}, {12, 17}, { 0,  6}, {-4, -6}, {16,  6},
                   {16, -7}, {16, -3}, {17, -4}, { 5, 19}, {19, -8},
                   { 3, 16}, {12, 13}, { 3, -4}, {17,  5}, {-3, 15},
                   {-3, -9}, { 0, 11}, {-9, -3}, {-4, -2}, {12, 10}, {1,-9}, {1,-9}},
                  {{0,0}, {5,5}, {5,0}, {0,5}},
                  {{0,0}, {0,1}, {0,2},
                   {1,0}, {1,1}, {1,2},
                   {2,0}, {2,1}, {2,2}},
                  {{-8,-8}, {-8,4}, {-8,16},
                   { 4,-8}, { 4,4}, { 4,16},
                   {16,-8}, {16,4}, {16,16}}}

integer t = 1 -- idx to tests, for the GTITLE.

enum x, y
function ccw(sequence a, b, c)
    return (b[x] - a[x]) * (c[y] - a[y])
         > (b[y] - a[y]) * (c[x] - a[x])
end function
 
function convex_hull(sequence points)
    sequence h = {}
    points = sort(deep_copy(points))
 
    /* lower hull */
    for i=1 to length(points) do
        while length(h)>=2
          and not ccw(h[$-1], h[$], points[i]) do
            h = h[1..$-1]
        end while
        h = append(h, points[i])
    end for
 
    /* upper hull */
    integer t = length(h)+1
    for i=length(points) to 1 by -1 do
        while length(h)>=t
          and not ccw(h[$-1],h[$],points[i]) do
            h = h[1..$-1]
        end while
        h = append(h, points[i])
    end for
 
    h = h[1..$-1]
    return h
end function

for i=1 to length(tests) do
    printf(1,"For test set: ")
    pp(tests[i],{pp_Indent,13,pp_Maxlen,111})
    printf(1,"Convex Hull is: ")
    pp(convex_hull(tests[i]))
end for

-- plots the test points in red crosses and the convex hull in green circles
include pGUI.e
include IupGraph.e
Ihandle dlg, graph

function get_data(Ihandle /*graph*/)
    IupSetStrAttribute(graph, "GTITLE", "Marks Mode (%d/%d)", {t, length(tests)})
    sequence tt = tests[t],
          {x,y} = columnize(tt),
        {cx,cy} = columnize(convex_hull(tt))
    -- and close the loop:
    cx &= cx[1]
    cy &= cy[1]
    return {{x,y,CD_RED},
            {cx,cy,CD_GREEN,"HOLLOW_CIRCLE","MARKLINE"}}
end function

function key_cb(Ihandle /*ih*/, atom c)
    if c=K_ESC then return IUP_CLOSE end if -- (standard practice for me)
    if c=K_F5 then return IUP_DEFAULT end if -- (let browser reload work)
    if c=K_LEFT then t = iff(t=1?length(tests):t-1) end if
    if c=K_RIGHT then t = iff(t=length(tests)?1:t+1) end if
    IupUpdate(graph)
    return IUP_CONTINUE
end function

IupOpen()
graph = IupGraph(get_data,"RASTERSIZE=640x480,MODE=MARK")
dlg = IupDialog(graph,`TITLE="Convex Hull",MINSIZE=270x430`)
IupSetAttributes(graph,"XTICK=2,XMIN=-12,XMAX=20,XMARGIN=25,YTICK=2,YMIN=-12,YMAX=20")
IupSetInt(graph,"GRIDCOLOR",CD_LIGHT_GREY)
IupShow(dlg)
IupSetCallback(dlg, "K_ANY", Icallback("key_cb"))
IupSetAttribute(graph,`RASTERSIZE`,NULL)
if platform()!=JS then
    IupMainLoop()
    IupClose()
end if
Output:
For test set: {{16,3}, {12,17}, {0,6}, {-4,-6}, {16,6}, {16,-7}, {16,-3}, {17,-4}, {5,19}, {19,-8}, {3,16},
              {12,13}, {3,-4}, {17,5}, {-3,15}, {-3,-9}, {0,11}, {-9,-3}, {-4,-2}, {12,10}}
Convex Hull is: {{-9,-3}, {-3,-9}, {19,-8}, {17,5}, {12,17}, {5,19}, {-3,15}}
For test set: {{16,3}, {12,17}, {0,6}, {-4,-6}, {16,6}, {16,-7}, {16,-3}, {17,-4}, {5,19}, {19,-8}, {3,16},
              {12,13}, {3,-4}, {17,5}, {-3,15}, {-3,-9}, {0,11}, {-9,-3}, {-4,-2}, {12,10}, {14,-9}, {1,-9}}
Convex Hull is: {{-9,-3}, {-3,-9}, {14,-9}, {19,-8}, {17,5}, {12,17}, {5,19}, {-3,15}}
For test set: {{0,0}, {5,5}, {5,0}, {0,5}}
Convex Hull is: {{0,0}, {5,0}, {5,5}, {0,5}}
For test set: {{0,0}, {0,1}, {0,2}, {1,0}, {1,1}, {1,2}, {2,0}, {2,1}, {2,2}}
Convex Hull is: {{0,0}, {2,0}, {2,2}, {0,2}}
For test set: {{-8,-8}, {-8,4}, {-8,16}, {4,-8}, {4,4}, {4,16}, {16,-8}, {16,4}, {16,16}}
Convex Hull is: {{-8,-8}, {16,-8}, {16,16}, {-8,16}}

Python[edit]

Library: Shapely

An approach that uses the shapely library:

from __future__ import print_function
from shapely.geometry import MultiPoint

if __name__=="__main__":
	pts = MultiPoint([(16,3), (12,17), (0,6), (-4,-6), (16,6), (16,-7), (16,-3), (17,-4), (5,19), (19,-8), (3,16), (12,13), (3,-4), (17,5), (-3,15), (-3,-9), (0,11), (-9,-3), (-4,-2), (12,10)])
	print (pts.convex_hull)
Output:
POLYGON ((-3 -9, -9 -3, -3 15, 5 19, 12 17, 17 5, 19 -8, -3 -9))

Racket[edit]

Also an implementation of https://en.wikibooks.org/wiki/Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain (therefore kinda
Translation of: Go
#lang typed/racket
(define-type Point (Pair Real Real))
(define-type Points (Listof Point))

(:  (Point Point Point -> Real))
(define/match ( o a b)
  [((cons o.x o.y) (cons a.x a.y) (cons b.x b.y))
   (- (* (- a.x o.x) (- b.y o.y)) (* (- a.y o.y) (- b.x o.x)))])

(: Point<? (Point Point -> Boolean))
(define (Point<? a b)
  (cond [(< (car a) (car b)) #t] [(> (car a) (car b)) #f] [else (< (cdr a) (cdr b))]))

;; Input: a list P of points in the plane.
(define (convex-hull [P:unsorted : Points])
  ;; Sort the points of P by x-coordinate (in case of a tie, sort by y-coordinate).
  (define P (sort P:unsorted Point<?))
  ;; for i = 1, 2, ..., n:
  ;;     while L contains at least two points and the sequence of last two points
  ;;             of L and the point P[i] does not make a counter-clockwise turn:
  ;;        remove the last point from L
  ;;     append P[i] to L
  ;; TB: U is identical with (reverse P) 
  (define (upper/lower-hull [P : Points])
    (reverse
     (for/fold ((rev : Points null))
       ((P.i (in-list P)))
       (let u/l : Points ((rev rev))
         (match rev
           [(list p-2 p-1 ps ...) #:when (not (positive? ( p-2 P.i p-1))) (u/l (list* p-1 ps))]
           [(list ps ...) (cons P.i ps)])))))

  ;; Initialize U and L as empty lists.
  ;; The lists will hold the vertices of upper and lower hulls respectively.
  (let ((U (upper/lower-hull (reverse P)))
        (L (upper/lower-hull P)))
    ;; Remove the last point of each list (it's the same as the first point of the other list).
    ;; Concatenate L and U to obtain the convex hull of P.
    (append (drop-right L 1) (drop-right U 1)))) ; Points in the result will be listed in CCW order.)

(module+ test
  (require typed/rackunit)
  (check-equal?
   (convex-hull
    (list '(16 . 3) '(12 . 17) '(0 . 6) '(-4 . -6) '(16 . 6) '(16 . -7) '(16 . -3) '(17 . -4)
          '(5 . 19) '(19 . -8) '(3 . 16) '(12 . 13) '(3 . -4) '(17 . 5) '(-3 . 15) '(-3 . -9)
          '(0 . 11) '(-9 . -3) '(-4 . -2) '(12 . 10)))
   (list '(-9 . -3) '(-3 . -9) '(19 . -8) '(17 . 5) '(12 . 17) '(5 . 19) '(-3 . 15))))
Output:

silence implies tests pass (and output is as expected)

Raku[edit]

(formerly Perl 6)

Works with: Rakudo version 2017.05
Translation of: zkl

Modified the angle sort method as the original could fail if there were multiple points on the same y coordinate as the starting point. Sorts on tangent rather than triangle area. Inexpensive since it still doesn't do any trigonometric math, just calculates the ratio of opposite over adjacent.

class Point {
    has Real $.x is rw;
    has Real $.y is rw;
    method gist { [~] '(', self.x,', ', self.y, ')' };
}

sub ccw (Point $a, Point $b, Point $c) {
    ($b.x - $a.x)*($c.y - $a.y) - ($b.y - $a.y)*($c.x - $a.x);
}

sub tangent (Point $a, Point $b) {
    my $opp = $b.x - $a.x;
    my $adj = $b.y - $a.y;
    $adj != 0 ?? $opp / $adj !! Inf
}

sub graham-scan (**@coords) {
    # sort points by y, secondary sort on x
    my @sp = @coords.map( { Point.new( :x($_[0]), :y($_[1]) ) } )
                    .sort: {.y, .x};

    # need at least 3 points to make a hull
    return @sp if +@sp < 3;

    # first point on hull is minimum y point
    my @h  = @sp.shift;

    # re-sort the points by angle, secondary on x
    @sp    = @sp.map( { $++ => [tangent(@h[0], $_), $_.x] } )
                .sort( {-$_.value[0], $_.value[1] } )
                .map: { @sp[$_.key] };

    # first point of re-sorted list is guaranteed to be on hull
    @h.push:  @sp.shift;

    # check through the remaining list making sure that
    # there is always a positive angle
    for @sp -> $point {
        if ccw( |@h.tail(2), $point ) > 0 {
            @h.push: $point;
        } else {
            @h.pop;
            @h.push: $point and next if +@h < 2;
            redo;
        }
    }
    @h
}

my @hull = graham-scan(
    (16, 3), (12,17), ( 0, 6), (-4,-6), (16, 6), (16,-7), (16,-3),
    (17,-4), ( 5,19), (19,-8), ( 3,16), (12,13), ( 3,-4), (17, 5),
    (-3,15), (-3,-9), ( 0,11), (-9,-3), (-4,-2), (12,10)
  );

say "Convex Hull ({+@hull} points): ", @hull;

@hull = graham-scan(
    (16, 3), (12,17), ( 0, 6), (-4,-6), (16, 6), (16,-7), (16,-3),
    (17,-4), ( 5,19), (19,-8), ( 3,16), (12,13), ( 3,-4), (17, 5),
    (-3,15), (-3,-9), ( 0,11), (-9,-3), (-4,-2), (12,10), (20,-9), (1,-9)
  );

say "Convex Hull ({+@hull} points): ", @hull;
Output:
Convex Hull (7 points): [(-3, -9) (19, -8) (17, 5) (12, 17) (5, 19) (-3, 15) (-9, -3)]
Convex Hull (7 points): [(-3, -9) (20, -9) (17, 5) (12, 17) (5, 19) (-3, 15) (-9, -3)]

RATFOR[edit]

Translation of: Fortran
Works with: ratfor77 version public domain 1.0
Works with: gfortran version 11.3.0
Works with: f2c version 20100827


#
# Convex hulls by Andrew's monotone chain algorithm.
#
# For a description of the algorithm, see
# https://en.wikibooks.org/w/index.php?title=Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain&stableid=40169
#
# As in the Fortran 2018 version upon which this code is based, I
# shall use the built-in "complex" type to represent "points" in the
# plane. This is merely for convenience, rather than to express a
# mathematical equivalence.
#

define(point, complex)

function x (u)

  # Return the x-coordinate of "point" u.

  implicit none

  point u
  real x

  x = real (u)
end

function y (u)

  # Return the y-coordinate of "point" u.

  implicit none

  point u
  real y

  y = aimag (u)
end

function cross (u, v)

  # Return, as a signed scalar, the cross product of "points" u and v
  # (regarded as "vectors" or multivectors).

  implicit none

  point u, v
  real cross, x, y

  cross = (x (u) * y (v)) - (y (u) * x (v))
end

subroutine sortpt (numpt, pt)

  # Sort "points" in ascending order, first by the x-coordinates and
  # then by the y-coordinates. Any decent sort algorithm will suffice;
  # for the sake of interest, here is the Shell sort of
  # https://en.wikipedia.org/w/index.php?title=Shellsort&oldid=1084744510

  implicit none

  integer numpt
  point pt(0:*)

  real x, y
  integer i, j, k, gap, offset
  point temp
  logical done

  integer gaps(1:8)
  data gaps / 701, 301, 132, 57, 23, 10, 4, 1 /

  for (k = 1; k <= 8; k = k + 1)
    {
      gap = gaps(k)
      for (offset = 0; offset <= gap - 1; offset = offset + 1)
        for (i = offset; i <= numpt - 1; i = i + gap)
          {
            temp = pt(i)
            j = i
            done = .false.
            while (!done)
              {
                if (j < gap)
                  done = .true.
                else if (x (pt(j - gap)) < x (temp))
                  done = .true.
                else if (x (pt(j - gap)) == x (temp)      _
                          && (y (pt(j - gap)) <= y (temp)))
                  done = .true.
                else
                  {
                    pt(j) = pt(j - gap)
                    j = j - gap
                  }
              }
            pt(j) = temp
          }
    }
end

subroutine deltrd (n, pt)

  # Delete trailing neighbor duplicates.

  implicit none

  integer n
  point pt(0:*)

  integer i
  logical done

  i = n - 1
  done = .false.
  while (!done)
    {
      if (i == 0)
        {
          n = 1
          done = .true.
        }
      else if (pt(i - 1) != pt(i))
        {
          n = i + 1
          done = .true.
        }
      else
        i = i - 1
    }
end

subroutine delntd (n, pt)

  # Delete non-trailing neighbor duplicates.

  implicit none

  integer n
  point pt(0:*)

  integer i, j, numdel
  logical done

  i = 0
  while (i < n - 1)
    {
      j = i + 1
      done = .false.
      while (!done)
        {
          if (j == n)
            done = .true.
          else if (pt(j) != pt(i))
            done = .true.
          else
            j = j + 1
        }
      if (j != i + 1)
        {
          numdel = j - i - 1
          while (j != n)
            {
              pt(j - numdel) = pt(j)
              j = j + 1
            }
          n = n - numdel
        }
      i = i + 1
    }
end

subroutine deldup (n, pt)

  # Delete neighbor duplicates.

  implicit none

  integer n
  point pt(0:*)

  call deltrd (n, pt)
  call delntd (n, pt)
end

subroutine cxlhul (n, pt, hullsz, hull)

  # Construct the lower hull.

  implicit none

  integer n                     # Number of points.
  point pt(0:*)
  integer hullsz                # Output.
  point hull(0:*)               # Output.

  real cross
  integer i, j
  logical done

  j = 1
  hull(0) = pt(0)
  hull(1) = pt(1)
  for (i = 2; i <= n - 1; i = i + 1)
    {
      done = .false.
      while (!done)
        {
          if (j == 0)
            {
              j = j + 1
              hull(j) = pt(i)
              done = .true.
            }
          else if (0.0 < cross (hull(j) - hull(j - 1), _
                                pt(i) - hull(j - 1)))
            {
              j = j + 1
              hull(j) = pt(i)
              done = .true.
            }
          else
            j = j - 1
        }
    }
  hullsz = j + 1
end

subroutine cxuhul (n, pt, hullsz, hull)

  # Construct the upper hull.

  implicit none

  integer n                     # Number of points.
  point pt(0:*)
  integer hullsz                # Output.
  point hull(0:*)               # Output.

  real cross
  integer i, j
  logical done

  j = 1
  hull(0) = pt(n - 1)
  hull(1) = pt(n - 2)
  for (i = n - 3; 0 <= i; i = i - 1)
    {
      done = .false.
      while (!done)
        {
          if (j == 0)
            {
              j = j + 1
              hull(j) = pt(i)
              done = .true.
            }
          else if (0.0 < cross (hull(j) - hull(j - 1), _
                                pt(i) - hull(j - 1)))
            {
              j = j + 1
              hull(j) = pt(i)
              done = .true.
            }
          else
            j = j - 1
        }
    }
  hullsz = j + 1
end

subroutine cxhull (n, pt, hullsz, lhull, uhull)

  # Construct the hull.

  implicit none

  integer n                     # Number of points.
  point pt(0:*)                 # Overwritten with hull.
  integer hullsz                # Output.
  point lhull(0:*)              # Workspace.
  point uhull(0:*)              # Workspace

  integer lhulsz, uhulsz, i

  # A side note: the calls to construct_lower_hull and
  # construct_upper_hull could be done in parallel.
  call cxlhul (n, pt, lhulsz, lhull)
  call cxuhul (n, pt, uhulsz, uhull)

  hullsz = lhulsz + uhulsz - 2

  for (i = 0; i <= lhulsz - 2; i = i + 1)
    pt(i) = lhull(i)
  for (i = 0; i <= uhulsz - 2; i = i + 1)
    pt(lhulsz - 1 + i) = uhull(i)
end

subroutine cvxhul (n, pt, hullsz, lhull, uhull)

  # Find a convex hull.

  implicit none

  integer n            # Number of points.
  point pt(0:*)        # The contents of pt is replaced with the hull.
  integer hullsz       # Output.
  point lhull(0:*)     # Workspace.
  point uhull(0:*)     # Workspace

  integer numpt

  numpt = n

  call sortpt (numpt, pt)
  call deldup (numpt, pt)

  if (numpt == 0)
    hullsz = 0
  else if (numpt <= 2)
    hullsz = numpt
  else
    call cxhull (numpt, pt, hullsz, lhull, uhull)
end

program cvxtsk

  # The Rosetta Code convex hull task.

  implicit none

  integer n, i
  point points(0:100)
  point lhull(0:100)
  point uhull(0:100)
  character*100 fmt

  point exampl(0:19)
  data exampl / (16, 3), (12, 17), (0, 6), (-4, -6), (16, 6),    _
                (16, -7), (16, -3), (17, -4), (5, 19), (19, -8), _
                (3, 16), (12, 13), (3, -4), (17, 5), (-3, 15),   _
                (-3, -9), (0, 11), (-9, -3), (-4, -2), (12, 10) /

  n = 20
  for (i = 0; i <= n - 1; i = i + 1)
    points(i) = exampl(i)
  call cvxhul (n, points, n, lhull, uhull)

  write (fmt, '("(", I20, ''("(", F3.0, 1X, F3.0, ") ")'', ")")') n
  write (*, fmt) (points(i), i = 0, n - 1)
end
Output:
$ ratfor77 convex_hull_task.r > cvxtsk.f && f2c -C cvxtsk.f && cc cvxtsk.c -lf2c && ./a.out
(-9. -3.) (-3. -9.) (19. -8.) (17.  5.) (12. 17.) ( 5. 19.) (-3. 15.)

REXX[edit]

version 1[edit]

/* REXX ---------------------------------------------------------------
* Compute the Convex Hull for a set of points
* Format of the input file:
* (16,3) (12,17) (0,6) (-4,-6) (16,6) (16,-7) (16,-3) (17,-4) (5,19)
* (19,-8) (3,16) (12,13) (3,-4) (17,5) (-3,15) (-3,-9) (0,11) (-9,-3)
* (-4,-2)
*--------------------------------------------------------------------*/
  Signal On Novalue
  Signal On Syntax
Parse Arg fid
If fid='' Then Do
  fid='chullmin.in'                 /* miscellaneous test data       */
  fid='chullx.in'
  fid='chullt.in'
  fid='chulla.in'
  fid='chullxx.in'
  fid='sq.in'
  fid='tri.in'
  fid='line.in'
  fid='point.in'
  fid='chull.in'                    /* data from task description    */
  End
g.0debug=''
g.0oid=fn(fid)'.txt'; 'erase' g.0oid
x.=0
yl.=''
Parse Value '1000 -1000' With g.0xmin g.0xmax
Parse Value '1000 -1000' With g.0ymin g.0ymax
/*---------------------------------------------------------------------
* First read the input and store the points' coordinates
* x.0 contains the number of points, x.i contains the x.coordinate
* yl.x contains the y.coordinate(s) of points (x/y)
*--------------------------------------------------------------------*/
Do while lines(fid)>0
  l=linein(fid)
  Do While l<>''
    Parse Var l '(' x ',' y ')' l
    Call store x,y
    End
  End
Call lineout fid
Do i=1 To x.0                       /* loop over points              */
  x=x.i
  yl.x=sortv(yl.x)                  /* sort y-coordinates            */
  End
Call sho

/*---------------------------------------------------------------------
* Now we look for special border points:
* lefthigh and leftlow: leftmost points with higheste and lowest y
* ritehigh and ritelow: rightmost points with higheste and lowest y
* yl.x contains the y.coordinate(s) of points (x/y)
*--------------------------------------------------------------------*/
leftlow=0
lefthigh=0
Do i=1 To x.0
  x=x.i
  If maxv(yl.x)=g.0ymax Then Do
    If lefthigh=0 Then lefthigh=x'/'g.0ymax
    ritehigh=x'/'g.0ymax
    End
  If minv(yl.x)=g.0ymin Then Do
    ritelow=x'/'g.0ymin
    If leftlow=0 Then leftlow=x'/'g.0ymin
    End
  End
Call o 'lefthigh='lefthigh
Call o 'ritehigh='ritehigh
Call o 'ritelow ='ritelow
Call o 'leftlow ='leftlow
/*---------------------------------------------------------------------
* Now we look for special border points:
* leftmost_n and leftmost_s: points with lowest x and highest/lowest y
* ritemost_n and ritemost_s: points with largest x and highest/lowest y
* n and s stand foNorth and South, respectively
*--------------------------------------------------------------------*/
x=g.0xmi; leftmost_n=x'/'maxv(yl.x)
x=g.0xmi; leftmost_s=x'/'minv(yl.x)
x=g.0xma; ritemost_n=x'/'maxv(yl.x)
x=g.0xma; ritemost_s=x'/'minv(yl.x)

/*---------------------------------------------------------------------
* Now we compute the paths from ritehigh to ritelow (n_end)
* and leftlow to lefthigh (s_end), respectively
*--------------------------------------------------------------------*/
x=g.0xma
n_end=''
Do i=words(yl.x) To 1 By -1
  n_end=n_end x'/'word(yl.x,i)
  End
Call o 'n_end='n_end
x=g.0xmi
s_end=''
Do i=1 To words(yl.x)
  s_end=s_end x'/'word(yl.x,i)
  End
Call o 's_end='s_end

n_high=''
s_low=''
/*---------------------------------------------------------------------
* Now we compute the upper part of the convex hull (nhull)
*--------------------------------------------------------------------*/
Call o 'leftmost_n='leftmost_n
Call o 'lefthigh  ='lefthigh
nhull=leftmost_n
res=mk_nhull(leftmost_n,lefthigh);
nhull=nhull res
Call o 'A nhull='nhull
Do While res<>lefthigh
  res=mk_nhull(res,lefthigh); nhull=nhull res
  Call o 'B nhull='nhull
  End
res=mk_nhull(lefthigh,ritemost_n); nhull=nhull res
Call o 'C nhull='nhull
Do While res<>ritemost_n
  res=mk_nhull(res,ritemost_n); nhull=nhull res
  Call o 'D nhull='nhull
  End

nhull=nhull n_end                /* attach the right vertical border */

/*---------------------------------------------------------------------
* Now we compute the lower part of the convex hull (shull)
*--------------------------------------------------------------------*/
res=mk_shull(ritemost_s,ritelow);
shull=ritemost_s res
Call o 'A shull='shull
Do While res<>ritelow
  res=mk_shull(res,ritelow)
  shull=shull res
  Call o 'B shull='shull
  End
res=mk_shull(ritelow,leftmost_s)
shull=shull res
Call o 'C shull='shull
Do While res<>leftmost_s
  res=mk_shull(res,leftmost_s);
  shull=shull res
  Call o 'D shull='shull
  End

shull=shull s_end

chull=nhull shull                 /* concatenate upper and lower part */
                                  /* eliminate duplicates             */
                                  /* too lazy to take care before :-) */
Parse Var chull chullx chull
have.=0
have.chullx=1
Do i=1 By 1 While chull>''
  Parse Var chull xy chull
  If have.xy=0 Then Do
    chullx=chullx xy
    have.xy=1
    End
  End
                                   /* show the result                */
Say 'Points of convex hull in clockwise order:'
Say    chullx
/**********************************************************************
* steps that were necessary in previous attempts
/*---------------------------------------------------------------------
* Final polish: Insert points that are not yet in chullx but should be
* First on the upper hull going from left to right
*--------------------------------------------------------------------*/
i=1
Do While i<words(chullx)
  xya=word(chullx,i)  ; Parse Var xya xa '/' ya
  If xa=g.0xmax Then Leave
  xyb=word(chullx,i+1); Parse Var xyb xb '/' yb
  Do j=1 To x.0
    If x.j>xa Then Do
      If x.j<xb Then Do
        xx=x.j
        parse Value kdx(xya,xyb) With k d x
        If (k*xx+d)=maxv(yl.xx) Then Do
          chullx=subword(chullx,1,i) xx'/'maxv(yl.xx),
                                                    subword(chullx,i+1)
          i=i+1
          End
        End
      End
    Else
      i=i+1
    End
  End

Say    chullx

/*---------------------------------------------------------------------
* Final polish: Insert points that are not yet in chullx but should be
* Then on the lower hull going from right to left
*--------------------------------------------------------------------*/
i=wordpos(ritemost_s,chullx)
Do While i<words(chullx)
  xya=word(chullx,i)  ; Parse Var xya xa '/' ya
  If xa=g.0xmin Then Leave
  xyb=word(chullx,i+1); Parse Var xyb xb '/' yb
  Do j=x.0 To 1 By -1
    If x.j<xa Then Do
      If x.j>xb Then Do
        xx=x.j
        parse Value kdx(xya,xyb) With k d x
        If (k*xx+d)=minv(yl.xx) Then Do
          chullx=subword(chullx,1,i) xx'/'minv(yl.xx),
                                                    subword(chullx,i+1)
          i=i+1
          End
        End
      End
    Else
      i=i+1
    End
  End
Say chullx
**********************************************************************/
Call lineout g.0oid

Exit

store: Procedure Expose x. yl. g.
/*---------------------------------------------------------------------
* arrange the points in ascending order of x (in x.) and,
* for each x in ascending order of y (in yl.x)
* g.0xmin is the smallest x-value, etc.
* g.0xmi  is the x-coordinate
* g.0ymin is the smallest y-value, etc.
* g.0ymi  is the x-coordinate of such a point
*--------------------------------------------------------------------*/
  Parse Arg x,y
  Call o 'store' x y
  If x<g.0xmin Then Do; g.0xmin=x; g.0xmi=x; End
  If x>g.0xmax Then Do; g.0xmax=x; g.0xma=x; End
  If y<g.0ymin Then Do; g.0ymin=y; g.0ymi=x; End
  If y>g.0ymax Then Do; g.0ymax=y; g.0yma=x; End
  Do i=1 To x.0
    Select
      When x.i>x Then
        Leave
      When x.i=x Then Do
        yl.x=yl.x y
        Return
        End
      Otherwise
        Nop
      End
    End
  Do j=x.0 To i By -1
    ja=j+1
    x.ja=x.j
    End
  x.i=x
  yl.x=y
  x.0=x.0+1
  Return

sho: Procedure Expose x. yl. g.
  Do i=1 To x.0
    x=x.i
    say  format(i,2) 'x='format(x,3) 'yl='yl.x
    End
  Say ''
  Return

maxv: Procedure Expose g.
  Call trace 'O'
  Parse Arg l
  res=-1000
  Do While l<>''
    Parse Var l v l
    If v>res Then res=v
    End
  Return res

minv: Procedure Expose g.
  Call trace 'O'
  Parse Arg l
  res=1000
  Do While l<>''
    Parse Var l v l
    If v<res Then res=v
    End
  Return res

sortv: Procedure Expose g.
  Call trace 'O'
  Parse Arg l
  res=''
  Do Until l=''
    v=minv(l)
    res=res v
    l=remove(v,l)
    End
  Return space(res)

lastword: return word(arg(1),words(arg(1)))

kdx: Procedure  Expose xy. g.
/*---------------------------------------------------------------------
* Compute slope and y-displacement of a straight line
* that is defined by two points:  y=k*x+d
* Specialty; k='*' x=xa if xb=xa
*--------------------------------------------------------------------*/
  Call trace 'O'
  Parse Arg xya,xyb
  Parse Var xya xa '/' ya
  Parse Var xyb xb '/' yb
  If xa=xb Then
    Parse Value '*' '-' xa With k d x
  Else Do
    k=(yb-ya)/(xb-xa)
    d=yb-k*xb
    x='*'
    End
  Return k d x

remove:
/*---------------------------------------------------------------------
* Remove a specified element (e) from a given string (s)
*--------------------------------------------------------------------*/
  Parse Arg e,s
  Parse Var s sa (e) sb
  Return space(sa sb)

o: Procedure Expose g.
/*---------------------------------------------------------------------
* Write a line to the debug file
*--------------------------------------------------------------------*/
  If arg(2)=1 Then say arg(1)
  Return lineout(g.0oid,arg(1))

is_ok: Procedure Expose x. yl. g. sigl
/*---------------------------------------------------------------------
* Test if a given point (b) is above/on/or below a straight line
* defined by two points (a and c)
*--------------------------------------------------------------------*/
  Parse Arg a,b,c,op
  Call o    'is_ok' a b c op
  Parse Value kdx(a,c) With k d x
  Parse Var b x'/'y
  If op='U' Then y=maxv(yl.x)
            Else y=minv(yl.x)
  Call o    y x (k*x+d)
  If (abs(y-(k*x+d))<1.e-8) Then Return 0
  If op='U' Then res=(y<=(k*x+d))
            Else res=(y>=(k*x+d))
  Return res

mk_nhull: Procedure Expose x. yl. g.
/*---------------------------------------------------------------------
* Compute the upper (north) hull between two points (xya and xyb)
* Move x from xyb back to xya until all points within the current
* range (x and xyb) are BELOW the straight line defined xya and x
* Then make x the new starting point
*--------------------------------------------------------------------*/
  Parse Arg xya,xyb
  Call o 'mk_nhull' xya xyb
  If xya=xyb Then Return xya
  Parse Var xya xa '/' ya
  Parse Var xyb xb '/' yb
  iu=0
  iv=0
  Do xi=1 To x.0
    if x.xi>=xa & iu=0 Then iu=xi
    if x.xi<=xb Then iv=xi
    If x.xi>xb Then Leave
    End
  Call o    iu iv
  xu=x.iu
  xyu=xu'/'maxv(yl.xu)
  Do h=iv To iu+1 By -1 Until good
    Call o 'iv='iv,g.0debug
    Call o ' h='h,g.0debug
    xh=x.h
    xyh=xh'/'maxv(yl.xh)
    Call o    'Testing' xyu xyh,g.0debug
    good=1
    Do hh=h-1 To iu+1 By -1 While good
      xhh=x.hh
      xyhh=xhh'/'maxv(yl.xhh)
      Call o 'iu hh iv=' iu hh h,g.0debug
      If is_ok(xyu,xyhh,xyh,'U') Then Do
        Call o    xyhh 'is under' xyu xyh,g.0debug
        Nop
        End
      Else Do
        good=0
        Call o    xyhh 'is above' xyu xyh '-' xyh 'ist nicht gut'
        End
      End
    End
  Call o xyh 'is the one'

  Return xyh

p: Return
Say arg(1)
Pull  .
Return

mk_shull: Procedure Expose x. yl. g.
/*---------------------------------------------------------------------
* Compute the lower (south) hull between two points (xya and xyb)
* Move x from xyb back to xya until all points within the current
* range (x and xyb) are ABOVE the straight line defined xya and x
* Then make x the new starting point
*-----
---------------------------------------------------------------*/
  Parse Arg xya,xyb
  Call o 'mk_shull' xya xyb
  If xya=xyb Then Return xya
  Parse Var xya xa '/' ya
  Parse Var xyb xb '/' yb
  iu=0
  iv=0
  Do xi=x.0 To 1 By -1
    if x.xi<=xa & iu=0 Then iu=xi
    if x.xi>=xb Then iv=xi
    If x.xi<xb Then Leave
    End
  Call o iu iv '_' x.iu x.iv
  Call o 'mk_shull iv iu' iv iu
  xu=x.iu
  xyu=xu'/'minv(yl.xu)
  good=0
  Do h=iv To iu-1 Until good
    xh=x.h
    xyh=xh'/'minv(yl.xh)
    Call o    'Testing' xyu xyh   h iu
    good=1
    Do hh=h+1 To iu-1 While good
      Call o 'iu hh h=' iu hh h
      xhh=x.hh
      xyhh=xhh'/'minv(yl.xhh)
      If is_ok(xyu,xyhh,xyh,'O') Then Do
        Call o xyhh 'is above' xyu xyh
        Nop
        End
      Else Do
        Call o xyhh 'is under' xyu xyh '-' xyh 'ist nicht gut'
        good=0
        End
      End
    End
  Call o xyh 'is the one'
  Return xyh

Novalue:
  Say 'Novalue raised in line' sigl
  Say sourceline(sigl)
  Say 'Variable' condition('D')
  Signal lookaround

Syntax:
  Say 'Syntax raised in line' sigl
  Say sourceline(sigl)
  Say 'rc='rc '('errortext(rc)')'

halt:
lookaround:
  Say 'You can look around now.'
  Trace ?R
  Nop
  Exit 12
Output:
 1 x= -9 yl=-3
 2 x= -4 yl=-6 -2
 3 x= -3 yl=-9 15
 4 x=  0 yl=6 11
 5 x=  3 yl=-4 16
 6 x=  5 yl=19
 7 x= 12 yl=13 17
 8 x= 16 yl=-7 -3 3 6
 9 x= 17 yl=-4 5
10 x= 19 yl=-8

Points of convex hull in clockwise order:
-9/-3 -3/15 5/19 12/17 17/5 19/-8 -3/-9

version 2[edit]

After learning from https://www.youtube.com/watch?v=wRTGDig3jx8

/* REXX ---------------------------------------------------------------
* Compute the Convex Hull for a set of points
* Format of the input file:
* (16,3) (12,17) (0,6) (-4,-6) (16,6) (16,-7) (16,-3) (17,-4) (5,19)
* (19,-8) (3,16) (12,13) (3,-4) (17,5) (-3,15) (-3,-9) (0,11) (-9,-3)
* (-4,-2)
* Alternate (better) method using slopes
* 1) Compute path from lowest/leftmost to leftmost/lowest
* 2) Compute leftmost vertical border
* 3) Compute path from rightmost/highest to highest/rightmost
* 4) Compute path from highest/rightmost to rightmost/highest
* 5) Compute rightmost vertical border
* 6) Compute path from rightmost/lowest to lowest_leftmost point
*--------------------------------------------------------------------*/
Parse Arg fid
If fid='' Then Do
  fid='line.in'
  fid='point.in'
  fid='chullmin.in'                 /* miscellaneous test data       */
  fid='chullxx.in'
  fid='chullx.in'
  fid='chullt.in'
  fid='chulla.in'
  fid='sq.in'
  fid='tri.in'
  fid='z.in'
  fid='chull.in'                    /* data from task description    */
  End
g.0debug=''
g.0oid=fn(fid)'.txt'; 'erase' g.0oid
x.=0
yl.=''
Parse Value '1000 -1000' With g.0xmin g.0xmax
Parse Value '1000 -1000' With g.0ymin g.0ymax
/*---------------------------------------------------------------------
* First read the input and store the points' coordinates
* x.0 contains the number of points, x.i contains the x.coordinate
* yl.x contains the y.coordinate(s) of points (x/y)
*--------------------------------------------------------------------*/
Do while lines(fid)>0
  l=linein(fid)
  Do While l<>''
    Parse Var l '(' x ',' y ')' l
    Call store x,y
    End
  End
Call lineout fid
g.0xlist=''
Do i=1 To x.0                       /* loop over points              */
  x=x.i
  g.0xlist=g.0xlist x
  yl.x=sortv(yl.x)                  /* sort y-coordinates            */
  End
Call sho
If x.0<3 Then Do
  Say 'We need at least three points!'
  Exit
  End
Call o 'g.0xmin='g.0xmin
Call o 'g.0xmi ='g.0xmi
Call o 'g.0ymin='g.0ymin
Call o 'g.0ymi ='g.0ymi

Do i=1 To x.0
  x=x.i
  If minv(yl.x)=g.0ymin Then Leave
  End
lowest_leftmost=i

highest_rightmost=0
Do i=1 To x.0
  x=x.i
  If maxv(yl.x)=g.0ymax Then
    highest_rightmost=i
  If maxv(yl.x)<g.0ymax Then
    If highest_rightmost>0 Then
      Leave
  End
Call o 'lowest_leftmost='lowest_leftmost
Call o 'highest_rightmost  ='highest_rightmost

x=x.lowest_leftmost
Call o 'We start at' from x'/'minv(yl.x)
path=x'/'minv(yl.x)
/*---------------------------------------------------------------------
* 1) Compute path from lowest/leftmost to leftmost/lowest
*--------------------------------------------------------------------*/
Call min_path lowest_leftmost,1
/*---------------------------------------------------------------------
* 2) Compute leftmost vertical border
*--------------------------------------------------------------------*/
Do i=2 To words(yl.x)
  path=path x'/'word(yl.x,i)
  End
cxy=x'/'maxv(yl.x)
/*---------------------------------------------------------------------
* 3) Compute path from rightmost/highest to highest/rightmost
*--------------------------------------------------------------------*/
Call max_path ci,highest_rightmost
/*---------------------------------------------------------------------
* 4) Compute path from highest/rightmost to rightmost/highest
*--------------------------------------------------------------------*/
Call max_path ci,x.0
/*---------------------------------------------------------------------
* 5) Compute rightmost vertical border
*--------------------------------------------------------------------*/
Do i=words(yl.x)-1 To 1 By -1
  cxy=x'/'word(yl.x,i)
  path=path cxy
  End
/*---------------------------------------------------------------------
* 6) Compute path from rightmost/lowest to lowest_leftmost
*--------------------------------------------------------------------*/
Call min_path ci,lowest_leftmost

Parse Var path pathx path
have.=0
Do i=1 By 1 While path>''
  Parse Var path xy path
  If have.xy=0 Then Do
    pathx=pathx xy
    have.xy=1
    End
  End
Say 'Points of convex hull in clockwise order:'
Say pathx
Call lineout g.0oid
Exit

min_path:
  Parse Arg from,tgt
  ci=from
  cxy=x.ci
  Do Until ci=tgt
    kmax=-1000
    Do i=ci-1 To 1 By sign(tgt-from)
      x=x.i
      k=k(cxy'/'minv(yl.cxy),x'/'minv(yl.x))
      If k>kmax Then Do
        kmax=k
        ii=i
        End
      End
    ci=ii
    cxy=x.ii
    path=path cxy'/'minv(yl.cxy)
    End
  Return

max_path:
  Parse Arg from,tgt
  Do Until ci=tgt
    kmax=-1000
    Do i=ci+1 To tgt
      x=x.i
      k=k(cxy,x'/'maxv(yl.x))
      If k>kmax Then Do
        kmax=k
        ii=i
        End
      End
    x=x.ii
    cxy=x'/'maxv(yl.x)
    path=path cxy
    ci=ii
    End
  Return

store: Procedure Expose x. yl. g.
/*---------------------------------------------------------------------
* arrange the points in ascending order of x (in x.) and,
* for each x in ascending order of y (in yl.x)
* g.0xmin is the smallest x-value, etc.
* g.0xmi  is the x-coordinate
* g.0ymin is the smallest y-value, etc.
* g.0ymi  is the x-coordinate of such a point
*--------------------------------------------------------------------*/
  Parse Arg x,y
  Call o 'store' x y
  If x<g.0xmin Then Do; g.0xmin=x; g.0xmi=x; End
  If x>g.0xmax Then Do; g.0xmax=x; g.0xma=x; End
  If y<g.0ymin Then Do; g.0ymin=y; g.0ymi=x; End
  If y>g.0ymax Then Do; g.0ymax=y; g.0yma=x; End
  Do i=1 To x.0
    Select
      When x.i>x Then
        Leave
      When x.i=x Then Do
        yl.x=yl.x y
        Return
        End
      Otherwise
        Nop
      End
    End
  Do j=x.0 To i By -1
    ja=j+1
    x.ja=x.j
    End
  x.i=x
  yl.x=y
  x.0=x.0+1
  Return

sho: Procedure Expose x. yl. g.
  Do i=1 To x.0
    x=x.i
    say  format(i,2) 'x='format(x,3) 'yl='yl.x
    End
  Say ''
  Return

maxv: Procedure Expose g.
  Call trace 'O'
  Parse Arg l
  res=-1000
  Do While l<>''
    Parse Var l v l
    If v>res Then res=v
    End
  Return res

minv: Procedure Expose g.
  Call trace 'O'
  Parse Arg l
  res=1000
  Do While l<>''
    Parse Var l v l
    If v<res Then res=v
    End
  Return res

sortv: Procedure Expose g.
  Call trace 'O'
  Parse Arg l
  res=''
  Do Until l=''
    v=minv(l)
    res=res v
    l=remove(v,l)
    End
  Return space(res)

lastword: return word(arg(1),words(arg(1)))

k: Procedure
/*---------------------------------------------------------------------
* Compute slope of a straight line
* that is defined by two points:  y=k*x+d
* Specialty; k='*' x=xa if xb=xa
*--------------------------------------------------------------------*/
  Call trace 'O'
  Parse Arg xya,xyb
  Parse Var xya xa '/' ya
  Parse Var xyb xb '/' yb
  If xa=xb Then
    k='*'
  Else
    k=(yb-ya)/(xb-xa)
  Return k

remove:
/*---------------------------------------------------------------------
* Remove a specified element (e) from a given string (s)
*--------------------------------------------------------------------*/
  Parse Arg e,s
  Parse Var s sa (e) sb
  Return space(sa sb)

o: Procedure Expose g.
/*---------------------------------------------------------------------
* Write a line to the debug file
*--------------------------------------------------------------------*/
  If arg(2)=1 Then say arg(1)
  Return lineout(g.0oid,arg(1))
Output:
 1 x= -9 yl=-3
 2 x= -4 yl=-6 -2
 3 x= -3 yl=-9 15
 4 x=  0 yl=6 11
 5 x=  3 yl=-4 16
 6 x=  5 yl=19
 7 x= 12 yl=13 17
 8 x= 16 yl=-7 -3 3 6
 9 x= 17 yl=-4 5
10 x= 19 yl=-8

Points of convex hull in clockwise order:
-3/-9 -9/-3 -3/15 5/19 12/17 17/5 19/-8 -3/-9

Ruby[edit]

Translation of: D
class Point
    include Comparable
    attr :x, :y

    def initialize(x, y)
        @x = x
        @y = y
    end

    def <=>(other)
        x <=> other.x
    end

    def to_s
        "(%d, %d)" % [@x, @y]
    end

    def to_str
        to_s()
    end
end

def ccw(a, b, c)
    ((b.x - a.x) * (c.y - a.y)) > ((b.y - a.y) * (c.x - a.x))
end

def convexHull(p)
    if p.length == 0 then
        return []
    end

    p = p.sort
    h = []

    # Lower hull
    p.each { |pt|
        while h.length >= 2 and not ccw(h[-2], h[-1], pt)
            h.pop()
        end
        h << pt
    }

    # upper hull
    t = h.length + 1
    p.reverse.each { |pt|
        while h.length >= t and not ccw(h[-2], h[-1], pt)
            h.pop()
        end
        h << pt
    }

    h.pop()
    h
end

def main
    points = [
        Point.new(16,  3), Point.new(12, 17), Point.new( 0,  6), Point.new(-4, -6), Point.new(16,  6),
        Point.new(16, -7), Point.new(16, -3), Point.new(17, -4), Point.new( 5, 19), Point.new(19, -8),
        Point.new( 3, 16), Point.new(12, 13), Point.new( 3, -4), Point.new(17,  5), Point.new(-3, 15),
        Point.new(-3, -9), Point.new( 0, 11), Point.new(-9, -3), Point.new(-4, -2), Point.new(12, 10)
    ]
    hull = convexHull(points)
    print "Convex Hull: [", hull.join(", "), "]\n"
end

main()
Output:
Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]

Rust[edit]

Calculates convex hull from list of points (f32, f32). This can be executed entirely in the Rust Playground.

#[derive(Debug, Clone)]
struct Point {
    x: f32,
    y: f32
}

fn calculate_convex_hull(points: &Vec<Point>) -> Vec<Point> {
    //There must be at least 3 points
    if points.len() < 3 { return points.clone(); }

    let mut hull = vec![];

    //Find the left most point in the polygon
    let (left_most_idx, _) = points.iter()
        .enumerate()
        .min_by(|lhs, rhs| lhs.1.x.partial_cmp(&rhs.1.x).unwrap())
        .expect("No left most point");

    
    let mut p = left_most_idx;
    let mut q = 0_usize;

    loop {
        //The left most point must be part of the hull
        hull.push(points[p].clone());

        q = (p + 1) % points.len();

        for i in 0..points.len() {
            if orientation(&points[p], &points[i], &points[q]) == 2 {
                q = i;
            }
        }

        p = q;

        //Break from loop once we reach the first point again
        if p == left_most_idx { break; }
    }

    return hull;
}

//Calculate orientation for 3 points
//0 -> Straight line
//1 -> Clockwise
//2 -> Counterclockwise
fn orientation(p: &Point, q: &Point, r: &Point) -> usize {
    let val = (q.y - p.y) * (r.x - q.x) -
        (q.x - p.x) * (r.y - q.y);

    if val == 0. { return 0 };
    if val > 0. { return 1; } else { return 2; }
}

fn main(){
    let points = vec![pt(16,3), pt(12,17), pt(0,6), pt(-4,-6), pt(16,6), pt(16,-7), pt(16,-3), pt(17,-4), pt(5,19), pt(19,-8), pt(3,16), pt(12,13), pt(3,-4), pt(17,5), pt(-3,15), pt(-3,-9), pt(0,11), pt(-9,-3), pt(-4,-2), pt(12,10)];
    let hull = calculate_convex_hull(&points);
    
    hull.iter()
        .for_each(|pt| println!("{:?}", pt));
}

fn pt(x: i32, y: i32) -> Point {
    return Point {x:x as f32, y:y as f32};
}
Output:
Point { x: -9.0, y: -3.0 }
Point { x: -3.0, y: -9.0 }
Point { x: 19.0, y: -8.0 }
Point { x: 17.0, y: 5.0 }
Point { x: 12.0, y: 17.0 }
Point { x: 5.0, y: 19.0 }
Point { x: -3.0, y: 15.0 }

Scala[edit]

Scala Implementation to find Convex hull of given points collection. Functional Paradigm followed

object convex_hull{
    def get_hull(points:List[(Double,Double)], hull:List[(Double,Double)]):List[(Double,Double)] = points match{
        case Nil  =>            join_tail(hull,hull.size -1)
        case head :: tail =>    get_hull(tail,reduce(head::hull))
    }
    def reduce(hull:List[(Double,Double)]):List[(Double,Double)] = hull match{
        case p1::p2::p3::rest => {
            if(check_point(p1,p2,p3))      hull
            else                           reduce(p1::p3::rest)
        }
        case _ =>                          hull
    }
    def check_point(pnt:(Double,Double), p2:(Double,Double),p1:(Double,Double)): Boolean = {
        val (x,y) = (pnt._1,pnt._2)
        val (x1,y1) = (p1._1,p1._2)
        val (x2,y2) = (p2._1,p2._2)
        ((x-x1)*(y2-y1) - (x2-x1)*(y-y1)) <= 0
    }
    def m(p1:(Double,Double), p2:(Double,Double)):Double = {
        if(p2._1 == p1._1 && p1._2>p2._2)       90
        else if(p2._1 == p1._1 && p1._2<p2._2)  -90
        else if(p1._1<p2._1)                    180 - Math.toDegrees(Math.atan(-(p1._2 - p2._2)/(p1._1 - p2._1)))
        else                                    Math.toDegrees(Math.atan((p1._2 - p2._2)/(p1._1 - p2._1)))
    }   
    def join_tail(hull:List[(Double,Double)],len:Int):List[(Double,Double)] = {
        if(m(hull(len),hull(0)) > m(hull(len-1),hull(0)))   join_tail(hull.slice(0,len),len-1)
        else                                                hull    
    }
    def main(args:Array[String]){
        val points = List[(Double,Double)]((16,3), (12,17), (0,6), (-4,-6), (16,6), (16,-7), (16,-3), (17,-4), (5,19), (19,-8), (3,16), (12,13), (3,-4), (17,5), (-3,15), (-3,-9), (0,11), (-9,-3), (-4,-2), (12,10))
        val sorted_points = points.sortWith(m(_,(0.0,0.0)) < m(_,(0.0,0.0)))
        println(f"Points:\n" + points + f"\n\nConvex Hull :\n" +get_hull(sorted_points,List[(Double,Double)]()))
    }
}
Output:
Points:
List((16.0,3.0), (12.0,17.0), (0.0,6.0), (-4.0,-6.0), (16.0,6.0), (16.0,-7.0), (16.0,-3.0), (17.0,-4.0), (5.0,19.0), (19.0,-8.0), (3.0,16.0), (12.0,13.0), (3.0,-4.0), (17.0,5.0), (-3.0,15.0), (-3.0,-9.0), (0.0,11.0), (-9.0,-3.0), (-4.0,-2.0), (12.0,10.0))

Convex Hull :
List((-3.0,-9.0), (-9.0,-3.0), (-3.0,15.0), (5.0,19.0), (12.0,17.0), (17.0,5.0), (19.0,-8.0))

Scheme[edit]

Works with: Gauche Scheme version 0.9.11-p1
Works with: CHICKEN Scheme version 5.3.0


The program is in R7RS Scheme. For CHICKEN, you need to install the r7rs and srfi-132 eggs.

See also Common Lisp, Standard ML, OCaml, and ATS. These implementations were based closely on the Scheme. The last includes proofs of various constraints and of termination of the recursions.

Interestingly, I also derived a Fortran implementation from this Scheme, which was itself based on Fortran code I wrote over a decade beforehand.


(define-library (convex-hulls)

  (export vector-convex-hull)
  (export list-convex-hull)

  (import (scheme base))
  (import (srfi 132))                   ; Sorting.

  (begin

    ;;
    ;; The implementation is based on Andrew's monotone chain
    ;; algorithm, and is adapted from a Fortran implementation I wrote
    ;; myself in 2011.
    ;;
    ;; For a description of the algorithm, see
    ;; https://en.wikibooks.org/w/index.php?title=Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain&stableid=4016938
    ;;

    (define x@ car)
    (define y@ cadr)

    (define (cross u v)
      ;; Cross product (as a signed scalar).
      (- (* (x@ u) (y@ v)) (* (y@ u) (x@ v))))

    (define (point- u v)
      (list (- (x@ u) (x@ v)) (- (y@ u) (y@ v))))

    (define (sort-points-vector points-vector)
      ;; Ascending sort on x-coordinates, followed by ascending sort
      ;; on y-coordinates.
      (vector-sort (lambda (u v)
                     (or (< (x@ u) (x@ v))
                         (and (= (x@ u) (x@ v))
                              (< (y@ u) (y@ v)))))
                   points-vector))

    (define (construct-lower-hull sorted-points-vector)
      (let* ((pt sorted-points-vector)
             (n (vector-length pt))
             (hull (make-vector n))
             (j 1))
        (vector-set! hull 0 (vector-ref pt 0))
        (vector-set! hull 1 (vector-ref pt 1))
        (do ((i 2 (+ i 1)))
            ((= i n))
          (let inner-loop ()
            (if (or (zero? j)
                    (positive?
                     (cross (point- (vector-ref hull j)
                                    (vector-ref hull (- j 1)))
                            (point- (vector-ref pt i)
                                    (vector-ref hull (- j 1))))))
                (begin
                  (set! j (+ j 1))
                  (vector-set! hull j (vector-ref pt i)))
                (begin
                  (set! j (- j 1))
                  (inner-loop)))))
        (values (+ j 1) hull)))         ; Hull size, hull points.

    (define (construct-upper-hull sorted-points-vector)
      (let* ((pt sorted-points-vector)
             (n (vector-length pt))
             (hull (make-vector n))
             (j 1))
        (vector-set! hull 0 (vector-ref pt (- n 1)))
        (vector-set! hull 1 (vector-ref pt (- n 2)))
        (do ((i (- n 3) (- i 1)))
            ((= i -1))
          (let inner-loop ()
            (if (or (zero? j)
                    (positive?
                     (cross (point- (vector-ref hull j)
                                    (vector-ref hull (- j 1)))
                            (point- (vector-ref pt i)
                                    (vector-ref hull (- j 1))))))
                (begin
                  (set! j (+ j 1))
                  (vector-set! hull j (vector-ref pt i)))
                (begin
                  (set! j (- j 1))
                  (inner-loop)))))
        (values (+ j 1) hull)))         ; Hull size, hull points.

    (define (construct-hull sorted-points-vector)
      ;; Notice that the lower and upper hulls could be constructed in
      ;; parallel.
      (let-values (((lower-hull-size lower-hull)
                    (construct-lower-hull sorted-points-vector))
                   ((upper-hull-size upper-hull)
                    (construct-upper-hull sorted-points-vector)))
        (let* ((hull-size (+ lower-hull-size upper-hull-size -2))
               (hull (make-vector hull-size)))
          (vector-copy! hull 0 lower-hull 0 (- lower-hull-size 1))
          (vector-copy! hull (- lower-hull-size 1) upper-hull
                        0 (- upper-hull-size 1))
          hull)))

    (define (vector-convex-hull points)
      (let* ((points-vector (if (vector? points)
                                points
                                (list->vector points)))
             (sorted-points-vector
              (vector-delete-neighbor-dups
               equal?
               (sort-points-vector points-vector))))
        (if (<= (vector-length sorted-points-vector) 2)
            sorted-points-vector
            (construct-hull sorted-points-vector))))

    (define (list-convex-hull points)
      (vector->list (vector-convex-hull points)))

    )) ;; end of library convex-hulls.

;;
;; A demonstration.
;;

(import (scheme base))
(import (scheme write))
(import (convex-hulls))

(define example-points
  '((16 3) (12 17) (0 6) (-4 -6) (16 6)
    (16 -7) (16 -3) (17 -4) (5 19) (19 -8)
    (3 16) (12 13) (3 -4) (17 5) (-3 15)
    (-3 -9) (0 11) (-9 -3) (-4 -2) (12 10)))

(write (list-convex-hull example-points))
(newline)
Output:

Using Gauche Scheme:

$ gosh convex-hull-task.scm
((-9 -3) (-3 -9) (19 -8) (17 5) (12 17) (5 19) (-3 15))

Compiling to native code with CHICKEN Scheme:

$ csc -O3 -R r7rs convex-hull-task.scm && ./convex-hull-task
((-9 -3) (-3 -9) (19 -8) (17 5) (12 17) (5 19) (-3 15))


A second implementation[edit]

Translation of: Mercury


From the first Scheme implementation, above, I derived an implementation in Standard ML. From that I derived one in Mercury. Now, starting from that Mercury implementation, I can come back and write a Scheme program more concise than the original.

One could pass these changes along to other languages as well.

If you are using CHICKEN Scheme you will need the srfi-1 egg, in addition to those you needed for the first Scheme implementation.


(define-library (convex-hulls)

  (export convex-hull)

  (import (scheme base))
  (import (only (srfi 1) fold))
  (import (only (srfi 1) append!))
  (import (only (srfi 132) list-sort))
  (import (only (srfi 132) list-delete-neighbor-dups))

  (begin

    ;;
    ;; Andrew's monotone chain algorithm for the convex hull in a
    ;; plane.
    ;;
    ;; For a description of the algorithm, see
    ;; https://en.wikibooks.org/w/index.php?title=Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain&stableid=4016938
    ;;

    (define x@ car)
    (define y@ cadr)

    (define (cross u v)
      ;; Cross product (as a signed scalar).
      (- (* (x@ u) (y@ v)) (* (y@ u) (x@ v))))

    (define (point- u v)
      (list (- (x@ u) (x@ v)) (- (y@ u) (y@ v))))

    (define (point=? u v)
      (and (= (x@ u) (x@ v)) (= (y@ u) (y@ v))))

    (define (point<? u v)
      (let ((xu (x@ u))
            (xv (x@ v)))
        (or (< xu xv) (and (= xu xv) (< (y@ u) (y@ v))))))

    (define (convex-hull points-list)
      (let* ((points (list-delete-neighbor-dups
                      point=? (list-sort point<? points-list)))
             (n (length points)))
        (cond
         ((<= n 2) points)
         (else
          (let ((half-hull (make-vector n)))
            (define (cross-test pt j)
              (or (zero? j)
                  (let ((elem-j (vector-ref half-hull j))
                        (elem-j1 (vector-ref half-hull (- j 1))))
                    (positive? (cross (point- elem-j elem-j1)
                                      (point- pt elem-j1))))))
            (define (construct-half-hull points)
              (vector-set! half-hull 0 (car points))
              (vector-set! half-hull 1 (cadr points))
              (fold (lambda (pt j)
                      (let loop ((j j))
                        (if (cross-test pt j)
                            (let ((j1 (+ j 1)))
                              (vector-set! half-hull j1 pt)
                              j1)
                            (loop (- j 1)))))
                    1 (cddr points)))
            (let* ((lower-hull
                    ;; Leave out the last point, which is the same
                    ;; as the first point of the upper hull.
                    (let ((j (construct-half-hull points)))
                      (vector->list half-hull 0 j)))
                   (upper-hull
                    ;; Leave out the last point, which is the same
                    ;; as the first point of the lower hull.
                    (let ((j (construct-half-hull (reverse points))))
                      (vector->list half-hull 0 j))))
              (append! lower-hull upper-hull)))))))

    )) ;; end of library convex-hulls.

;;
;; A demonstration.
;;

(import (scheme base))
(import (scheme write))
(import (convex-hulls))

(define example-points
  '((16 3) (12 17) (0 6) (-4 -6) (16 6)
    (16 -7) (16 -3) (17 -4) (5 19) (19 -8)
    (3 16) (12 13) (3 -4) (17 5) (-3 15)
    (-3 -9) (0 11) (-9 -3) (-4 -2) (12 10)))

(write (convex-hull example-points))
(newline)
Output:
$ gosh convex-hull-task-2.scm
((-9 -3) (-3 -9) (19 -8) (17 5) (12 17) (5 19) (-3 15))


Footnote: My Mercury implementation originally had a "fold" in it, the way this Scheme program does, but that got lost during debugging. (The bug turned out to be elsewhere.) The fold and inner loop became one loop that I think is easier to understand. However, here I wanted to show it could be done with a fold.

Sidef[edit]

Translation of: Raku
class Point(Number x, Number y) {
    method to_s {
        "(#{x}, #{y})"
    }
}

func ccw (Point a, Point b, Point c) {
    (b.x - a.x)*(c.y - a.y) - (b.y - a.y)*(c.x - a.x)
}

func tangent (Point a, Point b) {
    (b.x - a.x) / (b.y - a.y)
}

func graham_scan (*coords) {

    ## sort points by y, secondary sort on x
    var sp = coords.map { |a|
        Point(a...)
    }.sort { |a,b|
        (a.y <=> b.y) ||
        (a.x <=> b.x)
    }

    # need at least 3 points to make a hull
    if (sp.len < 3) {
        return sp
    }

    # first point on hull is minimum y point
    var h = [sp.shift]

    # re-sort the points by angle, secondary on x
    sp = sp.map_kv { |k,v|
        Pair(k, [tangent(h[0], v), v.x])
    }.sort { |a,b|
        (b.value[0] <=> a.value[0]) ||
        (a.value[1] <=> b.value[1])
    }.map { |a|
        sp[a.key]
    }

    # first point of re-sorted list is guaranteed to be on hull
    h << sp.shift

    # check through the remaining list making sure that
    # there is always a positive angle
    sp.each { |point|
        loop {
            if (ccw(h.last(2)..., point) >= 0) {
                h << point
                break
            } else {
                h.pop
            }
        }
    }

    return h
}

var hull = graham_scan(
    [16, 3], [12,17], [ 0, 6], [-4,-6], [16, 6], [16,-7], [16,-3],
    [17,-4], [ 5,19], [19,-8], [ 3,16], [12,13], [ 3,-4], [17, 5],
    [-3,15], [-3,-9], [ 0,11], [-9,-3], [-4,-2], [12,10])

say("Convex Hull (#{hull.len} points): ", hull.join(" "))

hull = graham_scan(
    [16, 3], [12,17], [ 0, 6], [-4,-6], [16, 6], [16,-7], [16,-3],
    [17,-4], [ 5,19], [19,-8], [ 3,16], [12,13], [ 3,-4], [17, 5],
    [-3,15], [-3,-9], [ 0,11], [-9,-3], [-4,-2], [12,10], [14,-9], [1,-9])

say("Convex Hull (#{hull.len} points): ", hull.join(" "))
Output:
Convex Hull (7 points): (-3, -9) (19, -8) (17, 5) (12, 17) (5, 19) (-3, 15) (-9, -3)
Convex Hull (9 points): (-3, -9) (1, -9) (14, -9) (19, -8) (17, 5) (12, 17) (5, 19) (-3, 15) (-9, -3)

Standard ML[edit]

Translation of: Scheme
Translation of: Fortran
Works with: MLton version 20210117
Works with: Poly/ML version 5.9


(*
 * Convex hulls by Andrew's monotone chain algorithm.
 *
 * For a description of the algorithm, see
 * https://en.wikibooks.org/w/index.php?title=Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain&stableid=40169
 *)

(*------------------------------------------------------------------*)
(*
 * Just enough plane geometry for our purpose.
 *)

signature PLANE_POINT =
sig
  type planePoint
  val planePoint : real * real -> planePoint
  val toTuple : planePoint -> real * real
  val x : planePoint -> real
  val y : planePoint -> real
  val == : planePoint * planePoint -> bool

  (* Impose a total order on points, making it one that will work for
     Andrew's monotone chain algorithm. *)
  val order : planePoint * planePoint -> bool

  (* Subtraction is really a vector or multivector operation. *)
  val subtract : planePoint * planePoint -> planePoint

  (* Cross product is really a multivector operation. *)
  val cross : planePoint * planePoint -> real

  val toString : planePoint -> string
end

structure PlanePoint : PLANE_POINT =
struct
type planePoint = real * real

fun planePoint xy = xy
fun toTuple (x, y) = (x, y)
fun x (x, _) = x
fun y (_, y) = y

fun == ((a : real, b : real), (c : real, d : real)) =
    Real.== (a, c) andalso Real.== (b, d)

fun order ((a : real, b : real), (c : real, d : real)) =
    Real.< (a, c) orelse (Real.== (a, c) andalso Real.< (b, d))

fun subtract ((a : real, b : real), (c : real, d : real)) =
    (a - c, b - d)

fun cross ((a : real, b : real), (c : real, d : real)) =
    (a * d) - (b * c)

fun toString (x, y) =
    "(" ^ Real.toString x ^ " " ^ Real.toString y ^ ")"
end

(*------------------------------------------------------------------*)
(*
 * Rather than rely on compiler extensions for sorting, let us write
 * our own.
 *
 * For no special reason, let us use the Shell sort of
 * https://en.wikipedia.org/w/index.php?title=Shellsort&oldid=1084744510
 *)

val ciura_gaps = Array.fromList [701, 301, 132, 57, 23, 10, 4, 1]

fun sort_in_place less arr =
    let
      open Array

      fun span_gap gap =
          let
            fun iloop i =
                if length arr <= i then
                  ()
                else
                  let
                    val temp = sub (arr, i)
                    fun jloop j =
                        if j < gap orelse
                           less (sub (arr, j - gap), temp) then
                          update (arr, j, temp)
                        else
                          (update (arr, j, sub (arr, j - gap));
                           jloop (j - gap))
                  in
                    jloop i;
                    iloop (i + gap)
                  end
          in
            iloop 0
          end
    in
      app span_gap ciura_gaps
    end

(*------------------------------------------------------------------*)
(*
 * To go with our sort routine, we want something akin to
 * array_delete_neighbor_dups of Scheme's SRFI-132.
 *)

fun array_delete_neighbor_dups equal arr =
    let
      open Array

      fun loop i lst =
          (* Cons a list of non-duplicates, going backwards through
             the array so the list will be in forwards order. *)
          if i = 0 then
            sub (arr, i) :: lst
          else if equal (sub (arr, i - 1), sub (arr, i)) then
            loop (i - 1) lst
          else
            loop (i - 1) (sub (arr, i) :: lst)

      val n = length arr
    in
      fromList (if n = 0 then [] else loop (n - 1) [])
    end

(*------------------------------------------------------------------*)
(*
 * The convex hull algorithm.
 *)

fun cross_test (pt_i, hull, j) =
    let
      open PlanePoint
      val hull_j = Array.sub (hull, j)
      val hull_j1 = Array.sub (hull, j - 1)
    in
      0.0 < cross (subtract (hull_j, hull_j1),
                   subtract (pt_i, hull_j1))
    end

fun construct_lower_hull (n, pt) =
    let
      open PlanePoint

      val hull = Array.array (n, planePoint (0.0, 0.0))
      val () = Array.update (hull, 0, Array.sub (pt, 0))
      val () = Array.update (hull, 1, Array.sub (pt, 1))

      fun outer_loop i j =
          if i = n then
            j + 1
          else
            let
              val pt_i = Array.sub (pt, i)

              fun inner_loop j =
                  if j = 0 orelse cross_test (pt_i, hull, j) then
                    (Array.update (hull, j + 1, pt_i);
                     j + 1)
                  else
                    inner_loop (j - 1)
            in
              outer_loop (i + 1) (inner_loop j)
            end

      val hull_size = outer_loop 2 1
    in
      (hull_size, hull)
    end

fun construct_upper_hull (n, pt) =
    let
      open PlanePoint

      val hull = Array.array (n, planePoint (0.0, 0.0))
      val () = Array.update (hull, 0, Array.sub (pt, n - 1))
      val () = Array.update (hull, 1, Array.sub (pt, n - 2))

      fun outer_loop i j =
          if i = ~1 then
            j + 1
          else
            let
              val pt_i = Array.sub (pt, i)

              fun inner_loop j =
                  if j = 0 orelse cross_test (pt_i, hull, j) then
                    (Array.update (hull, j + 1, pt_i);
                     j + 1)
                  else
                    inner_loop (j - 1)
            in
              outer_loop (i - 1) (inner_loop j)
            end

      val hull_size = outer_loop (n - 3) 1
    in
      (hull_size, hull)
    end

fun construct_hull (n, pt) =
    let
      (* Side note: Construction of the lower and upper hulls can be
         done in parallel. *)
      val (lower_hull_size, lower_hull) = construct_lower_hull (n, pt)
      and (upper_hull_size, upper_hull) = construct_upper_hull (n, pt)

      val hull_size = lower_hull_size + upper_hull_size - 2
      val hull =
          Array.array (hull_size, PlanePoint.planePoint (0.0, 0.0))

      fun copy_lower i =
          if i = lower_hull_size - 1 then
            ()
          else
            (Array.update (hull, i, Array.sub (lower_hull, i));
             copy_lower (i + 1))

      fun copy_upper i =
          if i = upper_hull_size - 1 then
            ()
          else
            (Array.update (hull, i + lower_hull_size - 1,
                           Array.sub (upper_hull, i));
             copy_upper (i + 1))
    in
      copy_lower 0;
      copy_upper 0;
      hull
    end

fun plane_convex_hull points_lst =
    (* Takes an arbitrary list of points, which may be in any order
       and may contain duplicates. Returns an ordered array of points
       that make up the convex hull. If the list of points is empty,
       the returned array is empty. *)
    let
      val pt = Array.fromList points_lst
      val () = sort_in_place PlanePoint.order pt
      val pt = array_delete_neighbor_dups (PlanePoint.==) pt
      val n = Array.length pt
    in
      if n <= 2 then
        pt
      else
        construct_hull (n, pt)
    end

(*------------------------------------------------------------------*)

fun main () =
    let
      open PlanePoint

      val example_points =
          [planePoint (16.0, 3.0),
           planePoint (12.0, 17.0),
           planePoint (0.0, 6.0),
           planePoint (~4.0, ~6.0),
           planePoint (16.0, 6.0),
           planePoint (16.0, ~7.0),
           planePoint (16.0, ~3.0),
           planePoint (17.0, ~4.0),
           planePoint (5.0, 19.0),
           planePoint (19.0, ~8.0),
           planePoint (3.0, 16.0),
           planePoint (12.0, 13.0),
           planePoint (3.0, ~4.0),
           planePoint (17.0, 5.0),
           planePoint (~3.0, 15.0),
           planePoint (~3.0, ~9.0),
           planePoint (0.0, 11.0),
           planePoint (~9.0, ~3.0),
           planePoint (~4.0, ~2.0),
           planePoint (12.0, 10.0)]

      val hull = plane_convex_hull example_points
    in
      Array.app (fn p => (print (toString p);
                          print " "))
                hull;
      print "\n"
    end;

main ();

(*------------------------------------------------------------------*)
(* local variables: *)
(* mode: sml *)
(* sml-indent-level: 2 *)
(* sml-indent-args: 2 *)
(* end: *)
Output:

Output with MLton:

$ mlton convex_hull_task.sml && ./convex_hull_task
(~9 ~3) (~3 ~9) (19 ~8) (17 5) (12 17) (5 19) (~3 15)

Output with Poly/ML (yes, the result is printed twice):

$ polyc convex_hull_task.sml && ./a.out
(~9.0 ~3.0) (~3.0 ~9.0) (19.0 ~8.0) (17.0 5.0) (12.0 17.0) (5.0 19.0) (~3.0 15.0) 
(~9.0 ~3.0) (~3.0 ~9.0) (19.0 ~8.0) (17.0 5.0) (12.0 17.0) (5.0 19.0) (~3.0 15.0)

When you use Poly/ML, if you want the result printed only once, comment out the call to main() near the bottom of the source file. The call is redundant, because a program compiled with Poly/ML automatically calls main().

Swift[edit]

Translation of: Rust
public struct Point: Equatable, Hashable {
  public var x: Double
  public var y: Double

  public init(fromTuple t: (Double, Double)) {
    self.x = t.0
    self.y = t.1
  }
}

public func calculateConvexHull(fromPoints points: [Point]) -> [Point] {
  guard points.count >= 3 else {
    return points
  }

  var hull = [Point]()
  let (leftPointIdx, _) = points.enumerated().min(by: { $0.element.x < $1.element.x })!

  var p = leftPointIdx
  var q = 0

  repeat {
    hull.append(points[p])

    q = (p + 1) % points.count

    for i in 0..<points.count where calculateOrientation(points[p], points[i], points[q]) == .counterClockwise {
      q = i
    }

    p = q
  } while p != leftPointIdx

  return hull
}

private func calculateOrientation(_ p: Point, _ q: Point, _ r: Point) -> Orientation {
  let val = (q.y - p.y) * (r.x - q.x) - (q.x - p.x) * (r.y - q.y)

  if val == 0 {
    return .straight
  } else if val > 0 {
    return .clockwise
  } else {
    return .counterClockwise
  }
}

private enum Orientation {
  case straight, clockwise, counterClockwise
}

let points = [
  (16,3),
  (12,17),
  (0,6),
  (-4,-6),
  (16,6),
  (16,-7),
  (16,-3),
  (17,-4),
  (5,19),
  (19,-8),
  (3,16),
  (12,13),
  (3,-4),
  (17,5),
  (-3,15),
  (-3,-9),
  (0,11),
  (-9,-3),
  (-4,-2),
  (12,10)
].map(Point.init(fromTuple:))

print("Input: \(points)")
print("Output: \(calculateConvexHull(fromPoints: points))")
Output:
Input: [Point(x: 16.0, y: 3.0), Point(x: 12.0, y: 17.0), Point(x: 0.0, y: 6.0), Point(x: -4.0, y: -6.0), Point(x: 16.0, y: 6.0), Point(x: 16.0, y: -7.0), Point(x: 16.0, y: -3.0), Point(x: 17.0, y: -4.0), Point(x: 5.0, y: 19.0), Point(x: 19.0, y: -8.0), Point(x: 3.0, y: 16.0), Point(x: 12.0, y: 13.0), Point(x: 3.0, y: -4.0), Point(x: 17.0, y: 5.0), Point(x: -3.0, y: 15.0), Point(x: -3.0, y: -9.0), Point(x: 0.0, y: 11.0), Point(x: -9.0, y: -3.0), Point(x: -4.0, y: -2.0), Point(x: 12.0, y: 10.0)]
Output: [Point(x: -9.0, y: -3.0), Point(x: -3.0, y: -9.0), Point(x: 19.0, y: -8.0), Point(x: 17.0, y: 5.0), Point(x: 12.0, y: 17.0), Point(x: 5.0, y: 19.0), Point(x: -3.0, y: 15.0)]

Tcl[edit]

Translation of: https://en.wikibooks.org/wiki/Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain#Python

catch {namespace delete test_convex_hull} ;# Start with a clean namespace

namespace eval test_convex_hull {
    package require Tcl 8.5 ;# maybe earlier?
    interp alias {} @ {} lindex;# An essential readability helper for list indexing

    proc cross {o a b} {
        ### 2D cross product of OA and OB vectors ###
        return [expr {([@ $a 0] - [@ $o 0]) * ([@ $b 1] - [@ $o 1]) - ([@ $a 1] - [@ $o 1]) * ([@ $b 0] - [@ $o 0]) }]
    }

    proc calc_hull_edge {points} {
        ### Build up hull edge ###
        set edge [list]
        foreach p $points {
            while {[llength $edge ] >= 2 && [cross [@ $edge end-1] [@ $edge end] $p] <= 0} {
                set edge [lreplace $edge end end] ;# pop
            }
            lappend edge $p
        }
        return $edge
    }

    proc convex_hull {points} {
        ### Convex hull of a set of 2D points ###

        # Unique points
        set points [lsort -unique $points]

        # Sorted points
        set points [lsort -real -index 0 [lsort -real -index 1 $points]]

        # No calculation necessary
        if {[llength $points] <= 1} {
            return $points
        }

        set lower [calc_hull_edge $points]
        set upper [calc_hull_edge [lreverse $points]]

        return [concat [