# Convex hull

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).

## 11l

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!

```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

minX=p.x
index=0
FOR i=1 TO pLen-1
DO
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
pDst.x=pSrc.x
pDst.y=pSrc.y
resLen^==+1

q=(p+1) MOD pLen
FOR i=0 TO pLen-1
DO
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
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:
```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)
```

Translation of: D
```with Ada.Text_IO;

procedure Convex_Hull is

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

package Point_Vectors is
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;
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
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));
end Convex_Hull;
```
Output:
```[ (-9,-3)(-3,-9)( 19,-8)( 17, 5)( 12, 17)( 5, 19)(-3, 15) ]
```

## Arturo

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

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.
//

#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

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

(* Make "cross" a left-associative infix operator with the same
precedence as "*". *)
infixl ( * ) 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]
@(@[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]
@(@[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]
@(@[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]
@(@[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]
@(@[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]
@(pf_lower_hull, pfgc_lower | p_lower_hull, lower_hull_size) =
construct_lower_hull (pt, n)
and [upper_hull_size : int]
@(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 =
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]
@(@[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

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#

```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);
}
}

// 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.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++

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

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)
)

(:use :cl))

;;;
;;; 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 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
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; 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

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

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;
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;
end;

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

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

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

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)]`

## EasyLang

Translation of: Nim
```func orientation p[] q[] r[] .
return (q[2] - p[2]) * (r[1] - q[1]) - (q[1] - p[1]) * (r[2] - q[2])
.
proc calcConvexHull . pts[][] res[][] .
res[][] = [ ]
indMinX = 1
for i to len pts[][]
if pts[i][1] < pts[indMinX][1]
indMinX = i
.
.
p = indMinX
repeat
res[][] &= pts[p][]
q = (p + 1) mod1 len pts[][]
for i to len pts[][]
if orientation pts[p][] pts[i][] pts[q][] < 0
q = i
.
.
p = q
until p = indMinX
.
.
#
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 ] ]
calcConvexHull pts[][] res[][]
print res[][]```
Output:
```[
[ -9 -3 ]
[ -3 -9 ]
[ 19 -8 ]
[ 17 5 ]
[ 12 17 ]
[ 5 19 ]
[ -3 15 ]
]
```

## F#

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

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

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)

```
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

```#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

```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

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)
}
}

// 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.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)]`

```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

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

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

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

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);
}
}

// 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.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

```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>

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

<body>
<table>
<tr>
<th><h1>Convex Hull</h4></th>
</tr>
</table>

</body>

</html>
```

## jq

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 ;```

```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

```# 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

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)
}
}

// 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.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

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

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

```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

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

```MODULE ConvexHull;
FROM FormatString IMPORT FormatString;
FROM Storage IMPORT ALLOCATE, DEALLOCATE;
FROM SYSTEM IMPORT TSIZE;

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

## Nim

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:

# 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.

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

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.

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

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

Translation of: Fortran
```{\$mode ISO}

{ 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
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

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;
}

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

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

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

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))

(: âŠ— (Point Point Point -> Real))
(define/match (âŠ— 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? (âŠ— 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

(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;
}

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

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

### version 1

```/* 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

```/* 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

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

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

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)
}
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

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 (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

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 (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))
(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

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

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<```