# Ray-casting algorithm

Ray-casting algorithm
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 This page uses content from Wikipedia. The original article was at Point_in_polygon. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance)

Given a point and a polygon, check if the point is inside or outside the polygon using the ray-casting algorithm.

A pseudocode can be simply:

 count ← 0
foreach side in polygon:
if ray_intersects_segment(P,side) then
count ← count + 1
if is_odd(count) then
return inside
else
return outside


Where the function ray_intersects_segment return true if the horizontal ray starting from the point P intersects the side (segment), false otherwise.

An intuitive explanation of why it works is that every time we cross a border, we change "country" (inside-outside, or outside-inside), but the last "country" we land on is surely outside (since the inside of the polygon is finite, while the ray continues towards infinity). So, if we crossed an odd number of borders we were surely inside, otherwise we were outside; we can follow the ray backward to see it better: starting from outside, only an odd number of crossing can give an inside: outside-inside, outside-inside-outside-inside, and so on (the - represents the crossing of a border).

So the main part of the algorithm is how we determine if a ray intersects a segment. The following text explain one of the possible ways.

Looking at the image on the right, we can easily be convinced of the fact that rays starting from points in the hatched area (like P1 and P2) surely do not intersect the segment AB. We also can easily see that rays starting from points in the greenish area surely intersect the segment AB (like point P3).

So the problematic points are those inside the white area (the box delimited by the points A and B), like P4.

Let us take into account a segment AB (the point A having y coordinate always smaller than B's y coordinate, i.e. point A is always below point B) and a point P. Let us use the cumbersome notation PAX to denote the angle between segment AP and AX, where X is always a point on the horizontal line passing by A with x coordinate bigger than the maximum between the x coordinate of A and the x coordinate of B. As explained graphically by the figures on the right, if PAX is greater than the angle BAX, then the ray starting from P intersects the segment AB. (In the images, the ray starting from PA does not intersect the segment, while the ray starting from PB in the second picture, intersects the segment).

Points on the boundary or "on" a vertex are someway special and through this approach we do not obtain coherent results. They could be treated apart, but it is not necessary to do so.

An algorithm for the previous speech could be (if P is a point, Px is its x coordinate):

 ray_intersects_segment:
P : the point from which the ray starts
A : the end-point of the segment with the smallest y coordinate
(A must be "below" B)
B : the end-point of the segment with the greatest y coordinate
(B must be "above" A)
if Py = Ay or Py = By then
Py ← Py + ε
end if
if Py < Ay or Py > By then
return false
else if Px >= max(Ax, Bx) then
return false
else
if Px < min(Ax, Bx) then
return true
else
if Ax ≠ Bx then
m_red ← (By - Ay)/(Bx - Ax)
else
m_red ← ∞
end if
if Ax ≠ Px then
m_blue ← (Py - Ay)/(Px - Ax)
else
m_blue ← ∞
end if
if m_blue ≥ m_red then
return true
else
return false
end if
end if
end if


(To avoid the "ray on vertex" problem, the point is moved upward of a small quantity   ε.)

## 11l

Translation of: Python
T Pt((Float x, Float y))
F String()
R ‘Pt(x=#., y=#.)’.format(.x, .y)

T Edge((Pt a, Pt b))
F String()
R ‘Edge(a=#., b=#.)’.format(.a, .b)

T Poly = (String name, [Edge] edges)

V _eps = 0.00001
V _huge = 1e+100
V _tiny = 1e-100

F rayintersectseg(=p, edge)
V a = edge.a
V b = edge.b
I a.y > b.y
swap(&a, &b)
I p.y == a.y | p.y == b.y
p = Pt(p.x, p.y + :_eps)

V intersect = 0B

I (p.y > b.y | p.y < a.y) | (p.x > max(a.x, b.x))
R 0B

I p.x < min(a.x, b.x)
intersect = 1B
E
Float m_red, m_blue
I abs(a.x - b.x) > :_tiny
m_red = (b.y - a.y) / Float(b.x - a.x)
E
m_red = :_huge
I abs(a.x - p.x) > :_tiny
m_blue = (p.y - a.y) / Float(p.x - a.x)
E
m_blue = :_huge
intersect = m_blue >= m_red
R intersect

F ispointinside(p, poly)
R sum(poly.edges.map(edge -> Int(rayintersectseg(@p, edge)))) % 2 == 1

F polypp(poly)
print("\n  Polygon(name='#.', edges=(".format(poly.name))
print(‘    ’(poly.edges.map(e -> String(e)).join(",\n    ")"\n    ))"))

V polys = [
Poly(name' ‘square’, edges' [Edge(Pt(0, 0), Pt(10, 0)), Edge(Pt(10, 0), Pt(10, 10)), Edge(Pt(10, 10), Pt(0, 10)), Edge(Pt(0, 10), Pt(0, 0))]),
Poly(name' ‘square_hole’, edges' [Edge(Pt(0, 0), Pt(10, 0)), Edge(Pt(10, 0), Pt(10, 10)), Edge(Pt(10, 10), Pt(0, 10)), Edge(Pt(0, 10), Pt(0, 0)), Edge(Pt(2.5, 2.5), Pt(7.5, 2.5)), Edge(Pt(7.5, 2.5), Pt(7.5, 7.5)), Edge(Pt(7.5, 7.5), Pt(2.5, 7.5)), Edge(Pt(2.5, 7.5), Pt(2.5, 2.5))]),
Poly(name' ‘strange’, edges' [Edge(Pt(0, 0), Pt(2.5, 2.5)), Edge(Pt(2.5, 2.5), Pt(0, 10)), Edge(Pt(0, 10), Pt(2.5, 7.5)), Edge(Pt(2.5, 7.5), Pt(7.5, 7.5)), Edge(Pt(7.5, 7.5), Pt(10, 10)), Edge(Pt(10, 10), Pt(10, 0)), Edge(Pt(10, 0), Pt(2.5, 2.5))]),
Poly(name' ‘exagon’, edges' [Edge(Pt(3, 0), Pt(7, 0)), Edge(Pt(7, 0), Pt(10, 5)), Edge(Pt(10, 5), Pt(7, 10)), Edge(Pt(7, 10), Pt(3, 10)), Edge(Pt(3, 10), Pt(0, 5)), Edge(Pt(0, 5), Pt(3, 0))])]

V testpoints = [Pt(5, 5), Pt(5, 8),
Pt(-10, 5), Pt(0, 5),
Pt(10, 5), Pt(8, 5),
Pt(10, 10)]

print("\n TESTING WHETHER POINTS ARE WITHIN POLYGONS")
L(poly) polys
polypp(poly)
print(‘    ’testpoints[0.<3].map(p -> ‘#.: #.’.format(p, I ispointinside(p, @poly) {‘True’} E ‘False’)).join("\t"))
print(‘    ’testpoints[3.<6].map(p -> ‘#.: #.’.format(p, I ispointinside(p, @poly) {‘True’} E ‘False’)).join("\t"))
print(‘    ’testpoints[6.. ].map(p -> ‘#.: #.’.format(p, I ispointinside(p, @poly) {‘True’} E ‘False’)).join("\t"))
Output:

TESTING WHETHER POINTS ARE WITHIN POLYGONS

Polygon(name='square', edges=(
Edge(a=Pt(x=0, y=0), b=Pt(x=10, y=0)),
Edge(a=Pt(x=10, y=0), b=Pt(x=10, y=10)),
Edge(a=Pt(x=10, y=10), b=Pt(x=0, y=10)),
Edge(a=Pt(x=0, y=10), b=Pt(x=0, y=0))
))
Pt(x=5, y=5): True	Pt(x=5, y=8): True	Pt(x=-10, y=5): False
Pt(x=0, y=5): False	Pt(x=10, y=5): True	Pt(x=8, y=5): True
Pt(x=10, y=10): False

Polygon(name='square_hole', edges=(
Edge(a=Pt(x=0, y=0), b=Pt(x=10, y=0)),
Edge(a=Pt(x=10, y=0), b=Pt(x=10, y=10)),
Edge(a=Pt(x=10, y=10), b=Pt(x=0, y=10)),
Edge(a=Pt(x=0, y=10), b=Pt(x=0, y=0)),
Edge(a=Pt(x=2.5, y=2.5), b=Pt(x=7.5, y=2.5)),
Edge(a=Pt(x=7.5, y=2.5), b=Pt(x=7.5, y=7.5)),
Edge(a=Pt(x=7.5, y=7.5), b=Pt(x=2.5, y=7.5)),
Edge(a=Pt(x=2.5, y=7.5), b=Pt(x=2.5, y=2.5))
))
Pt(x=5, y=5): False	Pt(x=5, y=8): True	Pt(x=-10, y=5): False
Pt(x=0, y=5): False	Pt(x=10, y=5): True	Pt(x=8, y=5): True
Pt(x=10, y=10): False

Polygon(name='strange', edges=(
Edge(a=Pt(x=0, y=0), b=Pt(x=2.5, y=2.5)),
Edge(a=Pt(x=2.5, y=2.5), b=Pt(x=0, y=10)),
Edge(a=Pt(x=0, y=10), b=Pt(x=2.5, y=7.5)),
Edge(a=Pt(x=2.5, y=7.5), b=Pt(x=7.5, y=7.5)),
Edge(a=Pt(x=7.5, y=7.5), b=Pt(x=10, y=10)),
Edge(a=Pt(x=10, y=10), b=Pt(x=10, y=0)),
Edge(a=Pt(x=10, y=0), b=Pt(x=2.5, y=2.5))
))
Pt(x=5, y=5): True	Pt(x=5, y=8): False	Pt(x=-10, y=5): False
Pt(x=0, y=5): False	Pt(x=10, y=5): True	Pt(x=8, y=5): True
Pt(x=10, y=10): False

Polygon(name='exagon', edges=(
Edge(a=Pt(x=3, y=0), b=Pt(x=7, y=0)),
Edge(a=Pt(x=7, y=0), b=Pt(x=10, y=5)),
Edge(a=Pt(x=10, y=5), b=Pt(x=7, y=10)),
Edge(a=Pt(x=7, y=10), b=Pt(x=3, y=10)),
Edge(a=Pt(x=3, y=10), b=Pt(x=0, y=5)),
Edge(a=Pt(x=0, y=5), b=Pt(x=3, y=0))
))
Pt(x=5, y=5): True	Pt(x=5, y=8): True	Pt(x=-10, y=5): False
Pt(x=0, y=5): False	Pt(x=10, y=5): True	Pt(x=8, y=5): True
Pt(x=10, y=10): False


package Polygons is

type Point is record
X, Y : Float;
end record;
type Point_List is array (Positive range <>) of Point;
subtype Segment is Point_List (1 .. 2);
type Polygon is array (Positive range <>) of Segment;

function Create_Polygon (List : Point_List) return Polygon;

function Is_Inside (Who : Point; Where : Polygon) return Boolean;

end Polygons;


package body Polygons is
EPSILON : constant := 0.00001;

function Ray_Intersects_Segment
(Who   : Point;
Where : Segment)
return  Boolean
is
The_Point        : Point   := Who;
Above            : Point;
Below            : Point;
M_Red            : Float;
Red_Is_Infinity  : Boolean := False;
M_Blue           : Float;
Blue_Is_Infinity : Boolean := False;
begin
if Where (1).Y < Where (2).Y then
Above := Where (2);
Below := Where (1);
else
Above := Where (1);
Below := Where (2);
end if;
if The_Point.Y = Above.Y or The_Point.Y = Below.Y then
The_Point.Y := The_Point.Y + EPSILON;
end if;
if The_Point.Y < Below.Y or The_Point.Y > Above.Y then
return False;
elsif The_Point.X > Above.X and The_Point.X > Below.X then
return False;
elsif The_Point.X < Above.X and The_Point.X < Below.X then
return True;
else
if Above.X /= Below.X then
M_Red := (Above.Y - Below.Y) / (Above.X - Below.X);
else
Red_Is_Infinity := True;
end if;
if Below.X /= The_Point.X then
M_Blue := (The_Point.Y - Below.Y) / (The_Point.X - Below.X);
else
Blue_Is_Infinity := True;
end if;
if Blue_Is_Infinity then
return True;
elsif Red_Is_Infinity then
return False;
elsif M_Blue >= M_Red then
return True;
else
return False;
end if;
end if;
end Ray_Intersects_Segment;

function Create_Polygon (List : Point_List) return Polygon is
Result : Polygon (List'Range);
Side   : Segment;
begin
for I in List'Range loop
Side (1) := List (I);
if I = List'Last then
Side (2) := List (List'First);
else
Side (2) := List (I + 1);
end if;
Result (I) := Side;
end loop;
return Result;
end Create_Polygon;

function Is_Inside (Who : Point; Where : Polygon) return Boolean is
Count : Natural := 0;
begin
for Side in Where'Range loop
if Ray_Intersects_Segment (Who, Where (Side)) then
Count := Count + 1;
end if;
end loop;
if Count mod 2 = 0 then
return False;
else
return True;
end if;
end Is_Inside;

end Polygons;


Example use:

with Ada.Text_IO;
with Polygons;
procedure Main is
package Float_IO is new Ada.Text_IO.Float_IO (Float);
Test_Points : Polygons.Point_List :=
((  5.0,  5.0),
(  5.0,  8.0),
(-10.0,  5.0),
(  0.0,  5.0),
( 10.0,  5.0),
(  8.0,  5.0),
( 10.0, 10.0));
Square      : Polygons.Polygon    :=
((( 0.0,  0.0), (10.0,  0.0)),
((10.0,  0.0), (10.0, 10.0)),
((10.0, 10.0), ( 0.0, 10.0)),
(( 0.0, 10.0), ( 0.0,  0.0)));
Square_Hole : Polygons.Polygon    :=
((( 0.0,  0.0), (10.0,  0.0)),
((10.0,  0.0), (10.0, 10.0)),
((10.0, 10.0), ( 0.0, 10.0)),
(( 0.0, 10.0), ( 0.0,  0.0)),
(( 2.5,  2.5), ( 7.5,  2.5)),
(( 7.5,  2.5), ( 7.5,  7.5)),
(( 7.5,  7.5), ( 2.5,  7.5)),
(( 2.5,  7.5), ( 2.5,  2.5)));
Strange     : Polygons.Polygon    :=
((( 0.0,  0.0), ( 2.5,  2.5)),
(( 2.5,  2.5), ( 0.0, 10.0)),
(( 0.0, 10.0), ( 2.5,  7.5)),
(( 2.5,  7.5), ( 7.5,  7.5)),
(( 7.5,  7.5), (10.0, 10.0)),
((10.0, 10.0), (10.0,  0.0)),
((10.0,  0.0), ( 2.5,  2.5)));
Exagon      : Polygons.Polygon    :=
((( 3.0,  0.0), ( 7.0,  0.0)),
(( 7.0,  0.0), (10.0,  5.0)),
((10.0,  5.0), ( 7.0, 10.0)),
(( 7.0, 10.0), ( 3.0, 10.0)),
(( 3.0, 10.0), ( 0.0,  5.0)),
(( 0.0,  5.0), ( 3.0,  0.0)));
begin
for Point in Test_Points'Range loop
Float_IO.Put (Test_Points (Point).X, 0, 0, 0);
Float_IO.Put (Test_Points (Point).Y, 0, 0, 0);
("): " &
Boolean'Image (Polygons.Is_Inside (Test_Points (Point), Square)));
end loop;
for Point in Test_Points'Range loop
Float_IO.Put (Test_Points (Point).X, 0, 0, 0);
Float_IO.Put (Test_Points (Point).Y, 0, 0, 0);
("): " &
Boolean'Image
(Polygons.Is_Inside (Test_Points (Point), Square_Hole)));
end loop;
for Point in Test_Points'Range loop
Float_IO.Put (Test_Points (Point).X, 0, 0, 0);
Float_IO.Put (Test_Points (Point).Y, 0, 0, 0);
("): " &
Boolean'Image (Polygons.Is_Inside (Test_Points (Point), Strange)));
end loop;
for Point in Test_Points'Range loop
Float_IO.Put (Test_Points (Point).X, 0, 0, 0);
Float_IO.Put (Test_Points (Point).Y, 0, 0, 0);
("): " &
Boolean'Image (Polygons.Is_Inside (Test_Points (Point), Exagon)));
end loop;
end Main;


Output:

Testing Square:
Point(5.0,5.0): TRUE
Point(5.0,8.0): TRUE
Point(-10.0,5.0): FALSE
Point(0.0,5.0): FALSE
Point(10.0,5.0): TRUE
Point(8.0,5.0): TRUE
Point(10.0,10.0): FALSE

Testing Square_Hole:
Point(5.0,5.0): FALSE
Point(5.0,8.0): TRUE
Point(-10.0,5.0): FALSE
Point(0.0,5.0): FALSE
Point(10.0,5.0): TRUE
Point(8.0,5.0): TRUE
Point(10.0,10.0): FALSE

Testing Strange:
Point(5.0,5.0): TRUE
Point(5.0,8.0): FALSE
Point(-10.0,5.0): FALSE
Point(0.0,5.0): FALSE
Point(10.0,5.0): TRUE
Point(8.0,5.0): TRUE
Point(10.0,10.0): FALSE

Testing Exagon:
Point(5.0,5.0): TRUE
Point(5.0,8.0): TRUE
Point(-10.0,5.0): FALSE
Point(0.0,5.0): FALSE
Point(10.0,5.0): TRUE
Point(8.0,5.0): TRUE
Point(10.0,10.0): FALSE

## ALGOL 68

Translation of: Lua
BEGIN

MODE POINT = STRUCT( REAL x, y );

MODE POLYGON = STRUCT( STRING name, FLEX[ 1 : 0 ]POINT points );
PROC contains = ( POLYGON self, POINT p )BOOL:
BEGIN
BOOL odd := FALSE, REAL eps = 1e-9;
PROC rayseg = ( POINT p in, a in, b in )BOOL:
BEGIN
PROC max = ( REAL m, n )REAL: IF m > n THEN m ELSE n FI;
PROC min = ( REAL m, n )REAL: IF m < n THEN m ELSE n FI;
POINT p := p in, a := a in, b := b in;
IF y OF a > y OF b THEN POINT t = a; a := b; b := t FI;
IF y OF p = y OF a OR y OF p = y OF b THEN y OF p+:= eps FI;
IF y OF p < y OF a OR y OF p > y OF b OR x OF p > max( x OF a, x OF b )
THEN FALSE
ELIF x OF p < min( x OF a, x OF b )
THEN TRUE
ELSE
REAL red = IF x OF a = x OF b THEN max real ELSE ( y OF b - y OF a ) / ( x OF b - x OF a ) FI;
REAL blu = IF x OF a = x OF p THEN max real ELSE ( y OF p - y OF a ) / ( x OF p - x OF a ) FI;
blu >= red
FI
END # rayseq # ;

INT len points = ( UPB points OF self - LWB points OF self ) + 1;
FOR i FROM LWB points OF self TO UPB points OF self DO
POINT a = ( points OF self )[ i ];
POINT b = ( points OF self )[ ( i MOD len points ) + 1 ];
IF rayseg( p, a, b ) THEN odd := NOT odd FI
OD;
odd
END # contains # ;

[]POLYGON polygons =
( ( "square"
, ( ( 0, 0 ), (  10,   0 ), ( 10, 10 ), ( 0,    10 ) )
)
, ( "squarehole"
, ( ( 0, 0 ), (  10,   0 ), ( 10, 10 ), ( 0,    10 ), ( 2.5, 2.5 ), ( 7.5, 2.5 ), ( 7.5, 7.5 ), ( 2.5, 7.5 ) )
)
, ( "strange"
, ( ( 0, 0 ), ( 2.5, 2.5 ), (  0, 10 ), ( 2.5, 7.5 ), ( 7.5, 7.5 ), (  10,  10 ), ( 10,    0 ), ( 2.5, 2.5 ) )
)
, ( "hexagon"
, ( ( 3, 0 ), (   7,   0 ), ( 10,  5 ), ( 7,    10 ), ( 3,    10 ), (   0,   5 ) )
)
);
[]POINT points = ( ( 5, 5 ), (5 , 8 ), ( -10, 5 ), ( 0, 5 ), ( 10, 5 ), ( 8, 5 ), ( 10, 10 ) );

FOR p FROM LWB polygons TO UPB polygons DO
POLYGON poly = polygons[ p ];
print(( "Does '", name OF poly, "' contain the point..", newline ) );
FOR i FROM LWB points TO UPB points DO
POINT pt = points[ i ];
print( ( "  ( ", fixed( x OF pt, -5, 1 ), ", ", fixed( y OF pt, -5, 1 ), " ) " ) );
print( ( IF contains( poly, pt ) THEN " true" ELSE " false" FI, newline ) )
OD;
print( ( newline ) )
OD

END
Output:
Does 'square' contain the point..
(   5.0,   5.0 )  true
(   5.0,   8.0 )  true
( -10.0,   5.0 )  false
(   0.0,   5.0 )  false
(  10.0,   5.0 )  true
(   8.0,   5.0 )  true
(  10.0,  10.0 )  false

Does 'squarehole' contain the point..
(   5.0,   5.0 )  false
(   5.0,   8.0 )  true
( -10.0,   5.0 )  false
(   0.0,   5.0 )  false
(  10.0,   5.0 )  true
(   8.0,   5.0 )  true
(  10.0,  10.0 )  false

Does 'strange' contain the point..
(   5.0,   5.0 )  true
(   5.0,   8.0 )  false
( -10.0,   5.0 )  false
(   0.0,   5.0 )  false
(  10.0,   5.0 )  true
(   8.0,   5.0 )  true
(  10.0,  10.0 )  false

Does 'hexagon' contain the point..
(   5.0,   5.0 )  true
(   5.0,   8.0 )  true
( -10.0,   5.0 )  false
(   0.0,   5.0 )  false
(  10.0,   5.0 )  true
(   8.0,   5.0 )  true
(  10.0,  10.0 )  false


## AutoHotkey

Works with: AutoHotkey L
Points :=[{x:  5.0, y: 5.0}
, {x:  5.0, y: 8.0}
, {x:-10.0, y: 5.0}
, {x:  0.0, y: 5.0}
, {x: 10.0, y: 5.0}
, {x:  8.0, y: 5.0}
, {x: 10.0, y:10.0}]
Square :=[{x: 0.0, y: 0.0}, {x:10.0, y: 0.0}
, {x:10.0, y: 0.0}, {x:10.0, y:10.0}
, {x:10.0, y:10.0}, {x: 0.0, y:10.0}
, {x: 0.0, y:10.0}, {x: 0.0, y: 0.0}]
Sq_Hole:=[{x: 0.0, y: 0.0}, {x:10.0, y: 0.0}
, {x:10.0, y: 0.0}, {x:10.0, y:10.0}
, {x:10.0, y:10.0}, {x: 0.0, y:10.0}
, {x: 0.0, y:10.0}, {x: 0.0, y: 0.0}
, {x: 2.5, y: 2.5}, {x: 7.5, y: 2.5}
, {x: 7.5, y: 2.5}, {x: 7.5, y: 7.5}
, {x: 7.5, y: 7.5}, {x: 2.5, y: 7.5}
, {x: 2.5, y: 7.5}, {x: 2.5, y: 2.5}]
Strange:=[{x: 0.0, y: 0.0}, {x: 2.5, y: 2.5}
, {x: 2.5, y: 2.5}, {x: 0.0, y:10.0}
, {x: 0.0, y:10.0}, {x: 2.5, y: 7.5}
, {x: 2.5, y: 7.5}, {x: 7.5, y: 7.5}
, {x: 7.5, y: 7.5}, {x:10.0, y:10.0}
, {x:10.0, y:10.0}, {x:10.0, y: 0.0}
, {x:10.0, y: 0.0}, {x: 2.5, y: 2.5}]
Exagon :=[{x: 3.0, y: 0.0}, {x: 7.0, y: 0.0}
, {x: 7.0, y: 0.0}, {x:10.0, y: 5.0}
, {x:10.0, y: 5.0}, {x: 7.0, y:10.0}
, {x: 7.0, y:10.0}, {x: 3.0, y:10.0}
, {x: 3.0, y:10.0}, {x: 0.0, y: 5.0}
, {x: 0.0, y: 5.0}, {x: 3.0, y: 0.0}]
Polygons := {"Square":Square, "Sq_Hole":Sq_Hole, "Strange":Strange, "Exagon":Exagon}
For j, Poly in Polygons
For i, Point in Points
If (point_in_polygon(Point,Poly))
s.= j " does contain point " i "n"
Else
s.= j " doesn't contain point " i "n"
Msgbox %s%

point_in_polygon(Point,Poly) {
n:=Poly.MaxIndex()
count:=0
loop, %n% {
if (ray_intersects_segment(Point,Poly[A_Index],Poly[mod(A_Index,n)+1])) {
count++
}
}
if (mod(count,2)) { ; true = inside, false = outside
return true		; P is in the polygon
} else {
return false	; P isn't in the polygon
}
}

ray_intersects_segment(P,A,B) {
;P = the point from which the ray starts
;A = the end-point of the segment with the smallest y coordinate
;B = the end-point of the segment with the greatest y coordinate
if (A.y > B.y) {
temp:=A
A:=B
B:=temp
}
if (P.y = A.y or P.y = B.y) {
P.y += 0.000001
}
if (P.y < A.y or P.y > B.y) {
return false
} else if (P.x > A.x && P.x > B.x) {
return false
} else {
if (P.x < A.x && P.x < B.x) {
return true
} else {
if (A.x != B.x) {
m_red := (B.y - A.y)/(B.x - A.x)
} else {
m_red := "inf"
}
if (A.x != P.x) {
m_blue := (P.y - A.y)/(P.x - A.x)
} else {
m_blue := "inf"
}
if (m_blue >= m_red) {
return true
} else {
return false
}
}
}
}

Output:
---------------------------
Ray-casting_algorithm.ahkl
---------------------------
Exagon does contain point 1
Exagon does contain point 2
Exagon doesn't contain point 3
Exagon doesn't contain point 4
Exagon does contain point 5
Exagon does contain point 6
Exagon doesn't contain point 7
Sq_Hole doesn't contain point 1
Sq_Hole does contain point 2
Sq_Hole doesn't contain point 3
Sq_Hole doesn't contain point 4
Sq_Hole does contain point 5
Sq_Hole does contain point 6
Sq_Hole doesn't contain point 7
Square does contain point 1
Square does contain point 2
Square doesn't contain point 3
Square doesn't contain point 4
Square does contain point 5
Square does contain point 6
Square doesn't contain point 7
Strange does contain point 1
Strange doesn't contain point 2
Strange doesn't contain point 3
Strange doesn't contain point 4
Strange does contain point 5
Strange does contain point 6
Strange doesn't contain point 7

---------------------------
OK
---------------------------

## BASIC

### ANSI BASIC

Translation of: FreeBASIC
Works with: Decimal BASIC
1000 PUBLIC NUMERIC x,y
1010 LET x=1
1020 LET y=2
1030 !
1040 DEF isLeft2(L(,),p()) = -SGN(  (L(1,x)-L(2,x))*(p(y)-L(2,y)) - (p(x)-L(2,x))*(L(1,y)-L(2,y)))
1050 !
1060 FUNCTION inpolygon(p1(,),p2())
1070    LET k=UBOUND(p1,1)+1
1080    DIM send (1 TO 2,2)
1090    LET wn=0
1100    FOR n=1 TO UBOUND(p1,1)
1110       LET index=MOD(n, k)
1120       LET nextindex=MOD(n+1, k)
1130       IF nextindex=0 THEN LET nextindex=1
1140       LET send(1,x)=p1(index,x)
1150       LET send(2,x)=p1(nextindex,x)
1160       LET send(1,y)=p1(index,y)
1170       LET send(2,y)=p1(nextindex,y)
1180       IF p1(index,y)<=p2(y) THEN
1190          IF p1(nextindex,y)>p2(y) THEN
1200             IF isleft2(send,p2)>=0 THEN !'=
1210                LET wn=wn+1
1220             END IF
1230          END IF
1240       ELSE
1250          IF p1(nextindex,y)<=p2(y) THEN
1260             IF isleft2(send,p2)<=0 THEN !'=
1270                LET wn=wn-1
1280             END IF
1290          END IF
1300       END IF
1310    NEXT n
1320    LET inpolygon = wn
1330 END FUNCTION
1340 !
1350 DIM type(1 TO 2)
1360 !
1370 DIM square(4,2)
1390 DATA 0,0,10,0,10,10,0,10
1400 !
1410 DIM hole(4,2)
1430 DATA 2.5,2.5,7.5,2.5,7.5,7.5,2.5,7.5
1440 !
1450 DIM strange(8,2)
1470 DATA 0,0,2.5,2.5,0,10,2.5,7.5,7.5,7.5,10,10,10,0,2.5,2.5
1480 !
1490 DIM exagon(6,2)
1510 DATA 3,0,7,0,10,5,7,10,3,10,0,5
1520 !
1530 ! printouts
1540 FOR z=1 TO 4
1550    SELECT CASE z
1560    CASE 1
1570       PRINT "squared"
1580       PRINT "(5,5)  ";TAB(12);
1600       DATA 5,5
1610       IF inpolygon(square,type) <> 0 THEN PRINT "in" ELSE PRINT "out"
1630       DATA 5,8
1640       PRINT "(5,8)  ";TAB(12);
1650       IF inpolygon(square,type) <> 0 THEN PRINT "in" ELSE PRINT "out"
1660       PRINT "(-10,5)  ";TAB(12);
1680       DATA -10,5
1690       IF inpolygon(square,type) <> 0 THEN PRINT "in" ELSE PRINT "out"
1700       Print "(0,5)  ";Tab(12);
1720       DATA 0,5
1730       IF inpolygon(square,type) <> 0 THEN PRINT "in" ELSE PRINT "out"
1740       Print "(10,5)  ";Tab(12);
1760       DATA 10,5
1770       IF inpolygon(square,type) <> 0 THEN PRINT "in" ELSE PRINT "out"
1780       PRINT "(8,5)  ";TAB(12);
1800       DATA 8,5
1810       IF inpolygon(square,Type) <> 0 THEN PRINT "in" ELSE PRINT "out"
1820       PRINT "(10,10)  ";TAB(12);
1840       DATA 10,10
1850       IF inpolygon(square,Type) <> 0 THEN PRINT "in" ELSE PRINT "out"
1860       PRINT
1870    CASE 2
1880       PRINT "squared hole"
1890       PRINT "(5,5)  ";TAB(12);
1910       DATA 5,5
1920       IF NOT inpolygon(hole,Type)<>0 AND inpolygon(square,Type)<>0 THEN PRINT "in" ELSE PRINT "out"
1930       Print "(5,8)  ";Tab(12);
1950       DATA 5,8
1960       IF NOT inpolygon(hole,Type)<>0 AND inpolygon(square,Type)<>0 THEN PRINT "in" ELSE PRINT "out"
1970       PRINT "(-10,5)  ";TAB(12);
1990       DATA -10,5
2000       IF NOT inpolygon(hole,Type)<>0 AND inpolygon(square,Type)<>0 THEN PRINT "in" ELSE PRINT "out"
2010       PRINT "(0,5)  ";TAB(12);
2030       DATA 0,5
2040       IF NOT inpolygon(hole,Type)<>0 AND inpolygon(square,Type)<>0 THEN PRINT "in" ELSE PRINT "out"
2050       PRINT "(10,5)  ";TAB(12);
2070       DATA 10,5
2080       IF NOT inpolygon(hole,Type)<>0 AND inpolygon(square,Type)<>0 THEN PRINT "in" ELSE PRINT "out"
2090       PRINT "(8,5)  ";TAB(12);
2110       DATA 8,5
2120       IF NOT inpolygon(hole,Type)<>0 AND inpolygon(square,Type)<>0 THEN PRINT "in" ELSE PRINT "out"
2130       PRINT "(10,10)  ";TAB(12);
2150       DATA 10,10
2160       IF NOT inpolygon(hole,Type)<>0 AND inpolygon(square,Type)<>0 THEN PRINT "in" ELSE PRINT "out"
2170       PRINT
2180    CASE 3
2190       PRINT "strange"
2200       PRINT "(5,5)  ";TAB(12);
2220       DATA 5,5
2230       IF inpolygon(strange,Type)<>0 THEN PRINT "in" ELSE PRINT "out"
2240       PRINT "(5,8)  ";TAB(12);
2260       DATA 5,8
2270       IF inpolygon(strange,Type)<>0 THEN PRINT "in" ELSE PRINT "out"
2280       PRINT "(-10,5)  ";TAB(12);
2300       DATA -10,5
2310       IF inpolygon(strange,Type)<>0 THEN PRINT "in" ELSE PRINT "out"
2320       PRINT "(0,5)  ";TAB(12);
2340       DATA 0,5
2350       IF inpolygon(strange,Type)<>0 THEN PRINT "in" ELSE PRINT "out"
2360       PRINT "(10,5)  ";TAB(12);
2380       DATA 10,5
2390       IF inpolygon(strange,Type)<>0 THEN PRINT "in" ELSE PRINT "out"
2400       PRINT "(8,5)  ";TAB(12);
2420       DATA 8,5
2430       IF inpolygon(strange,Type)<>0 THEN PRINT "in" ELSE PRINT "out"
2440       PRINT "(10,10)  ";TAB(12);
2460       DATA 10,10
2470       IF inpolygon(strange,Type)<>0 THEN PRINT "in" ELSE PRINT "out"
2480       PRINT
2490    CASE 4
2500       PRINT "exagon"
2510       PRINT "(5,5)  ";TAB(12);
2530       DATA 5,5
2540       IF inpolygon(exagon,Type)<>0 THEN PRINT "in" ELSE PRINT "out"
2550       PRINT "(5,8)  ";TAB(12);
2570       DATA 5,8
2580       IF inpolygon(exagon,Type)<>0 THEN PRINT "in" ELSE PRINT "out"
2590       PRINT "(-10,5)  ";TAB(12);
2610       DATA -10,5
2620       IF inpolygon(exagon,Type)<>0 THEN PRINT "in" ELSE PRINT "out"
2630       PRINT "(0,5)  ";TAB(12);
2650       DATA 0,5
2660       IF inpolygon(exagon,Type)<>0 THEN PRINT "in" ELSE PRINT "out"
2670       PRINT "(10,5)  ";TAB(12);
2690       DATA 10,5
2700       IF inpolygon(exagon,Type)<>0 THEN PRINT "in" ELSE PRINT "out"
2710       PRINT "(8,5)  ";TAB(12);
2730       DATA 8,5
2740       IF inpolygon(exagon,Type)<>0 THEN PRINT "in" ELSE PRINT "out"
2750       PRINT "(10,10)  ";TAB(12);
2770       DATA 10,10
2780       IF inpolygon(exagon,Type)<>0 THEN PRINT "in" ELSE PRINT "out"
2790       PRINT
2800    END SELECT
2810 NEXT z
2820 END

Output:
squared
(5,5)      in
(5,8)      in
(-10,5)    out
(0,5)      out
(10,5)     in
(8,5)      in
(10,10)    out

squared hole
(5,5)      out
(5,8)      in
(-10,5)    out
(0,5)      out
(10,5)     in
(8,5)      in
(10,10)    out

strange
(5,5)      in
(5,8)      out
(-10,5)    out
(0,5)      out
(10,5)     in
(8,5)      in
(10,10)    out

exagon
(5,5)      in
(5,8)      in
(-10,5)    out
(0,5)      out
(10,5)     in
(8,5)      in
(10,10)    out


## C

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

typedef struct { double x, y; } vec;
typedef struct { int n; vec* v; } polygon_t, *polygon;

#define BIN_V(op, xx, yy) vec v##op(vec a,vec b){vec c;c.x=xx;c.y=yy;return c;}
#define BIN_S(op, r) double v##op(vec a, vec b){ return r; }
BIN_V(sub, a.x - b.x, a.y - b.y);
BIN_V(add, a.x + b.x, a.y + b.y);
BIN_S(dot, a.x * b.x + a.y * b.y);
BIN_S(cross, a.x * b.y - a.y * b.x);

/* return a + s * b */
vec vmadd(vec a, double s, vec b)
{
vec c;
c.x = a.x + s * b.x;
c.y = a.y + s * b.y;
return c;
}

/* check if x0->x1 edge crosses y0->y1 edge. dx = x1 - x0, dy = y1 - y0, then
solve  x0 + a * dx == y0 + b * dy with a, b in real
cross both sides with dx, then: (remember, cross product is a scalar)
x0 X dx = y0 X dx + b * (dy X dx)
similarly,
x0 X dy + a * (dx X dy) == y0 X dy
there is an intersection iff 0 <= a <= 1 and 0 <= b <= 1

returns: 1 for intersect, -1 for not, 0 for hard to say (if the intersect
point is too close to y0 or y1)
*/
int intersect(vec x0, vec x1, vec y0, vec y1, double tol, vec *sect)
{
vec dx = vsub(x1, x0), dy = vsub(y1, y0);
double d = vcross(dy, dx), a;
if (!d) return 0; /* edges are parallel */

a = (vcross(x0, dx) - vcross(y0, dx)) / d;
if (sect)

if (a < -tol || a > 1 + tol) return -1;
if (a < tol || a > 1 - tol) return 0;

a = (vcross(x0, dy) - vcross(y0, dy)) / d;
if (a < 0 || a > 1) return -1;

return 1;
}

/* distance between x and nearest point on y0->y1 segment.  if the point
lies outside the segment, returns infinity */
double dist(vec x, vec y0, vec y1, double tol)
{
vec dy = vsub(y1, y0);
vec x1, s;
int r;

x1.x = x.x + dy.y; x1.y = x.y - dy.x;
r = intersect(x, x1, y0, y1, tol, &s);
if (r == -1) return HUGE_VAL;
s = vsub(s, x);
return sqrt(vdot(s, s));
}

#define for_v(i, z, p) for(i = 0, z = p->v; i < p->n; i++, z++)
/* returns 1 for inside, -1 for outside, 0 for on edge */
int inside(vec v, polygon p, double tol)
{
/* should assert p->n > 1 */
int i, k, crosses, intersectResult;
vec *pv;
double min_x, max_x, min_y, max_y;

for (i = 0; i < p->n; i++) {
k = (i + 1) % p->n;
min_x = dist(v, p->v[i], p->v[k], tol);
if (min_x < tol) return 0;
}

min_x = max_x = p->v[0].x;
min_y = max_y = p->v[1].y;

/* calculate extent of polygon */
for_v(i, pv, p) {
if (pv->x > max_x) max_x = pv->x;
if (pv->x < min_x) min_x = pv->x;
if (pv->y > max_y) max_y = pv->y;
if (pv->y < min_y) min_y = pv->y;
}
if (v.x < min_x || v.x > max_x || v.y < min_y || v.y > max_y)
return -1;

max_x -= min_x; max_x *= 2;
max_y -= min_y; max_y *= 2;
max_x += max_y;

vec e;
while (1) {
crosses = 0;
/* pick a rand point far enough to be outside polygon */
e.x = v.x + (1 + rand() / (RAND_MAX + 1.)) * max_x;
e.y = v.y + (1 + rand() / (RAND_MAX + 1.)) * max_x;

for (i = 0; i < p->n; i++) {
k = (i + 1) % p->n;
intersectResult = intersect(v, e, p->v[i], p->v[k], tol, 0);

/* picked a bad point, ray got too close to vertex.
re-pick */
if (!intersectResult) break;

if (intersectResult == 1) crosses++;
}
if (i == p->n) break;
}
return (crosses & 1) ? 1 : -1;
}

int main()
{
vec vsq[] = {	{0,0}, {10,0}, {10,10}, {0,10},
{2.5,2.5}, {7.5,0.1}, {7.5,7.5}, {2.5,7.5}};

polygon_t sq = { 4, vsq }, /* outer square */
sq_hole = { 8, vsq }; /* outer and inner square, ie hole */

vec c = { 10, 5 }; /* on edge */
vec d = { 5, 5 };

printf("%d\n", inside(c, &sq, 1e-10));
printf("%d\n", inside(c, &sq_hole, 1e-10));

printf("%d\n", inside(d, &sq, 1e-10));	/* in */
printf("%d\n", inside(d, &sq_hole, 1e-10));  /* out (in the hole) */

return 0;
}


## C#

Translation of: Java
using System;

class RayCasting {

static bool Intersects(int[] A, int[] B, double[] P) {
if (A[1] > B[1])
return Intersects(B, A, P);

if (P[1] == A[1] || P[1] == B[1])
P[1] += 0.0001;

if (P[1] > B[1] || P[1] < A[1] || P[0] >= Math.Max(A[0], B[0]))
return false;

if (P[0] < Math.Min(A[0], B[0]))
return true;

double red = (P[1] - A[1]) / (P[0] - A[0]);
double blue = (B[1] - A[1]) / (B[0] - A[0]);
return red >= blue;
}

static bool Contains(int[][] shape, double[] pnt) {
bool inside = false;
int len = shape.Length;
for (int i = 0; i < len; i++) {
if (Intersects(shape[i], shape[(i + 1) % len], pnt))
inside = !inside;
}
return inside;
}

public static void Main(string[] args) {
double[][] testPoints = new double[][] {
new double[] { 10, 10 }, new double[] { 10, 16 }, new double[] { -20, 10 },
new double[] { 0, 10 }, new double[] { 20, 10 }, new double[] { 16, 10 },
new double[] { 20, 20 }
};

foreach (int[][] shape in shapes) {
foreach (double[] pnt in testPoints)
Console.Write($"{Contains(shape, pnt),7} "); Console.WriteLine(); } } readonly static int[][] square = new int[][] { new int[] { 0, 0 }, new int[] { 20, 0 }, new int[] { 20, 20 }, new int[] { 0, 20 } }; readonly static int[][] squareHole = new int[][] { new int[] { 0, 0 }, new int[] { 20, 0 }, new int[] { 20, 20 }, new int[] { 0, 20 }, new int[] { 5, 5 }, new int[] { 15, 5 }, new int[] { 15, 15 }, new int[] { 5, 15 } }; readonly static int[][] strange = new int[][] { new int[] { 0, 0 }, new int[] { 5, 5 }, new int[] { 0, 20 }, new int[] { 5, 15 }, new int[] { 15, 15 }, new int[] { 20, 20 }, new int[] { 20, 0 } }; readonly static int[][] hexagon = new int[][] { new int[] { 6, 0 }, new int[] { 14, 0 }, new int[] { 20, 10 }, new int[] { 14, 20 }, new int[] { 6, 20 }, new int[] { 0, 10 } }; readonly static int[][][] shapes = new int[][][] { square, squareHole, strange, hexagon }; }  Output:  True True False True False True False False True False False False True False True False False False False True False True True False False False True False  ## C++ Works with: C++ version 11 Translation of: D #include <algorithm> #include <cstdlib> #include <iomanip> #include <iostream> #include <limits> using namespace std; const double epsilon = numeric_limits<float>().epsilon(); const numeric_limits<double> DOUBLE; const double MIN = DOUBLE.min(); const double MAX = DOUBLE.max(); struct Point { const double x, y; }; struct Edge { const Point a, b; bool operator()(const Point& p) const { if (a.y > b.y) return Edge{ b, a }(p); if (p.y == a.y || p.y == b.y) return operator()({ p.x, p.y + epsilon }); if (p.y > b.y || p.y < a.y || p.x > max(a.x, b.x)) return false; if (p.x < min(a.x, b.x)) return true; auto blue = abs(a.x - p.x) > MIN ? (p.y - a.y) / (p.x - a.x) : MAX; auto red = abs(a.x - b.x) > MIN ? (b.y - a.y) / (b.x - a.x) : MAX; return blue >= red; } }; struct Figure { const string name; const initializer_list<Edge> edges; bool contains(const Point& p) const { auto c = 0; for (auto e : edges) if (e(p)) c++; return c % 2 != 0; } template<unsigned char W = 3> void check(const initializer_list<Point>& points, ostream& os) const { os << "Is point inside figure " << name << '?' << endl; for (auto p : points) os << " (" << setw(W) << p.x << ',' << setw(W) << p.y << "): " << boolalpha << contains(p) << endl; os << endl; } }; int main() { const initializer_list<Point> points = { { 5.0, 5.0}, {5.0, 8.0}, {-10.0, 5.0}, {0.0, 5.0}, {10.0, 5.0}, {8.0, 5.0}, {10.0, 10.0} }; const Figure square = { "Square", { {{0.0, 0.0}, {10.0, 0.0}}, {{10.0, 0.0}, {10.0, 10.0}}, {{10.0, 10.0}, {0.0, 10.0}}, {{0.0, 10.0}, {0.0, 0.0}} } }; const Figure square_hole = { "Square hole", { {{0.0, 0.0}, {10.0, 0.0}}, {{10.0, 0.0}, {10.0, 10.0}}, {{10.0, 10.0}, {0.0, 10.0}}, {{0.0, 10.0}, {0.0, 0.0}}, {{2.5, 2.5}, {7.5, 2.5}}, {{7.5, 2.5}, {7.5, 7.5}}, {{7.5, 7.5}, {2.5, 7.5}}, {{2.5, 7.5}, {2.5, 2.5}} } }; const Figure strange = { "Strange", { {{0.0, 0.0}, {2.5, 2.5}}, {{2.5, 2.5}, {0.0, 10.0}}, {{0.0, 10.0}, {2.5, 7.5}}, {{2.5, 7.5}, {7.5, 7.5}}, {{7.5, 7.5}, {10.0, 10.0}}, {{10.0, 10.0}, {10.0, 0.0}}, {{10.0, 0}, {2.5, 2.5}} } }; const Figure exagon = { "Exagon", { {{3.0, 0.0}, {7.0, 0.0}}, {{7.0, 0.0}, {10.0, 5.0}}, {{10.0, 5.0}, {7.0, 10.0}}, {{7.0, 10.0}, {3.0, 10.0}}, {{3.0, 10.0}, {0.0, 5.0}}, {{0.0, 5.0}, {3.0, 0.0}} } }; for(auto f : {square, square_hole, strange, exagon}) f.check(points, cout); return EXIT_SUCCESS; }  Output: As D. ## CoffeeScript Takes a polygon as a list of points joining segments, and creates segments between them.  Point = (@x,@y) -> pointInPoly = (point,poly) -> segments = for pointA, index in poly pointB = poly[(index + 1) % poly.length] [pointA,pointB] intesected = (segment for segment in segments when rayIntesectsSegment(point,segment)) intesected.length % 2 != 0 rayIntesectsSegment = (p,segment) -> [p1,p2] = segment [a,b] = if p1.y < p2.y [p1,p2] else [p2,p1] if p.y == b.y || p.y == a.y p.y += Number.MIN_VALUE if p.y > b.y || p.y < a.y false else if p.x > a.x && p.x > b.x false else if p.x < a.x && p.x < b.x true else mAB = (b.y - a.y) / (b.x - a.x) mAP = (p.y - a.y) / (p.x - a.x) mAP > mAB  ## Common Lisp Points are represented as cons cells whose car is an x value and whose cdr is a y value. A line segment is a cons cell of two points. A polygon is a list of line segments. (defun point-in-polygon (point polygon) (do ((in-p nil)) ((endp polygon) in-p) (when (ray-intersects-segment point (pop polygon)) (setf in-p (not in-p))))) (defun ray-intersects-segment (point segment &optional (epsilon .001)) (destructuring-bind (px . py) point (destructuring-bind ((ax . ay) . (bx . by)) segment (when (< ay by) (rotatef ay by) (rotatef ax bx)) (when (or (= py ay) (= py by)) (incf py epsilon)) (cond ;; point is above, below, or to the right of the rectangle ;; determined by segment; ray does not intesect the segment. ((or (> px (max ax bx)) (> py (max ay by)) (< py (min ay by))) nil) ;; point is to left of the rectangle; ray intersects segment ((< px (min ax bx)) t) ;; point is within the rectangle... (t (let ((m-red (if (= ax bx) nil (/ (- by ay) (- bx ax)))) (m-blue (if (= px ax) nil (/ (- py ay) (- px ax))))) (cond ((null m-blue) t) ((null m-red) nil) (t (>= m-blue m-red)))))))))  Testing code (defparameter *points* #((0 . 0) (10 . 0) (10 . 10) (0 . 10) (2.5 . 2.5) (7.5 . 2.5) (7.5 . 7.5) (2.5 . 7.5) (0 . 5) (10 . 5) (3 . 0) (7 . 0) (7 . 10) (3 . 10))) (defun create-polygon (indices &optional (points *points*)) (loop for (a b) on indices by 'cddr collecting (cons (aref points (1- a)) (aref points (1- b))))) (defun square () (create-polygon '(1 2 2 3 3 4 4 1))) (defun square-hole () (create-polygon '(1 2 2 3 3 4 4 1 5 6 6 7 7 8 8 5))) (defun strange () (create-polygon '(1 5 5 4 4 8 8 7 7 3 3 2 2 5))) (defun exagon () (create-polygon '(11 12 12 10 10 13 13 14 14 9 9 11))) (defparameter *test-points* #((5 . 5) (5 . 8) (-10 . 5) (0 . 5) (10 . 5) (8 . 5) (10 . 10))) (defun test-pip () (dolist (shape '(square square-hole strange exagon)) (print shape) (loop with polygon = (funcall shape) for test-point across *test-points* do (format t "~&~w ~:[outside~;inside ~]." test-point (point-in-polygon test-point polygon)))))  ## D import std.stdio, std.math, std.algorithm; immutable struct Point { double x, y; } immutable struct Edge { Point a, b; } immutable struct Figure { string name; Edge[] edges; } bool contains(in Figure poly, in Point p) pure nothrow @safe @nogc { static bool raySegI(in Point p, in Edge edge) pure nothrow @safe @nogc { enum double epsilon = 0.00001; with (edge) { if (a.y > b.y) //swap(a, b); // if edge is mutable return raySegI(p, Edge(b, a)); if (p.y == a.y || p.y == b.y) //p.y += epsilon; // if p is mutable return raySegI(Point(p.x, p.y + epsilon), edge); if (p.y > b.y || p.y < a.y || p.x > max(a.x, b.x)) return false; if (p.x < min(a.x, b.x)) return true; immutable blue = (abs(a.x - p.x) > double.min_normal) ? ((p.y - a.y) / (p.x - a.x)) : double.max; immutable red = (abs(a.x - b.x) > double.min_normal) ? ((b.y - a.y) / (b.x - a.x)) : double.max; return blue >= red; } } return poly.edges.count!(e => raySegI(p, e)) % 2; } void main() { immutable Figure[] polys = [ {"Square", [ {{ 0.0, 0.0}, {10.0, 0.0}}, {{10.0, 0.0}, {10.0, 10.0}}, {{10.0, 10.0}, { 0.0, 10.0}}, {{ 0.0, 10.0}, { 0.0, 0.0}}]}, {"Square hole", [ {{ 0.0, 0.0}, {10.0, 0.0}}, {{10.0, 0.0}, {10.0, 10.0}}, {{10.0, 10.0}, { 0.0, 10.0}}, {{ 0.0, 10.0}, { 0.0, 0.0}}, {{ 2.5, 2.5}, { 7.5, 2.5}}, {{ 7.5, 2.5}, { 7.5, 7.5}}, {{ 7.5, 7.5}, { 2.5, 7.5}}, {{ 2.5, 7.5}, { 2.5, 2.5}}]}, {"Strange", [ {{ 0.0, 0.0}, { 2.5, 2.5}}, {{ 2.5, 2.5}, { 0.0, 10.0}}, {{ 0.0, 10.0}, { 2.5, 7.5}}, {{ 2.5, 7.5}, { 7.5, 7.5}}, {{ 7.5, 7.5}, {10.0, 10.0}}, {{10.0, 10.0}, {10.0, 0.0}}, {{10.0, 0}, { 2.5, 2.5}}]}, {"Exagon", [ {{ 3.0, 0.0}, { 7.0, 0.0}}, {{ 7.0, 0.0}, {10.0, 5.0}}, {{10.0, 5.0}, { 7.0, 10.0}}, {{ 7.0, 10.0}, { 3.0, 10.0}}, {{ 3.0, 10.0}, { 0.0, 5.0}}, {{ 0.0, 5.0}, { 3.0, 0.0}}]} ]; immutable Point[] testPoints = [{ 5, 5}, {5, 8}, {-10, 5}, {0, 5}, {10, 5}, {8, 5}, { 10, 10}]; foreach (immutable poly; polys) { writefln(Is point inside figure "%s"?, poly.name); foreach (immutable p; testPoints) writefln(" (%3s, %2s): %s", p.x, p.y, contains(poly, p)); writeln; } }  Output: Is point inside figure "Square"? ( 5, 5): true ( 5, 8): true (-10, 5): false ( 0, 5): false ( 10, 5): true ( 8, 5): true ( 10, 10): false Is point inside figure "Square hole"? ( 5, 5): false ( 5, 8): true (-10, 5): false ( 0, 5): false ( 10, 5): true ( 8, 5): true ( 10, 10): false Is point inside figure "Strange"? ( 5, 5): true ( 5, 8): false (-10, 5): false ( 0, 5): false ( 10, 5): true ( 8, 5): true ( 10, 10): false Is point inside figure "Exagon"? ( 5, 5): true ( 5, 8): true (-10, 5): false ( 0, 5): false ( 10, 5): true ( 8, 5): true ( 10, 10): false ## Factor To test whether a ray intersects a line, we test that the starting point is between the endpoints in y value, and that it is to the left of the point on the segment with the same y value. Note that this implementation does not support polygons with horizontal edges. USING: kernel prettyprint sequences arrays math math.vectors ; IN: raycasting : between ( a b x -- ? ) [ last ] tri@ [ < ] curry bi@ xor ; : lincomb ( a b x -- w ) 3dup [ last ] tri@ [ - ] curry bi@ [ drop ] 2dip neg 2dup + [ / ] curry bi@ [ [ v*n ] curry ] bi@ bi* v+ ; : leftof ( a b x -- ? ) dup [ lincomb ] dip [ first ] bi@ > ; : ray ( a b x -- ? ) [ between ] [ leftof ] 3bi and ; : raycast ( poly x -- ? ) [ dup first suffix [ rest-slice ] [ but-last-slice ] bi ] dip [ ray ] curry 2map f [ xor ] reduce ;  Usage: ( scratchpad ) CONSTANT: square { { -2 -1 } { 1 -2 } { 2 1 } { -1 2 } } ( scratchpad ) square { 0 0 } raycast . t ( scratchpad ) square { 5 5 } raycast . f ( scratchpad ) square { 2 0 } raycast . f  ## Fortran Works with: Fortran version 95 and later The following code uses the Points_Module defined here. This module defines "polygons". module Polygons use Points_Module implicit none type polygon type(point), dimension(:), allocatable :: points integer, dimension(:), allocatable :: vertices end type polygon contains function create_polygon(pts, vt) type(polygon) :: create_polygon type(point), dimension(:), intent(in) :: pts integer, dimension(:), intent(in) :: vt integer :: np, nv np = size(pts,1) nv = size(vt,1) allocate(create_polygon%points(np), create_polygon%vertices(nv)) create_polygon%points = pts create_polygon%vertices = vt end function create_polygon subroutine free_polygon(pol) type(polygon), intent(inout) :: pol deallocate(pol%points, pol%vertices) end subroutine free_polygon end module Polygons  The ray casting algorithm module: module Ray_Casting_Algo use Polygons implicit none real, parameter, private :: eps = 0.00001 private :: ray_intersects_seg contains function ray_intersects_seg(p0, a0, b0) result(intersect) type(point), intent(in) :: p0, a0, b0 logical :: intersect type(point) :: a, b, p real :: m_red, m_blue p = p0 ! let variable "a" be the point with smallest y coordinate if ( a0%y > b0%y ) then b = a0 a = b0 else a = a0 b = b0 end if if ( (p%y == a%y) .or. (p%y == b%y) ) p%y = p%y + eps intersect = .false. if ( (p%y > b%y) .or. (p%y < a%y) ) return if ( p%x > max(a%x, b%x) ) return if ( p%x < min(a%x, b%x) ) then intersect = .true. else if ( abs(a%x - b%x) > tiny(a%x) ) then m_red = (b%y - a%y) / (b%x - a%x) else m_red = huge(m_red) end if if ( abs(a%x - p%x) > tiny(a%x) ) then m_blue = (p%y - a%y) / (p%x - a%x) else m_blue = huge(m_blue) end if if ( m_blue >= m_red ) then intersect = .true. else intersect = .false. end if end if end function ray_intersects_seg function point_is_inside(p, pol) result(inside) logical :: inside type(point), intent(in) :: p type(polygon), intent(in) :: pol integer :: i, cnt, pa, pb cnt = 0 do i = lbound(pol%vertices,1), ubound(pol%vertices,1), 2 pa = pol%vertices(i) pb = pol%vertices(i+1) if ( ray_intersects_seg(p, pol%points(pa), pol%points(pb)) ) cnt = cnt + 1 end do inside = .true. if ( mod(cnt, 2) == 0 ) then inside = .false. end if end function point_is_inside end module Ray_Casting_Algo  Testing program Pointpoly use Points_Module use Ray_Casting_Algo implicit none character(len=16), dimension(4) :: names type(polygon), dimension(4) :: polys type(point), dimension(14) :: pts type(point), dimension(7) :: p integer :: i, j pts = (/ point(0,0), point(10,0), point(10,10), point(0,10), & point(2.5,2.5), point(7.5,2.5), point(7.5,7.5), point(2.5,7.5), & point(0,5), point(10,5), & point(3,0), point(7,0), point(7,10), point(3,10) /) polys(1) = create_polygon(pts, (/ 1,2, 2,3, 3,4, 4,1 /) ) polys(2) = create_polygon(pts, (/ 1,2, 2,3, 3,4, 4,1, 5,6, 6,7, 7,8, 8,5 /) ) polys(3) = create_polygon(pts, (/ 1,5, 5,4, 4,8, 8,7, 7,3, 3,2, 2,5 /) ) polys(4) = create_polygon(pts, (/ 11,12, 12,10, 10,13, 13,14, 14,9, 9,11 /) ) names = (/ "square", "square hole", "strange", "hexagon" /) p = (/ point(5,5), point(5, 8), point(-10, 5), point(0,5), point(10,5), & point(8,5), point(10,10) /) do j = 1, size(p) do i = 1, size(polys) write(*, "('point (',F8.2,',',F8.2,') is inside ',A,'? ', L)") & p(j)%x, p(j)%y, names(i), point_is_inside(p(j), polys(i)) end do print *, "" end do do i = 1, size(polys) call free_polygon(polys(i)) end do end program Pointpoly  ## FreeBASIC Inpolygon by Winding number method Type Point As Single x,y End Type Function inpolygon(p1() As Point,p2 As Point) As Integer #Macro isleft2(L,p) -Sgn( (L(1).x-L(2).x)*(p.y-L(2).y) - (p.x-L(2).x)*(L(1).y-L(2).y)) #EndMacro Dim As Integer index,nextindex Dim k As Integer=UBound(p1)+1 Dim send (1 To 2) As Point Dim wn As Integer=0 For n As Integer=1 To UBound(p1) index=n Mod k:nextindex=(n+1) Mod k If nextindex=0 Then nextindex=1 send(1).x=p1(index).x:send(2).x=p1(nextindex).x send(1).y=p1(index).y:send(2).y=p1(nextindex).y If p1(index).y<=p2.y Then If p1(nextindex).y>p2.y Then If isleft2(send,p2)>=0 Then '= wn=wn+1 End If End If Else If p1(nextindex).y<=p2.y Then If isleft2(send,p2)<=0 Then'= wn=wn-1 End If End If End If Next n Return wn End Function Dim As Point square(1 To 4) ={(0,0),(10,0),(10,10),(0,10)} Dim As Point hole(1 To 4) ={(2.5,2.5),(7.5,2.5),(7.5,7.5),(2.5,7.5)} Dim As Point strange(1 To 8) ={(0,0),(2.5,2.5),(0,10),(2.5,7.5),_ (7.5,7.5),(10,10),(10,0),(2.5,2.5)} Dim As Point exagon(1 To 6) ={(3,0),(7,0),(10,5),(7,10),(3,10),(0,5)} 'printouts For z As Integer=1 To 4 Select Case z Case 1: Print "squared" Print "(5,5) " ;Tab(12); If inpolygon(square(),Type<Point>(5,5)) Then Print "in" Else Print "out" Print "(5,8) " ;Tab(12); If inpolygon(square(),Type<Point>(5,8)) Then Print "in" Else Print "out" Print "(-10,5) " ;Tab(12); If inpolygon(square(),Type<Point>(-10,5)) Then Print "in" Else Print "out" Print "(0,5) " ;Tab(12); If inpolygon(square(),Type<Point>(0,5)) Then Print "in" Else Print "out" Print "(10,5) " ;Tab(12); If inpolygon(square(),Type<Point>(10,5)) Then Print "in" Else Print "out" Print "(8,5) " ;Tab(12); If inpolygon(square(),Type<Point>(8,5)) Then Print "in" Else Print "out" Print "(10,10) " ;Tab(12); If inpolygon(square(),Type<Point>(10,10)) Then Print "in" Else Print "out" Print Case 2:Print "squared hole" Print "(5,5) " ;Tab(12); If Not inpolygon(hole(),Type<Point>(5,5)) And inpolygon(square(),Type<Point>(5,5)) Then Print "in" Else Print "out" Print "(5,8) " ;Tab(12); If Not inpolygon(hole(),Type<Point>(5,8)) And inpolygon(square(),Type<Point>(5,8))Then Print "in" Else Print "out" Print "(-10,5) " ;Tab(12); If Not inpolygon(hole(),Type<Point>(-10,5))And inpolygon(square(),Type<Point>(-10,5)) Then Print "in" Else Print "out" Print "(0,5) " ;Tab(12); If Not inpolygon(hole(),Type<Point>(0,5))And inpolygon(square(),Type<Point>(0,5)) Then Print "in" Else Print "out" Print "(10,5) " ;Tab(12); If Not inpolygon(hole(),Type<Point>(10,5))And inpolygon(square(),Type<Point>(10,5)) Then Print "in" Else Print "out" Print "(8,5) " ;Tab(12); If Not inpolygon(hole(),Type<Point>(8,5))And inpolygon(square(),Type<Point>(8,5)) Then Print "in" Else Print "out" Print "(10,10) " ;Tab(12); If Not inpolygon(hole(),Type<Point>(10,10))And inpolygon(square(),Type<Point>(10,10)) Then Print "in" Else Print "out" Print Case 3:Print "strange" Print "(5,5) " ;Tab(12); If inpolygon(strange(),Type<Point>(5,5)) Then Print "in" Else Print "out" Print "(5,8) " ;Tab(12); If inpolygon(strange(),Type<Point>(5,8)) Then Print "in" Else Print "out" Print "(-10,5) " ;Tab(12); If inpolygon(strange(),Type<Point>(-10,5)) Then Print "in" Else Print "out" Print "(0,5) " ;Tab(12); If inpolygon(strange(),Type<Point>(0,5)) Then Print "in" Else Print "out" Print "(10,5) " ;Tab(12); If inpolygon(strange(),Type<Point>(10,5)) Then Print "in" Else Print "out" Print "(8,5) " ;Tab(12); If inpolygon(strange(),Type<Point>(8,5)) Then Print "in" Else Print "out" Print "(10,10) " ;Tab(12); If inpolygon(strange(),Type<Point>(10,10)) Then Print "in" Else Print "out" Print Case 4:Print "exagon" Print "(5,5) " ;Tab(12); If inpolygon(exagon(),Type<Point>(5,5)) Then Print "in" Else Print "out" Print "(5,8) " ;Tab(12); If inpolygon(exagon(),Type<Point>(5,8)) Then Print "in" Else Print "out" Print "(-10,5) " ;Tab(12); If inpolygon(exagon(),Type<Point>(-10,5)) Then Print "in" Else Print "out" Print "(0,5) " ;Tab(12); If inpolygon(exagon(),Type<Point>(0,5)) Then Print "in" Else Print "out" Print "(10,5) " ;Tab(12); If inpolygon(exagon(),Type<Point>(10,5)) Then Print "in" Else Print "out" Print "(8,5) " ;Tab(12); If inpolygon(exagon(),Type<Point>(8,5)) Then Print "in" Else Print "out" Print "(10,10) " ;Tab(12); If inpolygon(exagon(),Type<Point>(10,10)) Then Print "in" Else Print "out" Print End Select Next z Sleep Output: squared (5,5) in (5,8) in (-10,5) out (0,5) out (10,5) in (8,5) in (10,10) out squared hole (5,5) out (5,8) in (-10,5) out (0,5) out (10,5) in (8,5) in (10,10) out strange (5,5) in (5,8) out (-10,5) out (0,5) out (10,5) in (8,5) in (10,10) out exagon (5,5) in (5,8) in (-10,5) out (0,5) out (10,5) in (8,5) in (10,10) out ## Go Segment solution, task algorithm The first solution given here follows the model of most other solutions on the page in defining a polygon as a list of segments. Unfortunately this representation does not require that the polygon is closed. Input to the ray-casting algorithm, as noted in the WP article though, is specified to be a closed polygon. The "strange" shape defined here is not a closed polygon and so gives incorrect results against some points. (Graphically it may appear closed but mathematically it needs an additional segment returning to the starting point.) package main import ( "fmt" "math" ) type xy struct { x, y float64 } type seg struct { p1, p2 xy } type poly struct { name string sides []seg } func inside(pt xy, pg poly) (i bool) { for _, side := range pg.sides { if rayIntersectsSegment(pt, side) { i = !i } } return } func rayIntersectsSegment(p xy, s seg) bool { var a, b xy if s.p1.y < s.p2.y { a, b = s.p1, s.p2 } else { a, b = s.p2, s.p1 } for p.y == a.y || p.y == b.y { p.y = math.Nextafter(p.y, math.Inf(1)) } if p.y < a.y || p.y > b.y { return false } if a.x > b.x { if p.x > a.x { return false } if p.x < b.x { return true } } else { if p.x > b.x { return false } if p.x < a.x { return true } } return (p.y-a.y)/(p.x-a.x) >= (b.y-a.y)/(b.x-a.x) } var ( p1 = xy{0, 0} p2 = xy{10, 0} p3 = xy{10, 10} p4 = xy{0, 10} p5 = xy{2.5, 2.5} p6 = xy{7.5, 2.5} p7 = xy{7.5, 7.5} p8 = xy{2.5, 7.5} p9 = xy{0, 5} p10 = xy{10, 5} p11 = xy{3, 0} p12 = xy{7, 0} p13 = xy{7, 10} p14 = xy{3, 10} ) var tpg = []poly{ {"square", []seg{{p1, p2}, {p2, p3}, {p3, p4}, {p4, p1}}}, {"square hole", []seg{{p1, p2}, {p2, p3}, {p3, p4}, {p4, p1}, {p5, p6}, {p6, p7}, {p7, p8}, {p8, p5}}}, {"strange", []seg{{p1, p5}, {p5, p4}, {p4, p8}, {p8, p7}, {p7, p3}, {p3, p2}, {p2, p5}}}, {"exagon", []seg{{p11, p12}, {p12, p10}, {p10, p13}, {p13, p14}, {p14, p9}, {p9, p11}}}, } var tpt = []xy{ // test points common in other solutions on this page {5, 5}, {5, 8}, {-10, 5}, {0, 5}, {10, 5}, {8, 5}, {10, 10}, // test points that show the problem with "strange" {1, 2}, {2, 1}, } func main() { for _, pg := range tpg { fmt.Printf("%s:\n", pg.name) for _, pt := range tpt { fmt.Println(pt, inside(pt, pg)) } } }  Output: square: {5 5} true {5 8} true {-10 5} false {0 5} false {10 5} true {8 5} true {10 10} false {1 2} true {2 1} true square hole: {5 5} false {5 8} true {-10 5} false {0 5} false {10 5} true {8 5} true {10 10} false {1 2} true {2 1} true strange: {5 5} true {5 8} false {-10 5} false {0 5} false {10 5} true {8 5} true {10 10} false {1 2} true {2 1} false exagon: {5 5} true {5 8} true {-10 5} false {0 5} false {10 5} true {8 5} true {10 10} false {1 2} false {2 1} false  Closed polygon solution Here input is given as a list of N vertices defining N segments, where one segment extends from each vertex to the next, and one more extends from the last vertex to the first. In the case of the "strange" shape, this mathematically closes the polygon and allows the program to give correct results. package main import ( "math" "fmt" ) type xy struct { x, y float64 } type closedPoly struct { name string vert []xy } func inside(pt xy, pg closedPoly) bool { if len(pg.vert) < 3 { return false } in := rayIntersectsSegment(pt, pg.vert[len(pg.vert)-1], pg.vert[0]) for i := 1; i < len(pg.vert); i++ { if rayIntersectsSegment(pt, pg.vert[i-1], pg.vert[i]) { in = !in } } return in } func rayIntersectsSegment(p, a, b xy) bool { if a.y > b.y { a, b = b, a } for p.y == a.y || p.y == b.y { p.y = math.Nextafter(p.y, math.Inf(1)) } if p.y < a.y || p.y > b.y { return false } if a.x > b.x { if p.x > a.x { return false } if p.x < b.x { return true } } else { if p.x > b.x { return false } if p.x < a.x { return true } } return (p.y-a.y)/(p.x-a.x) >= (b.y-a.y)/(b.x-a.x) } var tpg = []closedPoly{ {"square", []xy{{0, 0}, {10, 0}, {10, 10}, {0, 10}}}, {"square hole", []xy{{0, 0}, {10, 0}, {10, 10}, {0, 10}, {0, 0}, {2.5, 2.5}, {7.5, 2.5}, {7.5, 7.5}, {2.5, 7.5}, {2.5, 2.5}}}, {"strange", []xy{{0, 0}, {2.5, 2.5}, {0, 10}, {2.5, 7.5}, {7.5, 7.5}, {10, 10}, {10, 0}, {2.5, 2.5}}}, {"exagon", []xy{{3, 0}, {7, 0}, {10, 5}, {7, 10}, {3, 10}, {0, 5}}}, } var tpt = []xy{{1, 2}, {2, 1}} func main() { for _, pg := range tpg { fmt.Printf("%s:\n", pg.name) for _, pt := range tpt { fmt.Println(pt, inside(pt, pg)) } } }  Output: square: {1 2} true {2 1} true square hole: {1 2} true {2 1} true strange: {1 2} false {2 1} false exagon: {1 2} false {2 1} false  PNPoly algorithm This solution replaces the rayIntersectsSegment function above with the expression from the popular PNPoly algorithm described at https://www.ecse.rpi.edu/Homepages/wrf/Research/Short_Notes/pnpoly.html. The expression is not only simpler but more accurate. This solution is preferred over the two above. package main import "fmt" type xy struct { x, y float64 } type closedPoly struct { name string vert []xy } func inside(pt xy, pg closedPoly) bool { if len(pg.vert) < 3 { return false } in := rayIntersectsSegment(pt, pg.vert[len(pg.vert)-1], pg.vert[0]) for i := 1; i < len(pg.vert); i++ { if rayIntersectsSegment(pt, pg.vert[i-1], pg.vert[i]) { in = !in } } return in } func rayIntersectsSegment(p, a, b xy) bool { return (a.y > p.y) != (b.y > p.y) && p.x < (b.x-a.x)*(p.y-a.y)/(b.y-a.y)+a.x } var tpg = []closedPoly{ {"square", []xy{{0, 0}, {10, 0}, {10, 10}, {0, 10}}}, {"square hole", []xy{{0, 0}, {10, 0}, {10, 10}, {0, 10}, {0, 0}, {2.5, 2.5}, {7.5, 2.5}, {7.5, 7.5}, {2.5, 7.5}, {2.5, 2.5}}}, {"strange", []xy{{0, 0}, {2.5, 2.5}, {0, 10}, {2.5, 7.5}, {7.5, 7.5}, {10, 10}, {10, 0}, {2.5, 2.5}}}, {"exagon", []xy{{3, 0}, {7, 0}, {10, 5}, {7, 10}, {3, 10}, {0, 5}}}, } var tpt = []xy{{1, 2}, {2, 1}} func main() { for _, pg := range tpg { fmt.Printf("%s:\n", pg.name) for _, pt := range tpt { fmt.Println(pt, inside(pt, pg)) } } }  ## Haskell import Data.Ratio type Point = (Rational, Rational) type Polygon = [Point] data Line = Sloped {lineSlope, lineYIntercept :: Rational} | Vert {lineXIntercept :: Rational} polygonSides :: Polygon -> [(Point, Point)] polygonSides poly@(p1 : ps) = zip poly$ ps ++ [p1]

intersects :: Point -> Line -> Bool
{- @intersects (px, py) l@ is true if the ray {(x, py) | x ≥ px}
intersects l. -}
intersects (px, _)  (Vert xint)  = px <= xint
intersects (px, py) (Sloped m b) | m < 0     = py <= m * px + b
| otherwise = py >= m * px + b

onLine :: Point -> Line -> Bool
{- Is the point on the line? -}
onLine (px, _)  (Vert xint)  = px == xint
onLine (px, py) (Sloped m b) = py == m * px + b

carrier :: (Point, Point) -> Line
{- Finds the line containing the given line segment. -}
carrier ((ax, ay), (bx, by)) | ax == bx  = Vert ax
| otherwise = Sloped slope yint
where slope = (ay - by) / (ax - bx)
yint = ay - slope * ax

between :: Ord a => a -> a -> a -> Bool
between x a b | a > b     = b <= x && x <= a
| otherwise = a <= x && x <= b

inPolygon :: Point -> Polygon -> Bool
inPolygon p@(px, py) = f 0 . polygonSides
where f n []                             = odd n
f n (side : sides) | far           = f n       sides
| onSegment     = True
| rayIntersects = f (n + 1) sides
| otherwise     = f n       sides
where far = not $between py ay by onSegment | ay == by = between px ax bx | otherwise = p onLine line rayIntersects = intersects p line && (py /= ay || by < py) && (py /= by || ay < py) ((ax, ay), (bx, by)) = side line = carrier side  ## J NB.*crossPnP v point in closed polygon, crossing number NB. bool=. points crossPnP polygon crossPnP=: 4 : 0"2 'X Y'=. |:x 'x0 y0 x1 y1'=. |:2 ,/\^:(2={:@$@]) y
p1=. ((y0<:/Y)*. y1>/Y) +. (y0>/Y)*. y1<:/Y
p2=. (x0-/X) < (x0-x1) * (y0-/Y) % (y0 - y1)
2|+/ p1*.p2
)


Sample data:

SQUAREV=:          0   0   , 10  0   , 10  10  ,: 0   10
SQUAREV=: SQUAREV, 2.5 2.5 , 7.5 0.1 , 7.5 7.5 ,: 2.5 7.5

ESAV=: 3 0 , 7 0 , 10 5 , 7 10 , 3 10 ,: 0 5

ESA=:        (0 1,1 2,2 3,3 4,4 5,:5 0) , .{ ESAV
SQUARE=:     (0 1,1 2,2 3,:3 0)         , .{ SQUAREV
SQUAREHOLE=: (0 1,1 2,2 3,3 0,4 5,5 6,6 7,:7 4) , .{ SQUAREV
STRANGE=:    (0 4,4 3,3 7,7 6,6 2,2 1,1 5,:5 0) , .{ SQUAREV

POINTS=: 5 5,5 8,2 2,0 0,10 10,2.5 2.5,0.01 5,2.2 7.4,0 5,10 5,:_4 10


Testing:

   (<POINTS) crossPnP every ESA;SQUARE;SQUAREHOLE;STRANGE
1 1 1 0 0 1 1 1 0 1 0
1 1 1 0 0 1 1 1 0 1 0
0 1 1 0 0 1 1 1 0 1 0
1 0 0 0 0 0 0 1 0 1 0


## Java

import static java.lang.Math.*;

public class RayCasting {

static boolean intersects(int[] A, int[] B, double[] P) {
if (A[1] > B[1])
return intersects(B, A, P);

if (P[1] == A[1] || P[1] == B[1])
P[1] += 0.0001;

if (P[1] > B[1] || P[1] < A[1] || P[0] >= max(A[0], B[0]))
return false;

if (P[0] < min(A[0], B[0]))
return true;

double red = (P[1] - A[1]) / (double) (P[0] - A[0]);
double blue = (B[1] - A[1]) / (double) (B[0] - A[0]);
return red >= blue;
}

static boolean contains(int[][] shape, double[] pnt) {
boolean inside = false;
int len = shape.length;
for (int i = 0; i < len; i++) {
if (intersects(shape[i], shape[(i + 1) % len], pnt))
inside = !inside;
}
return inside;
}

public static void main(String[] a) {
double[][] testPoints = {{10, 10}, {10, 16}, {-20, 10}, {0, 10},
{20, 10}, {16, 10}, {20, 20}};

for (int[][] shape : shapes) {
for (double[] pnt : testPoints)
System.out.printf("%7s ", contains(shape, pnt));
System.out.println();
}
}

final static int[][] square = {{0, 0}, {20, 0}, {20, 20}, {0, 20}};

final static int[][] squareHole = {{0, 0}, {20, 0}, {20, 20}, {0, 20},
{5, 5}, {15, 5}, {15, 15}, {5, 15}};

final static int[][] strange = {{0, 0}, {5, 5}, {0, 20}, {5, 15}, {15, 15},
{20, 20}, {20, 0}};

final static int[][] hexagon = {{6, 0}, {14, 0}, {20, 10}, {14, 20},
{6, 20}, {0, 10}};

final static int[][][] shapes = {square, squareHole, strange, hexagon};
}


true    true   false    true   false    true   false
false    true   false   false   false    true   false
true   false   false   false   false    true   false
true    true   false   false   false    true   false


## JavaScript

/**
* @return {boolean} true if (lng, lat) is in bounds
*/
function contains(bounds, lat, lng) {
//https://rosettacode.org/wiki/Ray-casting_algorithm
var count = 0;
for (var b = 0; b < bounds.length; b++) {
var vertex1 = bounds[b];
var vertex2 = bounds[(b + 1) % bounds.length];
if (west(vertex1, vertex2, lng, lat))
++count;
}
return count % 2;

/**
* @return {boolean} true if (x,y) is west of the line segment connecting A and B
*/
function west(A, B, x, y) {
if (A.y <= B.y) {
if (y <= A.y || y > B.y ||
x >= A.x && x >= B.x) {
return false;
} else if (x < A.x && x < B.x) {
return true;
} else {
return (y - A.y) / (x - A.x) > (B.y - A.y) / (B.x - A.x);
}
} else {
return west(B, A, x, y);
}
}
}

var square = {name: 'square', bounds: [{x: 0, y: 0}, {x: 20, y: 0}, {x: 20, y: 20}, {x: 0, y: 20}]};
var squareHole = {
name: 'squareHole',
bounds: [{x: 0, y: 0}, {x: 20, y: 0}, {x: 20, y: 20}, {x: 0, y: 20}, {x: 5, y: 5}, {x: 15, y: 5}, {x: 15, y: 15}, {x: 5, y: 15}]
};
var strange = {
name: 'strange',
bounds: [{x: 0, y: 0}, {x: 5, y: 5}, {x: 0, y: 20}, {x: 5, y: 15}, {x: 15, y: 15}, {x: 20, y: 20}, {x: 20, y: 0}]
};
var hexagon = {
name: 'hexagon',
bounds: [{x: 6, y: 0}, {x: 14, y: 0}, {x: 20, y: 10}, {x: 14, y: 20}, {x: 6, y: 20}, {x: 0, y: 10}]
};

var shapes = [square, squareHole, strange, hexagon];
var testPoints = [{lng: 10, lat: 10}, {lng: 10, lat: 16}, {lng: -20, lat: 10},
{lng: 0, lat: 10}, {lng: 20, lat: 10}, {lng: 16, lat: 10}, {lng: 20, lat: 20}];

for (var s = 0; s < shapes.length; s++) {
var shape = shapes[s];
for (var tp = 0; tp < testPoints.length; tp++) {
var testPoint = testPoints[tp];
console.log(JSON.stringify(testPoint) + '\tin ' + shape.name + '\t' + contains(shape.bounds, testPoint.lat, testPoint.lng));
}
}


## Julia

Translation of: Python

Module:

module RayCastings

export Point

struct Point{T}
x::T
y::T
end
Base.show(io::IO, p::Point) = print(io, "($(p.x),$(p.y))")

const Edge = Tuple{Point{T}, Point{T}} where T
Base.show(io::IO, e::Edge) = print(io, "$(e[1]) ∘-∘$(e[2])")

function rayintersectseg(p::Point{T}, edge::Edge{T}) where T
a, b = edge
if a.y > b.y
a, b = b, a
end
if p.y ∈ (a.y, b.y)
p = Point(p.x, p.y + eps(p.y))
end

rst = false
if (p.y > b.y || p.y < a.y) || (p.x > max(a.x, b.x))
return false
end

if p.x < min(a.x, b.x)
rst = true
else
mred = (b.y - a.y) / (b.x - a.x)
mblu = (p.y - a.y) / (p.x - a.x)
rst = mblu ≥ mred
end

return rst
end

isinside(poly::Vector{Tuple{Point{T}, Point{T}}}, p::Point{T}) where T =
isodd(count(edge -> rayintersectseg(p, edge), poly))

connect(a::Point{T}, b::Point{T}...) where T =
[(a, b) for (a, b) in zip(vcat(a, b...), vcat(b..., a))]

end  # module RayCastings


Main:

using Printf

let A = Point(0.0, 0.0),
B = Point(0.0, 10.0),
C = Point(10.0, 10.0),
D = Point(10.0, 0.0),
E = Point(2.5, 2.5),
F = Point(2.5, 7.5),
G = Point(7.5, 7.5),
H = Point(7.5, 2.5),
I = Point(3.0, 0.0),
J = Point(7.0, 0.0),
K = Point(10.0, 5.0),
L = Point(7.0, 10.0),
M = Point(3.0, 10.0),
N = Point(0.0, 5.0),
testpts = (Point(5.0, 5.0), Point(5.0, 8.0), Point(-10.0, 5.0), Point(0.0, 5.0),
Point(10.0, 5.0), Point(8.0, 5.0), Point(10.0, 10.0))

square = RayCastings.connect(A, B, C, D)
square_withhole = vcat(square, RayCastings.connect(E, F, G, H))
strange = RayCastings.connect(A, E, B, F, G, C, D, E)
exagon = RayCastings.connect(I, J, K, L, M, N)

println("\n# TESTING WHETHER POINTS ARE WITHIN POLYGONS")
for poly in (square, square_withhole, strange, exagon)
println("\nEdges: \n - ", join(poly, "\n - "))
println("Inside/outside:")
for p in testpts
@printf(" - %-12s is %s\n", p, RayCastings.isinside(poly, p) ? "inside" : "outside")
end
end
end

Output:
# TESTING WHETHER POINTS ARE WITHIN POLYGONS

Edges:
- (0.0, 0.0) ∘-∘ (0.0, 10.0)
- (0.0, 10.0) ∘-∘ (10.0, 10.0)
- (10.0, 10.0) ∘-∘ (10.0, 0.0)
- (10.0, 0.0) ∘-∘ (0.0, 0.0)
Inside/outside:
- (5.0, 5.0)   is inside
- (5.0, 8.0)   is inside
- (-10.0, 5.0) is outside
- (0.0, 5.0)   is outside
- (10.0, 5.0)  is inside
- (8.0, 5.0)   is inside
- (10.0, 10.0) is outside

Edges:
- (0.0, 0.0) ∘-∘ (0.0, 10.0)
- (0.0, 10.0) ∘-∘ (10.0, 10.0)
- (10.0, 10.0) ∘-∘ (10.0, 0.0)
- (10.0, 0.0) ∘-∘ (0.0, 0.0)
- (2.5, 2.5) ∘-∘ (2.5, 7.5)
- (2.5, 7.5) ∘-∘ (7.5, 7.5)
- (7.5, 7.5) ∘-∘ (7.5, 2.5)
- (7.5, 2.5) ∘-∘ (2.5, 2.5)
Inside/outside:
- (5.0, 5.0)   is outside
- (5.0, 8.0)   is inside
- (-10.0, 5.0) is outside
- (0.0, 5.0)   is outside
- (10.0, 5.0)  is inside
- (8.0, 5.0)   is inside
- (10.0, 10.0) is outside

Edges:
- (0.0, 0.0) ∘-∘ (2.5, 2.5)
- (2.5, 2.5) ∘-∘ (0.0, 10.0)
- (0.0, 10.0) ∘-∘ (2.5, 7.5)
- (2.5, 7.5) ∘-∘ (7.5, 7.5)
- (7.5, 7.5) ∘-∘ (10.0, 10.0)
- (10.0, 10.0) ∘-∘ (10.0, 0.0)
- (10.0, 0.0) ∘-∘ (2.5, 2.5)
- (2.5, 2.5) ∘-∘ (0.0, 0.0)
Inside/outside:
- (5.0, 5.0)   is inside
- (5.0, 8.0)   is outside
- (-10.0, 5.0) is outside
- (0.0, 5.0)   is outside
- (10.0, 5.0)  is inside
- (8.0, 5.0)   is inside
- (10.0, 10.0) is outside

Edges:
- (3.0, 0.0) ∘-∘ (7.0, 0.0)
- (7.0, 0.0) ∘-∘ (10.0, 5.0)
- (10.0, 5.0) ∘-∘ (7.0, 10.0)
- (7.0, 10.0) ∘-∘ (3.0, 10.0)
- (3.0, 10.0) ∘-∘ (0.0, 5.0)
- (0.0, 5.0) ∘-∘ (3.0, 0.0)
Inside/outside:
- (5.0, 5.0)   is inside
- (5.0, 8.0)   is inside
- (-10.0, 5.0) is outside
- (0.0, 5.0)   is outside
- (10.0, 5.0)  is inside
- (8.0, 5.0)   is inside
- (10.0, 10.0) is outside

## Kotlin

Translation of: D
import java.lang.Double.MAX_VALUE
import java.lang.Double.MIN_VALUE
import java.lang.Math.abs

data class Point(val x: Double, val y: Double)

data class Edge(val s: Point, val e: Point) {
operator fun invoke(p: Point) : Boolean = when {
s.y > e.y -> Edge(e, s).invoke(p)
p.y == s.y || p.y == e.y -> invoke(Point(p.x, p.y + epsilon))
p.y > e.y || p.y < s.y || p.x > Math.max(s.x, e.x) -> false
p.x < Math.min(s.x, e.x) -> true
else -> {
val blue = if (abs(s.x - p.x) > MIN_VALUE) (p.y - s.y) / (p.x - s.x) else MAX_VALUE
val red = if (abs(s.x - e.x) > MIN_VALUE) (e.y - s.y) / (e.x - s.x) else MAX_VALUE
blue >= red
}
}

val epsilon = 0.00001
}

class Figure(val name: String, val edges: Array<Edge>) {
operator fun contains(p: Point) = edges.count({ it(p) }) % 2 != 0
}

object Ray_casting {
fun check(figures : Array<Figure>, points : List<Point>) {
println("points: " + points)
figures.forEach { f ->
println("figure: " + f.name)
f.edges.forEach { println("        " + it) }
println("result: " + (points.map { it in f }))
}
}
}

fun main(args: Array<String>) {
val figures = arrayOf(Figure("Square", arrayOf(Edge(Point(0.0, 0.0), Point(10.0, 0.0)), Edge(Point(10.0, 0.0), Point(10.0, 10.0)),
Edge(Point(10.0, 10.0), Point(0.0, 10.0)),Edge(Point(0.0, 10.0), Point(0.0, 0.0)))),
Figure("Square hole", arrayOf(Edge(Point(0.0, 0.0), Point(10.0, 0.0)), Edge(Point(10.0, 0.0), Point(10.0, 10.0)),
Edge(Point(10.0, 10.0), Point(0.0, 10.0)), Edge(Point(0.0, 10.0), Point(0.0, 0.0)), Edge(Point(2.5, 2.5), Point(7.5, 2.5)),
Edge(Point(7.5, 2.5), Point(7.5, 7.5)),Edge(Point(7.5, 7.5), Point(2.5, 7.5)), Edge(Point(2.5, 7.5), Point(2.5, 2.5)))),
Figure("Strange", arrayOf(Edge(Point(0.0, 0.0), Point(2.5, 2.5)), Edge(Point(2.5, 2.5), Point(0.0, 10.0)),
Edge(Point(0.0, 10.0), Point(2.5, 7.5)), Edge(Point(2.5, 7.5), Point(7.5, 7.5)), Edge(Point(7.5, 7.5), Point(10.0, 10.0)),
Edge(Point(10.0, 10.0), Point(10.0, 0.0)), Edge(Point(10.0, 0.0), Point(2.5, 2.5)))),
Figure("Exagon", arrayOf(Edge(Point(3.0, 0.0), Point(7.0, 0.0)), Edge(Point(7.0, 0.0), Point(10.0, 5.0)), Edge(Point(10.0, 5.0), Point(7.0, 10.0)),
Edge(Point(7.0, 10.0), Point(3.0, 10.0)), Edge(Point(3.0, 10.0), Point(0.0, 5.0)), Edge(Point(0.0, 5.0), Point(3.0, 0.0)))))

val points = listOf(Point(5.0, 5.0), Point(5.0, 8.0), Point(-10.0, 5.0), Point(0.0, 5.0),
Point(10.0, 5.0), Point(8.0, 5.0), Point(10.0, 10.0))

Ray_casting.check(figures, points)
}

Output:
points: [Point(x=5.0, y=5.0), Point(x=5.0, y=8.0), Point(x=-10.0, y=5.0), Point(x=0.0, y=5.0), Point(x=10.0, y=5.0), Point(x=8.0, y=5.0), Point(x=10.0, y=10.0)]
figure: Square
Edge(s=Point(x=0.0, y=0.0), e=Point(x=10.0, y=0.0))
Edge(s=Point(x=10.0, y=0.0), e=Point(x=10.0, y=10.0))
Edge(s=Point(x=10.0, y=10.0), e=Point(x=0.0, y=10.0))
Edge(s=Point(x=0.0, y=10.0), e=Point(x=0.0, y=0.0))
result: [true, true, false, false, true, true, false]
figure: Square hole
Edge(s=Point(x=0.0, y=0.0), e=Point(x=10.0, y=0.0))
Edge(s=Point(x=10.0, y=0.0), e=Point(x=10.0, y=10.0))
Edge(s=Point(x=10.0, y=10.0), e=Point(x=0.0, y=10.0))
Edge(s=Point(x=0.0, y=10.0), e=Point(x=0.0, y=0.0))
Edge(s=Point(x=2.5, y=2.5), e=Point(x=7.5, y=2.5))
Edge(s=Point(x=7.5, y=2.5), e=Point(x=7.5, y=7.5))
Edge(s=Point(x=7.5, y=7.5), e=Point(x=2.5, y=7.5))
Edge(s=Point(x=2.5, y=7.5), e=Point(x=2.5, y=2.5))
result: [false, true, false, false, true, true, false]
figure: Strange
Edge(s=Point(x=0.0, y=0.0), e=Point(x=2.5, y=2.5))
Edge(s=Point(x=2.5, y=2.5), e=Point(x=0.0, y=10.0))
Edge(s=Point(x=0.0, y=10.0), e=Point(x=2.5, y=7.5))
Edge(s=Point(x=2.5, y=7.5), e=Point(x=7.5, y=7.5))
Edge(s=Point(x=7.5, y=7.5), e=Point(x=10.0, y=10.0))
Edge(s=Point(x=10.0, y=10.0), e=Point(x=10.0, y=0.0))
Edge(s=Point(x=10.0, y=0.0), e=Point(x=2.5, y=2.5))
result: [true, false, false, false, true, true, false]
figure: Exagon
Edge(s=Point(x=3.0, y=0.0), e=Point(x=7.0, y=0.0))
Edge(s=Point(x=7.0, y=0.0), e=Point(x=10.0, y=5.0))
Edge(s=Point(x=10.0, y=5.0), e=Point(x=7.0, y=10.0))
Edge(s=Point(x=7.0, y=10.0), e=Point(x=3.0, y=10.0))
Edge(s=Point(x=3.0, y=10.0), e=Point(x=0.0, y=5.0))
Edge(s=Point(x=0.0, y=5.0), e=Point(x=3.0, y=0.0))
result: [true, true, false, false, true, true, false]


## Liberty BASIC

Translated from C code at: http://alienryderflex.com/polygon/

Displays interactively on-screen.

NoMainWin
Global sw, sh, verts

sw = 640 :   sh = 480
WindowWidth  = sw+8 : WindowHeight = sh+31
UpperLeftX = (DisplayWidth -sw)/2
UpperLeftY = (DisplayHeight-sh)/2
Open"Ray Casting Algorithm" For Graphics_nf_nsb As #g
#g "Down; TrapClose [halt]"
h$= "#g" Dim xp(15),yp(15) #g "when leftButtonDown [halt];when mouseMove checkPoint" #g "when rightButtonDown [Repeat]" [Repeat] #g "Cls;Fill 32 160 255; Color white;BackColor 32 160 255" #g "Place 5 460;\L-click to exit" #g "Place 485 460;\R-click for new polygon" 'generate polygon from random points numPoints = rand(4,15) verts = numPoints For i = 0 To numPoints-1 xp(i) = rand(20,620) yp(i) = rand(40,420) Next Call drawPoly h$, verts, "white"
#g "Flush"
Wait

[halt]
Close #g
End

'Point In Polygon Function
Function pnp(x, y, numSides)
j= numSides-1: oddNodes = 0
For i = 0 To numSides-1
If ((yp(i)<y) And (yp(j)>=y)) Or ((yp(j)<y) And (yp(i)>=y)) Then
f1 = y - yp(i):f2 = yp(j) - yp(i): f3 = xp(j) - xp(i)
If (xp(i) + f1 / f2 * f3) < x Then oddNodes = 1 - oddNodes
End If
j = i
Next
pnp = oddNodes
End Function

'draw the polygon
Sub drawPoly h$, verts, colour$
#h$, "Color ";colour$
j = verts-1
For i = 0 To verts-1
#h$"Line ";xp(j);" ";yp(j);" ";xp(i);" ";yp(i) j = i Next End Sub 'change message and color of polygon Sub checkPoint h$, x, y
If pnp(x,y,verts) Then
#h$"Color 32 160 255;BackColor 32 160 255" #h$ "Place 5 0;BoxFilled 150 20;Color white"
#h$"Place 7 15;\Mouse In Polygon" Call drawPoly h$, verts, "red"
Else
#h$"Color 32 160 255;BackColor 32 160 255" #h$ "Place 5 0;BoxFilled 150 20;Color white"
#h$"Place 7 15;\Mouse Not In Polygon" Call drawPoly h$, verts, "white"
End If
End Sub

Function rand(loNum,hiNum)
rand = Int(Rnd(0)*(hiNum-loNum+1)+loNum)
End Function

## Lua

function Point(x,y) return {x=x, y=y} end

function Polygon(name, points)
local function contains(self, p)
local odd, eps = false, 1e-9
local function rayseg(p, a, b)
if a.y > b.y then a, b = b, a end
if p.y == a.y or p.y == b.y then p.y = p.y + eps end
if p.y < a.y or p.y > b.y or p.x > math.max(a.x, b.x) then return false end
if p.x < math.min(a.x, b.x) then return true end
local red = a.x == b.x and math.huge or (b.y-a.y)/(b.x-a.x)
local blu = a.x == p.x and math.huge or (p.y-a.y)/(p.x-a.x)
return blu >= red
end
for i, a in ipairs(self.points) do
local b = self.points[i%#self.points+1]
if rayseg(p, a, b) then odd = not odd end
end
return odd
end
return {name=name, points=points, contains=contains}
end

polygons = {
Polygon("square", { Point(0,0), Point(10,0), Point(10,10), Point(0,10) }),
Polygon("squarehole", { Point(0,0), Point(10,0), Point(10,10), Point(0,10), Point(2.5,2.5), Point(7.5,2.5), Point(7.5,7.5), Point(2.5,7.5) }),
Polygon("strange", { Point(0,0), Point(2.5,2.5),  Point(0, 10), Point(2.5,7.5), Point(7.5,7.5), Point(10,10), Point(10,0), Point(2.5,2.5) }),
Polygon("hexagon", { Point(3,0), Point(7,0), Point(10,5), Point(7,10), Point(3,10), Point(0,5) })
}
points = { Point(5,5), Point(5,8), Point(-10,5), Point(0,5), Point(10,5), Point(8,5), Point(10,10) }

for _,poly in ipairs(polygons) do
print("Does '"..poly.name.."' contain the point..")
for _,pt in ipairs(points) do
print(string.format("  (%3.f, %2.f)?  %s", pt.x, pt.y, tostring(poly:contains(pt))))
end
print()
end

Output:
Does 'square' contain..
(  5,  5)?  true
(  5,  8)?  true
(-10,  5)?  false
(  0,  5)?  false
( 10,  5)?  true
(  8,  5)?  true
( 10, 10)?  false

Does 'squarehole' contain..
(  5,  5)?  false
(  5,  8)?  true
(-10,  5)?  false
(  0,  5)?  false
( 10,  5)?  true
(  8,  5)?  true
( 10, 10)?  false

Does 'strange' contain..
(  5,  5)?  true
(  5,  8)?  false
(-10,  5)?  false
(  0,  5)?  false
( 10,  5)?  true
(  8,  5)?  true
( 10, 10)?  false

Does 'hexagon' contain..
(  5,  5)?  true
(  5,  8)?  true
(-10,  5)?  false
(  0,  5)?  false
( 10,  5)?  true
(  8,  5)?  true
( 10, 10)?  false

## Nim

Translation of: D
import fenv, sequtils, strformat

type
Point = tuple[x, y: float]
Edge = tuple[a, b: Point]
Figure = tuple[name: string; edges: seq[Edge]]

func contains(poly: Figure; p: Point): bool =

func raySegI(p: Point; edge: Edge): bool =
const Epsilon = 0.00001
if edge.a.y > edge.b.y:
# Swap "a" and "b".
return p.raySegI((edge.b, edge.a))
if p.y == edge.a.y or p.y == edge.b.y:
# p.y += Epsilon.
return (p.x, p.y + Epsilon).raySegI(edge)
if p.y > edge.b.y or p.y < edge.a.y or p.x > max(edge.a.x, edge.b.x):
return false
if p.x < min(edge.a.x, edge.b.x):
return true
let blue = if abs(edge.a.x - p.x) > minimumPositiveValue(float):
(p.y - edge.a.y) / (p.x - edge.a.x)
else:
maximumPositiveValue(float)
let red = if abs(edge.a.x - edge.b.x) > minimumPositiveValue(float):
(edge.b.y - edge.a.y) / (edge.b.x - edge.a.x)
else:
maximumPositiveValue(float)
result = blue >= red

result = (poly.edges.filterIt(p.raySegI(it)).len and 1) != 0

when isMainModule:

const
Polys: array[4, Figure] =
[("Square",
@[(( 0.0,  0.0), (10.0,  0.0)),  ((10.0,  0.0), (10.0, 10.0)),
((10.0, 10.0), ( 0.0, 10.0)),  (( 0.0, 10.0), ( 0.0,  0.0))]),
("Square hole",
@[(( 0.0,  0.0), (10.0,  0.0)),  ((10.0,  0.0), (10.0, 10.0)),
((10.0, 10.0), ( 0.0, 10.0)),  (( 0.0, 10.0), ( 0.0,  0.0)),
(( 2.5,  2.5), ( 7.5,  2.5)),  (( 7.5,  2.5), ( 7.5,  7.5)),
(( 7.5,  7.5), ( 2.5,  7.5)),  (( 2.5,  7.5), ( 2.5,  2.5))]),
("Strange",
@[(( 0.0,  0.0), ( 2.5,  2.5)),  (( 2.5,  2.5), ( 0.0, 10.0)),
(( 0.0, 10.0), ( 2.5,  7.5)),  (( 2.5,  7.5), ( 7.5,  7.5)),
(( 7.5,  7.5), (10.0, 10.0)),  ((10.0, 10.0), (10.0,  0.0)),
((10.0,  0.0),   ( 2.5,  2.5))]),
("Hexagon",
@[(( 3.0,  0.0), ( 7.0,  0.0)),  (( 7.0,  0.0), (10.0,  5.0)),
((10.0,  5.0), ( 7.0, 10.0)),  (( 7.0, 10.0), ( 3.0, 10.0)),
(( 3.0, 10.0), ( 0.0,  5.0)),  (( 0.0,  5.0), ( 3.0,  0.0))])
]

TestPoints: array[7, Point] =
[(5.0, 5.0), (5.0, 8.0), (-10.0, 5.0), (0.0, 5.0), (10.0, 5.0), (8.0, 5.0), (10.0, 10.0)]

for poly in Polys:
echo &"Is point inside figure {poly.name}?"
for p in TestPoints:
echo &"  ({p.x:3},{p.y:3}): {poly.contains(p)}"

Output:
Is point inside figure Square?
(  5,  5): true
(  5,  8): true
(-10,  5): false
(  0,  5): false
( 10,  5): true
(  8,  5): true
( 10, 10): false
Is point inside figure Square hole?
(  5,  5): false
(  5,  8): true
(-10,  5): false
(  0,  5): false
( 10,  5): true
(  8,  5): true
( 10, 10): false
Is point inside figure Strange?
(  5,  5): true
(  5,  8): false
(-10,  5): false
(  0,  5): false
( 10,  5): true
(  8,  5): true
( 10, 10): false
Is point inside figure Hexagon?
(  5,  5): true
(  5,  8): true
(-10,  5): false
(  0,  5): false
( 10,  5): true
(  8,  5): true
( 10, 10): false

## OCaml

Translation of: C
type point = { x:float; y:float }

type polygon = {
vertices: point array;
edges: (int * int) list;
}

let p x y = { x=x; y=y }

let square_v = [|
(p 0. 0.); (p 10. 0.); (p 10. 10.); (p 0. 10.);
(p 2.5 2.5); (p 7.5 0.1); (p 7.5 7.5); (p 2.5 7.5)
|]

let esa_v = [|
(p 3. 0.); (p 7. 0.); (p 10. 5.); (p 7. 10.); (p 3. 10.); (p 0. 5.)
|]

let esa = {
vertices = esa_v;
edges = [ (0,1); (1,2); (2,3); (3,4); (4,5); (5,0) ]
}

let square = {
vertices = square_v;
edges = [ (0,1); (1,2); (2,3); (3,0) ]
}

let squarehole = {
vertices = square_v;
edges = [ (0,1); (1,2); (2,3); (3,0); (4,5); (5,6); (6,7); (7,4) ]
}

let strange = {
vertices = square_v;
edges = [ (0,4); (4,3); (3,7); (7,6); (6,2); (2,1); (1,5); (5,0) ]
}

let min_y ~a ~b = if a.y > b.y then (b) else (a)

let coeff_ang ~pa ~pb = (pb.y -. pa.y) /. (pb.x -. pa.x)

let huge_val = infinity

let hseg_intersect_seg ~s ~a ~b =
let _eps =
if s.y = (max a.y b.y) ||
s.y = (min a.y b.y) then 0.00001 else 0.0
in
if  (s.y +. _eps) > (max a.y b.y) ||
(s.y +. _eps) < (min a.y b.y) ||
s.x > (max a.x b.x) then (false)
else if s.x <= (min a.x b.x) then (true)
else
let ca = if a.x <> b.x then (coeff_ang a b) else (huge_val) in
let my = min_y ~a ~b in
let cp = if (s.x -. my.x) <> 0.0 then (coeff_ang my s) else (huge_val) in
(cp >= ca)
;;

let point_is_inside ~poly ~pt =
let cross = ref 0 in
List.iter (fun (a,b) ->
if hseg_intersect_seg pt
poly.vertices.(a)
poly.vertices.(b)
then incr cross
) poly.edges;
( (!cross mod 2) <> 0)
;;

let make_test p label s =
Printf.printf "point (%.5f,%.5f) is " p.x p.y;
print_string (if point_is_inside s p
then "INSIDE "
else "OUTSIDE ");
print_endline label;
;;

let () =
let test_points = [
(p 5. 5.); (p 5. 8.); (p 2. 2.); (p 0. 0.);
(p 10. 10.); (p 2.5 2.5); (p 0.01 5.);
(p 2.2 7.4); (p 0. 5.); (p 10. 5.); (p (-4.) 10.) ] in

List.iter (fun p ->
make_test p "square"     square;
make_test p "squarehole" squarehole;
make_test p "strange"    strange;
make_test p "esa"        esa;
print_newline()
) test_points;
;;


## Perl

use strict;
use List::Util qw(max min);

sub point_in_polygon
{
my ( $point,$polygon ) = @_;

my $count = 0; foreach my$side ( @$polygon ) {$count++ if ray_intersect_segment($point,$side);
}
return ($count % 2 == 0) ? 0 : 1; } my$eps = 0.0001;
my $inf = 1e600; sub ray_intersect_segment { my ($point, $segment) = @_; my ($A, $B) = @$segment;

my @P = @$point; # copy it ($A, $B) = ($B, $A) if$A->[1] > $B->[1];$P[1] += $eps if ($P[1] == $A->[1]) || ($P[1] == $B->[1]); return 0 if ($P[1] < $A->[1]) || ($P[1] > $B->[1]) || ($P[0] > max($A->[0],$B->[1]) );
return 1 if $P[0] < min($A->[0], $B->[0]); my$m_red = ($A->[0] !=$B->[0]) ? ( $B->[1] -$A->[1] )/($B->[0] -$A->[0]) : $inf; my$m_blue = ($A->[0] !=$P[0]) ? ( $P[1] -$A->[1] )/($P[0] -$A->[0]) : $inf; return ($m_blue >= $m_red) ? 1 : 0; }  Testing: # the following are utilities to use the same Fortran data... sub point { [shift, shift]; } sub create_polygon { my ($pts, $sides) = @_; my @poly; for(my$i = 0; $i <$#$sides;$i += 2) {
push @poly, [ $pts->[$sides->[$i]-1],$pts->[$sides->[$i+1]-1] ];
}
\@poly;
}

my @pts = ( point(0,0), point(10,0), point(10,10), point(0,10),
point(2.5,2.5), point(7.5,2.5), point(7.5,7.5), point(2.5,7.5),
point(0,5), point(10,5),
point(3,0), point(7,0), point(7,10), point(3,10) );

my %pgs = (
squared => create_polygon(\@pts, [ 1,2, 2,3, 3,4, 4,1 ] ),
squaredhole => create_polygon(\@pts, [ 1,2, 2,3, 3,4, 4,1, 5,6, 6,7, 7,8, 8,5 ] ),
strange => create_polygon(\@pts, [ 1,5, 5,4, 4,8, 8,7, 7,3, 3,2, 2,5 ] ),
exagon => create_polygon(\@pts, [ 11,12, 12,10, 10,13, 13,14, 14,9, 9,11 ]) ,
);

my @p = ( point(5,5), point(5, 8), point(-10, 5), point(0,5), point(10,5), &
point(8,5), point(10,10) );

foreach my $pol ( sort keys %pgs ) { no strict 'refs'; print "$pol\n";
my $rp =$pgs{$pol}; foreach my$tp ( @p ) {
print "\t($tp->[0],$tp->[1]) " .
( point_in_polygon($tp,$rp) ? "INSIDE" : "OUTSIDE" ) . "\n";
}
}


## Phix

constant inf = 1e300*1e300

function rayIntersectsSegment(sequence point, sequence segment)
sequence {a, b} = segment
atom {pX,pY} = point,
{aX,aY} = a,
{bX,bY} = b,
m_red,m_blue
-- ensure {aX,aY} is the segment end-point with the smallest y coordinate
if aY>bY then
{bX,bY} = a
{aX,aY} = b
end if
if pY=aY or pY=bY then
--
-- Consider a ray passing through the top or left corner of a diamond:
--          /
--  --- or  -
--   ^      \
-- In both cases it touches both edges, but ends up outside in the
--  top case, whereas it ends up inside in the left case. Just move
--  the y co-ordinate down a smidge and it'll work properly, by
--  missing one line in the left/right cases and hitting both/none
--  in the top/bottom cases.
--
pY += 0.000001
end if
if pY<aY or pY>bY then return false end if
if pX>max(aX,bX) then return false end if
if pX<min(aX,bX) then return true end if
if aX!=bX then
m_red = (bY-aY)/(bX-aX)
else
m_red = inf
end if
if aX!=pX then
m_blue = (pY-aY)/(pX-aX)
else
m_blue = inf
end if
return m_blue >= m_red
end function

function inside(sequence point, sequence poly)
bool res = false
for i=1 to length(poly) do
if rayIntersectsSegment(point,poly[i]) then
res = not res
end if
end for
return res
end function

function instr(integer flag, integer expected)
string res = {"in", "out"}[2-flag]
if flag!=expected then
res &= "*** ERROR ***"
end if
return res
end function

function instar(integer flag)
return "* "[2-flag]
end function

constant test_points = {{5,5},{5,8},{-10,5},{0,5},{10,5},{8,5},{10,10}}

--constant NAME = 1, POLY = 2, EXPECTED = 3
constant test_polygons = {
{"square",      {{{0,0},{10,0}},{{10,0},{10,10}},{{10,10},{0,10}},{{0,10},{0,0}}},
{true,true,false,false,true,true,false}},
{"square hole", {{{0,0},{10,0}},{{10,0},{10,10}},{{10,10},{0,10}},{{0,10},{0,0}},
{{2.5,2.5},{7.5,2.5}},{{7.5,2.5},{7.5,7.5}},{{7.5,7.5},{2.5,7.5}},{{2.5,7.5},{2.5,2.5}}},
{false,true,false,false,true,true,false}},
{"strange",     {{{0,5},{2.5,2.5}},{{2.5,2.5},{0,10}},{{0,10},{2.5,7.5}},{{2.5,7.5},{7.5,7.5}},
{{7.5,7.5},{10,10}},{{10,10},{10,0}},{{10,0},{2.5,2.5}}},
{true,false,false,false,true,true,false}},
{"exagon",      {{{3,0},{7,0}},{{7,0},{10,5}},{{10,5},{7,10}},{{7,10},{3,10}},{{3,10},{0,5}},{{0,5},{3,0}}},
{true,true,false,false,true,true,false}}
}

sequence name, poly, expected, tp

for i=1 to length(test_polygons) do
{name,poly,expected} = test_polygons[i]
printf(1,"\n%12s:",{name})
for j=1 to length(test_points) do
tp = test_points[j]
printf(1," %s, %-4s",{sprint(tp),instr(inside(tp,poly),expected[j])})
end for
end for

puts(1,"\n\n\n")

for i=0 to 10 do
for j=1 to length(test_polygons) do
puts(1,"     ")
poly = test_polygons[j][2]
for k=0 to 10 do
puts(1,instar(inside({k+0.5,10.5-i},poly)))
end for
end for
puts(1,"\n")
end for

Output:
      square: {5,5}, in   {5,8}, in   {-10,5}, out  {0,5}, out  {10,5}, in   {8,5}, in   {10,10}, out
square hole: {5,5}, out  {5,8}, in   {-10,5}, out  {0,5}, out  {10,5}, in   {8,5}, in   {10,10}, out
strange: {5,5}, in   {5,8}, out  {-10,5}, out  {0,5}, out  {10,5}, in   {8,5}, in   {10,10}, out
exagon: {5,5}, in   {5,8}, in   {-10,5}, out  {0,5}, out  {10,5}, in   {8,5}, in   {10,10}, out

**********      **********      *                  ****
**********      **********       *       *        ******
**********      **********       *      **        *******
**********      ***     **       * *******       ********
**********      ***     **         *******      **********
**********      ***     **      * ********      **********
**********      ***     **      **********       ********
**********      ***     **      **********        *******
**********      **********            ****        ******
**********      **********               *         ****


## PHP

<?php

function contains($bounds,$lat, $lng) {$count = 0;
$bounds_count = count($bounds);
for ($b = 0;$b < $bounds_count;$b++) {
$vertex1 =$bounds[$b];$vertex2 = $bounds[($b + 1) % $bounds_count]; if (west($vertex1, $vertex2,$lng, $lat))$count++;
}

return $count % 2; } function west($A, $B,$x, $y) { if ($A['y'] <= $B['y']) { if ($y <= $A['y'] ||$y > $B['y'] ||$x >= $A['x'] &&$x >= $B['x']) { return false; } if ($x < $A['x'] &&$x < $B['x']) { return true; } if ($x == $A['x']) { if ($y == $A['y']) {$result1 = NAN;
} else {
$result1 = INF; } } else {$result1 = ($y -$A['y']) / ($x -$A['x']);
}
if ($B['x'] ==$A['x']) {
if ($B['y'] ==$A['y']) {
$result2 = NAN; } else {$result2 = INF;
}
} else {
$result2 = ($B['y'] - $A['y']) / ($B['x'] - $A['x']); } return$result1 > $result2; } return west($B, $A,$x, $y); }$square = [
'name' => 'square',
'bounds' => [['x' => 0, 'y' => 0], ['x' => 20, 'y' => 0], ['x' => 20, 'y' => 20], ['x' => 0, 'y' => 20]]
];
$squareHole = [ 'name' => 'squareHole', 'bounds' => [['x' => 0, 'y' => 0], ['x' => 20, 'y' => 0], ['x' => 20, 'y' => 20], ['x' => 0, 'y' => 20], ['x' => 5, 'y' => 5], ['x' => 15, 'y' => 5], ['x' => 15, 'y' => 15], ['x' => 5, 'y' => 15]] ];$strange = [
'name' => 'strange',
'bounds' => [['x' => 0, 'y' => 0], ['x' => 5, 'y' => 5], ['x' => 0, 'y' => 20], ['x' => 5, 'y' => 15], ['x' => 15, 'y' => 15], ['x' => 20, 'y' => 20], ['x' => 20, 'y' => 0]]
];
$hexagon = [ 'name' => 'hexagon', 'bounds' => [['x' => 6, 'y' => 0], ['x' => 14, 'y' => 0], ['x' => 20, 'y' => 10], ['x' => 14, 'y' => 20], ['x' => 6, 'y' => 20], ['x' => 0, 'y' => 10]] ];$shapes = [$square,$squareHole, $strange,$hexagon];

$testPoints = [ ['lng' => 10, 'lat' => 10], ['lng' => 10, 'lat' => 16], ['lng' => -20, 'lat' => 10], ['lng' => 0, 'lat' => 10], ['lng' => 20, 'lat' => 10], ['lng' => 16, 'lat' => 10], ['lng' => 20, 'lat' => 20] ]; for ($s = 0; $s < count($shapes); $s++) {$shape = $shapes[$s];
for ($tp = 0;$tp < count($testPoints);$tp++) {
$testPoint =$testPoints[$tp]; echo json_encode($testPoint) . "\tin " . $shape['name'] . "\t" . contains($shape['bounds'], $testPoint['lat'],$testPoint['lng']) . PHP_EOL;
}
}

Output:
{"lng":10,"lat":10}     in square       1
{"lng":10,"lat":16}     in square       1
{"lng":-20,"lat":10}    in square       0
{"lng":0,"lat":10}      in square       1
{"lng":20,"lat":10}     in square       0
{"lng":16,"lat":10}     in square       1
{"lng":20,"lat":20}     in square       0
{"lng":10,"lat":10}     in squareHole   0
{"lng":10,"lat":16}     in squareHole   1
{"lng":-20,"lat":10}    in squareHole   0
{"lng":0,"lat":10}      in squareHole   0
{"lng":20,"lat":10}     in squareHole   0
{"lng":16,"lat":10}     in squareHole   1
{"lng":20,"lat":20}     in squareHole   0
{"lng":10,"lat":10}     in strange      1
{"lng":10,"lat":16}     in strange      0
{"lng":-20,"lat":10}    in strange      0
{"lng":0,"lat":10}      in strange      0
{"lng":20,"lat":10}     in strange      0
{"lng":16,"lat":10}     in strange      1
{"lng":20,"lat":20}     in strange      0
{"lng":10,"lat":10}     in hexagon      1
{"lng":10,"lat":16}     in hexagon      1
{"lng":-20,"lat":10}    in hexagon      0
{"lng":0,"lat":10}      in hexagon      1
{"lng":20,"lat":10}     in hexagon      0
{"lng":16,"lat":10}     in hexagon      1
{"lng":20,"lat":20}     in hexagon      0



## PicoLisp

(scl 4)

(de intersects (Px Py Ax Ay Bx By)
(when (> Ay By)
(xchg 'Ax 'Bx)
(xchg 'Ay 'By) )
(when (or (= Py Ay) (= Py By))
(inc 'Py) )
(and
(>= Py Ay)
(>= By Py)
(>= (max Ax Bx) Px)
(or
(> (min Ax Bx) Px)
(= Ax Px)
(and
(<> Ax Bx)
(>=
(*/ (- Py Ay) 1.0 (- Px Ax))            # Blue
(*/ (- By Ay) 1.0 (- Bx Ax)) ) ) ) ) )  # Red

(de inside (Pt Poly)
(let Res NIL
(for Edge Poly
(when (apply intersects Edge (car Pt) (cdr Pt))
(onOff Res) ) )
Res ) )

Test data:

(de Square
( 0.0  0.0  10.0  0.0)
(10.0  0.0  10.0 10.0)
(10.0 10.0   0.0 10.0)
( 0.0 10.0   0.0  0.0) )

(de SquareHole
( 0.0  0.0  10.0  0.0)
(10.0  0.0  10.0 10.0)
(10.0 10.0   0.0 10.0)
( 0.0 10.0   0.0  0.0)
( 2.5  2.5   7.5  2.5)
( 7.5  2.5   7.5  7.5)
( 7.5  7.5   2.5  7.5)
( 2.5  7.5   2.5  2.5) )

(de Strange
( 0.0  0.0   2.5  2.5)
( 2.5  2.5   0.0 10.0)
( 0.0 10.0   2.5  7.5)
( 2.5  7.5   7.5  7.5)
( 7.5  7.5  10.0 10.0)
(10.0 10.0  10.0  0.0)
(10.0  0.0   2.5  2.5) )

(de Exagon
( 3.0  0.0   7.0  0.0)
( 7.0  0.0  10.0  5.0)
(10.0  5.0   7.0 10.0)
( 7.0 10.0   3.0 10.0)
( 3.0 10.0   0.0  5.0)
( 0.0  5.0   3.0  0.0) )

Output:

: (inside (5.0 . 5.0) Square)
-> T
: (inside (5.0 . 8.0) Square)
-> T
: (inside (-10.0 . 5.0) Square)
-> NIL
: (inside (0.0 . 5.0) Square)
-> NIL
: (inside (10.0 . 5.0) Square)
-> T
: (inside (8.0 . 5.0) Square)
-> T
: (inside (10.0 . 10.0) Square)
-> NIL

: (inside (5.0 . 5.0) SquareHole)
-> NIL
: (inside (5.0 . 8.0) SquareHole)
-> T
: (inside (-10.0 . 5.0) SquareHole)
-> NIL
: (inside (0 . 5.0) SquareHole)
-> NIL
: (inside (10.0 . 5.0) SquareHole)
-> T
: (inside (8.0 . 5.0) SquareHole)
-> T
: (inside (10.0 . 10.0) SquareHole)
-> NIL

: (inside (5.0 . 5.0) Strange)
-> T
: (inside (5.0 . 8.0) Strange)
-> NIL
: (inside (-10.0 . 5.0) Strange)
-> NIL
: (inside (0 . 5.0) Strange)
-> NIL
: (inside (10.0 . 5.0) Strange)
-> T
: (inside (8.0 . 5.0) Strange)
-> T
: (inside (10.0 . 10.0) Strange)
-> NIL

: (inside (5.0 . 5.0) Exagon)
-> T
: (inside (5.0 . 8.0) Exagon)
-> T
: (inside (-10.0 . 5.0) Exagon)
-> NIL
: (inside (0.0 . 5.0) Exagon)
-> NIL
: (inside (10.0 . 5.0) Exagon)
-> T
: (inside (8.0 . 5.0) Exagon)
-> T
: (inside (10.0 . 10.0) Exagon)
-> NIL

## PureBasic

The code below is includes a GUI for drawing a polygon with the mouse that constantly tests whether the mouse is inside or outside the polygon. It displays a message and changes the windows color slightly to indicate if the pointer is inside or outside the polygon being drawn. The routine that does the checking is called inpoly() and it returns a value of one if the point is with the polygon and zero if it isn't.

Structure point_f
x.f
y.f
EndStructure
Procedure inpoly(*p.point_f, List poly.point_f())
Protected.point_f new, old, lp, rp
Protected inside
If ListSize(poly()) < 3: ProcedureReturn 0: EndIf
LastElement(poly()): old = poly()
ForEach poly()
;find leftmost endpoint 'lp' and the rightmost endpoint 'rp' based on x value
If poly()\x > old\x
lp = old
rp = poly()
Else
lp = poly()
rp = old
EndIf
If lp\x < *p\x And *p\x <= rp\x And (*p\y - lp\y) * (rp\x - lp\x) < (rp\y - lp\y) * (*p\x - lp\x)
inside = ~inside
EndIf
old = poly()
Next
ProcedureReturn inside & 1
EndProcedure

If InitSprite()
If InitKeyboard() And InitMouse()
OpenWindow(0, 0, 0, 800, 600, "Press [Esc] to close, [Left mouse button] Add Point, [Right mouse button] Clear All Points.", #PB_Window_ScreenCentered | #PB_Window_SystemMenu)
OpenWindowedScreen(WindowID(0), 0, 0, 800, 600, 1, 0, 0)
SetFrameRate(60)
EndIf
Else
MessageRequester("", "Unable to initsprite"): End
EndIf

NewList v.point_f()
Define.point_f pvp, mp
Define Col, EventID, mode.b, modetxt.s
Repeat
Delay(1)
EventID = WindowEvent()
ExamineKeyboard()
ExamineMouse()
ClearScreen(Col)

mp\x = MouseX()
mp\y = MouseY()
If MouseButton(#PB_MouseButton_Left)
v()\x = mp\x
v()\y = mp\y
Delay(100)
EndIf

If MouseButton(#PB_MouseButton_Right)
ClearList(v())
Delay(100)
EndIf

StartDrawing(ScreenOutput())
If LastElement(v())
pvp = v()
ForEach v()
LineXY(pvp\x, pvp\y, v()\x, v()\y, RGB(0, $FF, 0)) ;Green Circle(pvp\x, pvp\y, 5, RGB($FF, 0, 0)) ;Red
pvp = v()
Next
EndIf
Circle(MouseX(), MouseY(), 5, RGB($C0,$C0, $FF)) ;LightBlue If inpoly(mp, v()) modetxt = "You are in the polygon." Col = RGB(0, 0, 0) Else modetxt = "You are not in the polygon." Col = RGB($50, $50,$50)
EndIf
DrawText((800 - TextWidth(modetxt)) / 2, 0, modetxt)
StopDrawing()

FlipBuffers()
Until KeyboardReleased(#PB_Key_Escape) Or EventID = #PB_Event_CloseWindow

## Python

from collections import namedtuple
from pprint import pprint as pp
import sys

Pt = namedtuple('Pt', 'x, y')               # Point
Edge = namedtuple('Edge', 'a, b')           # Polygon edge from a to b
Poly = namedtuple('Poly', 'name, edges')    # Polygon

_eps = 0.00001
_huge = sys.float_info.max
_tiny = sys.float_info.min

def rayintersectseg(p, edge):
''' takes a point p=Pt() and an edge of two endpoints a,b=Pt() of a line segment returns boolean
'''
a,b = edge
if a.y > b.y:
a,b = b,a
if p.y == a.y or p.y == b.y:
p = Pt(p.x, p.y + _eps)

intersect = False

if (p.y > b.y or p.y < a.y) or (
p.x > max(a.x, b.x)):
return False

if p.x < min(a.x, b.x):
intersect = True
else:
if abs(a.x - b.x) > _tiny:
m_red = (b.y - a.y) / float(b.x - a.x)
else:
m_red = _huge
if abs(a.x - p.x) > _tiny:
m_blue = (p.y - a.y) / float(p.x - a.x)
else:
m_blue = _huge
intersect = m_blue >= m_red
return intersect

def _odd(x): return x%2 == 1

def ispointinside(p, poly):
ln = len(poly)
return _odd(sum(rayintersectseg(p, edge)
for edge in poly.edges ))

def polypp(poly):
print ("\n  Polygon(name='%s', edges=(" % poly.name)
print ('   ', ',\n    '.join(str(e) for e in poly.edges) + '\n    ))')

if __name__ == '__main__':
polys = [
Poly(name='square', edges=(
Edge(a=Pt(x=0, y=0), b=Pt(x=10, y=0)),
Edge(a=Pt(x=10, y=0), b=Pt(x=10, y=10)),
Edge(a=Pt(x=10, y=10), b=Pt(x=0, y=10)),
Edge(a=Pt(x=0, y=10), b=Pt(x=0, y=0))
)),
Poly(name='square_hole', edges=(
Edge(a=Pt(x=0, y=0), b=Pt(x=10, y=0)),
Edge(a=Pt(x=10, y=0), b=Pt(x=10, y=10)),
Edge(a=Pt(x=10, y=10), b=Pt(x=0, y=10)),
Edge(a=Pt(x=0, y=10), b=Pt(x=0, y=0)),
Edge(a=Pt(x=2.5, y=2.5), b=Pt(x=7.5, y=2.5)),
Edge(a=Pt(x=7.5, y=2.5), b=Pt(x=7.5, y=7.5)),
Edge(a=Pt(x=7.5, y=7.5), b=Pt(x=2.5, y=7.5)),
Edge(a=Pt(x=2.5, y=7.5), b=Pt(x=2.5, y=2.5))
)),
Poly(name='strange', edges=(
Edge(a=Pt(x=0, y=0), b=Pt(x=2.5, y=2.5)),
Edge(a=Pt(x=2.5, y=2.5), b=Pt(x=0, y=10)),
Edge(a=Pt(x=0, y=10), b=Pt(x=2.5, y=7.5)),
Edge(a=Pt(x=2.5, y=7.5), b=Pt(x=7.5, y=7.5)),
Edge(a=Pt(x=7.5, y=7.5), b=Pt(x=10, y=10)),
Edge(a=Pt(x=10, y=10), b=Pt(x=10, y=0)),
Edge(a=Pt(x=10, y=0), b=Pt(x=2.5, y=2.5))
)),
Poly(name='exagon', edges=(
Edge(a=Pt(x=3, y=0), b=Pt(x=7, y=0)),
Edge(a=Pt(x=7, y=0), b=Pt(x=10, y=5)),
Edge(a=Pt(x=10, y=5), b=Pt(x=7, y=10)),
Edge(a=Pt(x=7, y=10), b=Pt(x=3, y=10)),
Edge(a=Pt(x=3, y=10), b=Pt(x=0, y=5)),
Edge(a=Pt(x=0, y=5), b=Pt(x=3, y=0))
)),
]
testpoints = (Pt(x=5, y=5), Pt(x=5, y=8),
Pt(x=-10, y=5), Pt(x=0, y=5),
Pt(x=10, y=5), Pt(x=8, y=5),
Pt(x=10, y=10))

print ("\n TESTING WHETHER POINTS ARE WITHIN POLYGONS")
for poly in polys:
polypp(poly)
print ('   ', '\t'.join("%s: %s" % (p, ispointinside(p, poly))
for p in testpoints[:3]))
print ('   ', '\t'.join("%s: %s" % (p, ispointinside(p, poly))
for p in testpoints[3:6]))
print ('   ', '\t'.join("%s: %s" % (p, ispointinside(p, poly))
for p in testpoints[6:]))


Sample output

 TESTING WHETHER POINTS ARE WITHIN POLYGONS

Polygon(name='square', edges=(
Edge(a=Pt(x=0, y=0), b=Pt(x=10, y=0)),
Edge(a=Pt(x=10, y=0), b=Pt(x=10, y=10)),
Edge(a=Pt(x=10, y=10), b=Pt(x=0, y=10)),
Edge(a=Pt(x=0, y=10), b=Pt(x=0, y=0))
))
Pt(x=5, y=5): True	Pt(x=5, y=8): True	Pt(x=-10, y=5): False
Pt(x=0, y=5): False	Pt(x=10, y=5): True	Pt(x=8, y=5): True
Pt(x=10, y=10): False

Polygon(name='square_hole', edges=(
Edge(a=Pt(x=0, y=0), b=Pt(x=10, y=0)),
Edge(a=Pt(x=10, y=0), b=Pt(x=10, y=10)),
Edge(a=Pt(x=10, y=10), b=Pt(x=0, y=10)),
Edge(a=Pt(x=0, y=10), b=Pt(x=0, y=0)),
Edge(a=Pt(x=2.5, y=2.5), b=Pt(x=7.5, y=2.5)),
Edge(a=Pt(x=7.5, y=2.5), b=Pt(x=7.5, y=7.5)),
Edge(a=Pt(x=7.5, y=7.5), b=Pt(x=2.5, y=7.5)),
Edge(a=Pt(x=2.5, y=7.5), b=Pt(x=2.5, y=2.5))
))
Pt(x=5, y=5): False	Pt(x=5, y=8): True	Pt(x=-10, y=5): False
Pt(x=0, y=5): False	Pt(x=10, y=5): True	Pt(x=8, y=5): True
Pt(x=10, y=10): False

Polygon(name='strange', edges=(
Edge(a=Pt(x=0, y=0), b=Pt(x=2.5, y=2.5)),
Edge(a=Pt(x=2.5, y=2.5), b=Pt(x=0, y=10)),
Edge(a=Pt(x=0, y=10), b=Pt(x=2.5, y=7.5)),
Edge(a=Pt(x=2.5, y=7.5), b=Pt(x=7.5, y=7.5)),
Edge(a=Pt(x=7.5, y=7.5), b=Pt(x=10, y=10)),
Edge(a=Pt(x=10, y=10), b=Pt(x=10, y=0)),
Edge(a=Pt(x=10, y=0), b=Pt(x=2.5, y=2.5))
))
Pt(x=5, y=5): True	Pt(x=5, y=8): False	Pt(x=-10, y=5): False
Pt(x=0, y=5): False	Pt(x=10, y=5): True	Pt(x=8, y=5): True
Pt(x=10, y=10): False

Polygon(name='exagon', edges=(
Edge(a=Pt(x=3, y=0), b=Pt(x=7, y=0)),
Edge(a=Pt(x=7, y=0), b=Pt(x=10, y=5)),
Edge(a=Pt(x=10, y=5), b=Pt(x=7, y=10)),
Edge(a=Pt(x=7, y=10), b=Pt(x=3, y=10)),
Edge(a=Pt(x=3, y=10), b=Pt(x=0, y=5)),
Edge(a=Pt(x=0, y=5), b=Pt(x=3, y=0))
))
Pt(x=5, y=5): True	Pt(x=5, y=8): True	Pt(x=-10, y=5): False
Pt(x=0, y=5): False	Pt(x=10, y=5): True	Pt(x=8, y=5): True
Pt(x=10, y=10): False

Helper routine to convert Fortran Polygons and points to Python

def _convert_fortran_shapes():
point = Pt
pts = (point(0,0), point(10,0), point(10,10), point(0,10),
point(2.5,2.5), point(7.5,2.5), point(7.5,7.5), point(2.5,7.5),
point(0,5), point(10,5),
point(3,0), point(7,0), point(7,10), point(3,10))
p = (point(5,5), point(5, 8), point(-10, 5), point(0,5), point(10,5),
point(8,5), point(10,10) )

def create_polygon(pts,vertexindex):
return [tuple(Edge(pts[vertexindex[i]-1], pts[vertexindex[i+1]-1])
for i in range(0, len(vertexindex), 2) )]
polys=[]
polys += create_polygon(pts, ( 1,2, 2,3, 3,4, 4,1 ) )
polys += create_polygon(pts, ( 1,2, 2,3, 3,4, 4,1, 5,6, 6,7, 7,8, 8,5 ) )
polys += create_polygon(pts, ( 1,5, 5,4, 4,8, 8,7, 7,3, 3,2, 2,5 ) )
polys += create_polygon(pts, ( 11,12, 12,10, 10,13, 13,14, 14,9, 9,11 ) )

names = ( "square", "square_hole", "strange", "exagon" )
polys = [Poly(name, edges)
for name, edges in zip(names, polys)]
print 'polys = ['
for p in polys:
print "  Poly(name='%s', edges=(" % p.name
print '   ', ',\n    '.join(str(e) for e in p.edges) + '\n    )),'
print '  ]'
_convert_fortran_shapes()


## R

point_in_polygon <- function(polygon, p) {
count <- 0
for(side in polygon) {
if ( ray_intersect_segment(p, side) ) {
count <- count + 1
}
}
if ( count %% 2 == 1 )
"INSIDE"
else
"OUTSIDE"
}

ray_intersect_segment <- function(p, side) {
eps <- 0.0001
a <- side$A b <- side$B
if ( a$y > b$y ) {
a <- side$B b <- side$A
}
if ( (p$y == a$y) || (p$y == b$y) ) {
p$y <- p$y + eps
}
if ( (p$y < a$y) || (p$y > b$y) )
return(FALSE)
else if ( p$x > max(a$x, b$x) ) return(FALSE) else { if ( p$x < min(a$x, b$x) )
return(TRUE)
else {
if ( a$x != b$x )
m_red <- (b$y - a$y) / (b$x - a$x)
else
m_red <- Inf
if ( a$x != p$x )
m_blue <- (p$y - a$y) / (p$x - a$x)
else
m_blue <- Inf
return( m_blue >= m_red )
}
}
}

######## utility functions #########

point <- function(x,y) list(x=x, y=y)

# pts = list(p1, p2, ... )... coords
# segs = list(c(1,2), c(2,1) ...) indices
createPolygon <- function(pts, segs) {
pol <- list()
for(pseg in segs) {
pol <- c(pol, list(list(A=pts[[pseg[1]]], B=pts[[pseg[2]]])))
}
pol
}

#### testing ####

pts <- list(point(0,0), point(10,0), point(10,10), point(0,10),
point(2.5,2.5), point(7.5,2.5), point(7.5,7.5), point(2.5,7.5),
point(0,5), point(10,5),
point(3,0), point(7,0), point(7,10), point(3,10))

polygons <-
list(
square = createPolygon(pts, list(c(1,2), c(2,3), c(3,4), c(4,1))),
squarehole = createPolygon(pts, list(c(1,2), c(2,3), c(3,4), c(4,1), c(5,6), c(6,7), c(7,8), c(8,5))),
exagon = createPolygon(pts, list(c(11,12), c(12,10), c(10,13), c(13,14), c(14,9), c(9,11)))
)

testpoints <-
list(
point(5,5), point(5, 8), point(-10, 5), point(0,5), point(10,5),
point(8,5), point(9.9,9.9)
)

for(p in testpoints) {
for(polysi in 1:length(polygons)) {
cat(sprintf("point (%lf, %lf) is %s polygon (%s)\n",
p$x, p$y, point_in_polygon(polygons[[polysi]], p), names(polygons[polysi])))
}
}


## Racket

Straightforward implementation of pseudocode

#lang racket

(module pip racket
(require racket/contract)

(provide point)
(provide seg)
(provide (contract-out [point-in-polygon? (->
point?
list?
boolean?)]))

(struct point (x y) #:transparent)
(struct seg (Ax Ay Bx By) #:transparent)
(define ε 0.000001)
(define (neq? x y) (not (eq? x y)))

(define (ray-cross-seg? r s)
(let* ([Ax (seg-Ax s)] [Ay (seg-Ay s)]
[Bx (seg-Bx s)] [By (seg-By s)]
[Px (point-x r)] [Pyo (point-y r)]
[Py (+ Pyo (if (or (eq? Pyo Ay)
(eq? Pyo By))
ε 0))])

(define Ax2 (if (< Ay By) Ax Bx))
(define Ay2 (if (< Ay By) Ay By))
(define Bx2 (if (< Ay By) Bx Ax))
(define By2 (if (< Ay By) By Ay))

(cond [(or (> Py (max Ay By)) (< Py (min Ay By))) #f]
[(> Px (max Ax Bx)) #f]
[else (cond
[(< Px (min Ax Bx)) #t]
[else
(let ([red (if (neq? Ax2 Bx2)
(/ (- By2 Ay2) (- Bx2 Ax2))
+inf.0)]
[blue (if (neq? Ax2 Px)
(/ (- Py Ay2) (- Px Ax2))
+inf.0)])
(if (>= blue red) #t #f))])])))

(define (point-in-polygon? point polygon)
(odd?
(for/fold ([c 0]) ([seg polygon])
(+ c (if (ray-cross-seg? point seg) 1 0))))))

(require 'pip)

(define test-point-list
(list
(point 5.0    5.0)
(point 5.0    8.0)
(point -10.0  5.0)
(point  0.0   5.0)
(point 10.0   5.0)
(point  8.0   5.0)
(point 10.0  10.0)))

(define square
(list (seg 0.0   0.0  10.0   0.0)
(seg 10.0  0.0  10.0  10.0)
(seg 10.0  10.0  0.0  10.0)
(seg 0.0   0.0  0.0   10.0)))

(define exagon
(list (seg  3.0   0.0   7.0   0.0)
(seg  7.0   0.0  10.0   5.0)
(seg 10.0   5.0   7.0  10.0)
(seg  7.0  10.0   3.0  10.0)
(seg  0.0   5.0   3.0   10.0)
(seg  3.0   0.0 0.0   5.0)))

(define (test-figure fig name)
(printf "\ntesting ~a: \n" name)
(for ([p test-point-list])
(printf "testing ~v: ~a\n"  p (point-in-polygon? p fig))))

(test-figure square "square")
(test-figure exagon "exagon")

Output:
testing square:
testing (point 5.0 5.0): #t
testing (point 5.0 8.0): #t
testing (point -10.0 5.0): #f
testing (point 0.0 5.0): #f
testing (point 10.0 5.0): #t
testing (point 8.0 5.0): #t
testing (point 10.0 10.0): #f

testing exagon:
testing (point 5.0 5.0): #t
testing (point 5.0 8.0): #t
testing (point -10.0 5.0): #f
testing (point 0.0 5.0): #f
testing (point 10.0 5.0): #t
testing (point 8.0 5.0): #t
testing (point 10.0 10.0): #f


## Raku

(formerly Perl 6)

constant ε = 0.0001;

sub ray-hits-seg([\Px,\Py], [[\Ax,\Ay], [\Bx,\By]] --> Bool) {
Py += ε if Py == Ay | By;

if Py < Ay or Py > By or Px > (Ax max Bx) {
False;
}
elsif Px < (Ax min Bx) {
True;
}
else {
my \red  = Ax == Bx ?? Inf !! (By - Ay) / (Bx - Ax);
my \blue = Ax == Px ?? Inf !! (Py - Ay) / (Px - Ax);
blue >= red;
}
}

sub point-in-poly(@point, @polygon --> Bool) {
so 2 R% [+] gather for @polygon -> @side {
take ray-hits-seg @point, @side.sort(*.[1]);
}
}

my %poly =
squared =>
[[[ 0.0,  0.0], [10.0,  0.0]],
[[10.0,  0.0], [10.0, 10.0]],
[[10.0, 10.0], [ 0.0, 10.0]],
[[ 0.0, 10.0], [ 0.0,  0.0]]],
squaredhole =>
[[[ 0.0,  0.0], [10.0,  0.0]],
[[10.0,  0.0], [10.0, 10.0]],
[[10.0, 10.0], [ 0.0, 10.0]],
[[ 0.0, 10.0], [ 0.0,  0.0]],
[[ 2.5,  2.5], [ 7.5,  2.5]],
[[ 7.5,  2.5], [ 7.5,  7.5]],
[[ 7.5,  7.5], [ 2.5,  7.5]],
[[ 2.5,  7.5], [ 2.5,  2.5]]],
strange =>
[[[ 0.0,  0.0], [ 2.5,  2.5]],
[[ 2.5,  2.5], [ 0.0, 10.0]],
[[ 0.0, 10.0], [ 2.5,  7.5]],
[[ 2.5,  7.5], [ 7.5,  7.5]],
[[ 7.5,  7.5], [10.0, 10.0]],
[[10.0, 10.0], [10.0,  0.0]],
[[10.0,  0.0], [ 2.5,  2.5]],
[[ 2.5,  2.5], [ 0.0,  0.0]]],  # conjecturally close polygon
exagon =>
[[[ 3.0,  0.0], [ 7.0,  0.0]],
[[ 7.0,  0.0], [10.0,  5.0]],
[[10.0,  5.0], [ 7.0, 10.0]],
[[ 7.0, 10.0], [ 3.0, 10.0]],
[[ 3.0, 10.0], [ 0.0,  5.0]],
[[ 0.0,  5.0], [ 3.0,  0.0]]];

my @test-points =
[  5.0,  5.0],
[  5.0,  8.0],
[-10.0,  5.0],
[  0.0,  5.0],
[ 10.0,  5.0],
[  8.0,  5.0],
[ 10.0, 10.0];

for <squared squaredhole strange exagon> -> $polywanna { say "$polywanna";
my @poly = %poly{$polywanna}[]; for @test-points -> @point { say "\t(@point.fmt('%.1f',','))\t{ point-in-poly(@point, @poly) ?? 'IN' !! 'OUT' }"; } }  Output: squared (5.0,5.0) IN (5.0,8.0) IN (-10.0,5.0) OUT (0.0,5.0) OUT (10.0,5.0) IN (8.0,5.0) IN (10.0,10.0) OUT squaredhole (5.0,5.0) OUT (5.0,8.0) IN (-10.0,5.0) OUT (0.0,5.0) OUT (10.0,5.0) IN (8.0,5.0) IN (10.0,10.0) OUT strange (5.0,5.0) IN (5.0,8.0) OUT (-10.0,5.0) OUT (0.0,5.0) OUT (10.0,5.0) IN (8.0,5.0) IN (10.0,10.0) OUT exagon (5.0,5.0) IN (5.0,8.0) IN (-10.0,5.0) OUT (0.0,5.0) OUT (10.0,5.0) IN (8.0,5.0) IN (10.0,10.0) OUT ## REXX Over half of the REXX program is devoted to specifying/defining/assigning the points for the test cases and for the various polygons. Code was added to facilitate easier specification of polygon sides by just specifying their vertices instead of specifying their line segments. /*REXX program verifies if a horizontal ray from point P intersects a polygon. */ call points 5 5, 5 8, -10 5, 0 5, 10 5, 8 5, 10 10 A= 2.5; B= 7.5 /* ◄───── used for shorter arguments (below).*/ call poly 0 0, 10 0, 10 10, 0 10 ; call test 'square' call poly 0 0, 10 0, 10 10, 0 10, A A, B A, B B, A B ; call test 'square hole' call poly 0 0, A A, 0 10, A B, B B, 10 10, 10 0 ; call test 'irregular' call poly 3 0, 7 0, 10 5, 7 10, 3 10, 0 5 ; call test 'hexagon' exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ in$out: procedure expose point. poly.;     parse arg p;              #= 0
do side=1  to poly.0  by 2;  #= # +intersect(p, side);   end /*side*/
return # // 2                            /*ODD  is inside.     EVEN  is outside.*/
/*──────────────────────────────────────────────────────────────────────────────────────*/
intersect: procedure expose point. poly.;         parse arg ?,s;                 sp= s + 1
epsilon= '1e'  ||  (-digits() % 2)
Px= point.?.x;     Ax= poly.s.x;       Bx= poly.sp.x
Py= point.?.y;     Ay= poly.s.y;       By= poly.sp.y  /* [↓]  do vertex swap.*/
if Ay>By           then parse  value   Ax Ay Bx By    with    Bx By Ax Ay
if Py=Ay | Py=By   then Py= Py + epsilon
if Py<Ay | Py>By | Px>max(Ax, Bx)  then  return 0
if                 Px<min(Ax, Bx)  then  return 1
if Ax\=Bx          then red = (By-Ay) / (Bx-Ax)
else red = i"1e"  ||  (digits() *2)      /* ◄─── infinity.*/
if Ax\=Px          then return  (Py-Ay) / (Px-Ax)  >=  red
else return  1
/*──────────────────────────────────────────────────────────────────────────────────────*/
points: wx= 0;  wy= 0;   do j=1  for arg();         parse value  arg(j)    with   xx  yy
wx= max(wx, length(xx) );  call  value  'POINT.'j".X",   xx
wy= max(wy, length(yy) );  call  value  'POINT.'j".Y",       yy
end   /*j*/
call value point.0,  j-1                         /*define the number of points. */
return                                           /* [↑]  adjust J  (for DO loop)*/
/*──────────────────────────────────────────────────────────────────────────────────────*/
poly:   @= 'POLY.';      parse arg Fx Fy                 /* [↓]  process the X,Y points.*/
n= 0
do j=1  for arg();      n= n + 1;  parse value arg(j)   with   xx yy
call value @ || n'.X',  xx ;       call value  @ || n".Y", yy
if n//2  then iterate;  n= n + 1        /*Inside? Then skip this point.*/
call value @ || n'.X',  xx ;       call value  @ || n".Y", yy
end   /*j*/
n= n + 1                                         /*POLY.0  is # segments(sides).*/
call value @ || n'.X', Fx;       call value @ || n".Y", Fy;     call value @'0', n
return
/*──────────────────────────────────────────────────────────────────────────────────────*/
test:   say;     do k=1  for point.0;    w= wx + wy + 2  /*W, WX, WY ≡are various widths*/
say right('  ['arg(1)"]  point:",  30),
right( right(point.k.x, wx)', 'right(point.k.y, wy), w)     "  is  ",
right( word('outside inside',  in$out(k) + 1), 7) end /*k*/ return /* [↑] format the output nicely*/  output when using the default inputs:  [square] point: 5, 5 is inside [square] point: 5, 8 is inside [square] point: -10, 5 is outside [square] point: 0, 5 is outside [square] point: 10, 5 is inside [square] point: 8, 5 is inside [square] point: 10, 10 is outside [square hole] point: 5, 5 is outside [square hole] point: 5, 8 is inside [square hole] point: -10, 5 is outside [square hole] point: 0, 5 is outside [square hole] point: 10, 5 is inside [square hole] point: 8, 5 is inside [square hole] point: 10, 10 is outside [irregular] point: 5, 5 is inside [irregular] point: 5, 8 is outside [irregular] point: -10, 5 is outside [irregular] point: 0, 5 is outside [irregular] point: 10, 5 is inside [irregular] point: 8, 5 is inside [irregular] point: 10, 10 is outside [hexagon] point: 5, 5 is inside [hexagon] point: 5, 8 is inside [hexagon] point: -10, 5 is outside [hexagon] point: 0, 5 is outside [hexagon] point: 10, 5 is inside [hexagon] point: 8, 5 is inside [hexagon] point: 10, 10 is outside  ## Rust Translation of: Python use std::f64; const _EPS: f64 = 0.00001; const _MIN: f64 = f64::MIN_POSITIVE; const _MAX: f64 = f64::MAX; #[derive(Clone)] struct Point { x: f64, y: f64, } #[derive(Clone)] struct Edge { pt1: Point, pt2: Point, } impl Edge { fn new(pt1: (f64, f64), pt2: (f64, f64)) -> Edge { Edge { pt1: Point { x: pt1.0, y: pt1.1 }, pt2: Point { x: pt2.0, y: pt2.1 }, } } } struct Polygon { edges: Vec<Edge>, // Polygon has to be created with counter-clockwise coordinates } fn pt_in_polygon(pt: &Point, poly: &Polygon) -> bool { let count = poly.edges .iter() .filter(|edge| ray_intersect_seg(pt, edge)) .count(); count % 2 == 1 } fn ray_intersect_seg(p: &Point, edge: &Edge) -> bool { let mut pt = p.clone(); let (mut a, mut b): (&Point, &Point) = (&edge.pt1, &edge.pt2); if a.y > b.y { std::mem::swap(&mut a, &mut b); } if pt.y == a.y || pt.y == b.y { pt.y += _EPS; } if (pt.y > b.y || pt.y < a.y) || pt.x > a.x.max(b.x) { false } else if pt.x < a.x.min(b.x) { true } else { let m_red = if (a.x - b.x).abs() > _MIN { (b.y - a.y) / (b.x - a.x) } else { _MAX }; let m_blue = if (a.x - pt.x).abs() > _MIN { (pt.y - a.y) / (pt.x - a.x) } else { _MAX }; m_blue >= m_red } } fn main() { let p = |x, y| Point { x, y }; let testpoints = [p(5.0, 5.0), p(5.0, 8.0), p(-10.0, 5.0), p(0.0, 5.0), p(10.0, 5.0), p(8.0, 5.0), p(10.0, 10.0)]; let poly_square = Polygon { edges: vec![ Edge::new((0.0, 0.0), (10.0, 0.0)), Edge::new((10.0, 0.0), (10.0, 10.0)), Edge::new((10.0, 10.0), (0.0, 10.0)), Edge::new((0.0, 10.0), (0.0, 0.0)), ], }; let poly_square_hole = Polygon { edges: vec![ Edge::new((0.0, 0.0), (10.0, 0.0)), Edge::new((10.0, 0.0), (10.0, 10.0)), Edge::new((10.0, 10.0), (0.0, 10.0)), Edge::new((0.0, 10.0), (0.0, 0.0)), Edge::new((2.5, 2.5), (7.5, 2.5)), Edge::new((7.5, 2.5), (7.5, 7.5)), Edge::new((7.5, 7.5), (2.5, 7.5)), Edge::new((2.5, 7.5), (2.5, 2.5)), ], }; let poly_strange = Polygon { edges: vec![ Edge::new((0.0, 0.0), (2.5, 2.5)), Edge::new((2.5, 2.5), (0.0, 10.0)), Edge::new((0.0, 10.0), (2.5, 7.5)), Edge::new((2.5, 7.5), (7.5, 7.5)), Edge::new((7.5, 7.5), (10.0, 10.0)), Edge::new((10.0, 10.0), (10.0, 0.0)), Edge::new((10.0, 0.0), (2.5, 2.5)), ], }; let poly_hexagon = Polygon { edges: vec![ Edge::new((3.0, 0.0), (7.0, 0.0)), Edge::new((7.0, 0.0), (10.0, 5.0)), Edge::new((10.0, 5.0), (7.0, 10.0)), Edge::new((7.0, 10.0), (3.0, 10.0)), Edge::new((3.0, 10.0), (0.0, 5.0)), Edge::new((0.0, 5.0), (3.0, 0.0)), ], }; print!("\nSquare :"); for pt in &testpoints { print!(" {:?}", pt_in_polygon(pt, &poly_square)); } print!("\nSquare with hole:"); for pt in &testpoints { print!(" {:?}", pt_in_polygon(pt, &poly_square_hole)); } print!("\nStrange polygon :"); for pt in &testpoints { print!(" {:?}", pt_in_polygon(pt, &poly_strange)); } print!("\nHexagon :"); for pt in &testpoints { print!(" {:?}", pt_in_polygon(pt, &poly_hexagon)); } println!(); }  Output: Square : true true false false true true false Square with hole: false true false false true true false Strange polygon : true false false false true true false Hexagon : true true false false true true false  ## Scala Translation of: D package scala.ray_casting case class Edge(_1: (Double, Double), _2: (Double, Double)) { import Math._ import Double._ def raySegI(p: (Double, Double)): Boolean = { if (_1._2 > _2._2) return Edge(_2, _1).raySegI(p) if (p._2 == _1._2 || p._2 == _2._2) return raySegI((p._1, p._2 + epsilon)) if (p._2 > _2._2 || p._2 < _1._2 || p._1 > max(_1._1, _2._1)) return false if (p._1 < min(_1._1, _2._1)) return true val blue = if (abs(_1._1 - p._1) > MinValue) (p._2 - _1._2) / (p._1 - _1._1) else MaxValue val red = if (abs(_1._1 - _2._1) > MinValue) (_2._2 - _1._2) / (_2._1 - _1._1) else MaxValue blue >= red } final val epsilon = 0.00001 } case class Figure(name: String, edges: Seq[Edge]) { def contains(p: (Double, Double)) = edges.count(_.raySegI(p)) % 2 != 0 } object Ray_casting extends App { val figures = Seq(Figure("Square", Seq(((0.0, 0.0), (10.0, 0.0)), ((10.0, 0.0), (10.0, 10.0)), ((10.0, 10.0), (0.0, 10.0)),((0.0, 10.0), (0.0, 0.0)))), Figure("Square hole", Seq(((0.0, 0.0), (10.0, 0.0)), ((10.0, 0.0), (10.0, 10.0)), ((10.0, 10.0), (0.0, 10.0)), ((0.0, 10.0), (0.0, 0.0)), ((2.5, 2.5), (7.5, 2.5)), ((7.5, 2.5), (7.5, 7.5)),((7.5, 7.5), (2.5, 7.5)), ((2.5, 7.5), (2.5, 2.5)))), Figure("Strange", Seq(((0.0, 0.0), (2.5, 2.5)), ((2.5, 2.5), (0.0, 10.0)), ((0.0, 10.0), (2.5, 7.5)), ((2.5, 7.5), (7.5, 7.5)), ((7.5, 7.5), (10.0, 10.0)), ((10.0, 10.0), (10.0, 0.0)), ((10.0, 0.0), (2.5, 2.5)))), Figure("Exagon", Seq(((3.0, 0.0), (7.0, 0.0)), ((7.0, 0.0), (10.0, 5.0)), ((10.0, 5.0), (7.0, 10.0)), ((7.0, 10.0), (3.0, 10.0)), ((3.0, 10.0), (0.0, 5.0)), ((0.0, 5.0), (3.0, 0.0))))) val points = Seq((5.0, 5.0), (5.0, 8.0), (-10.0, 5.0), (0.0, 5.0), (10.0, 5.0), (8.0, 5.0), (10.0, 10.0)) println("points: " + points) for (f <- figures) { println("figure: " + f.name) println(" " + f.edges) println("result: " + (points map f.contains)) } private implicit def to_edge(p: ((Double, Double), (Double, Double))): Edge = Edge(p._1, p._2) }  Output: points: List((5.0,5.0), (5.0,8.0), (-10.0,5.0), (0.0,5.0), (10.0,5.0), (8.0,5.0), (10.0,10.0)) figure: Square List(Edge((0.0,0.0),(10.0,0.0)), Edge((10.0,0.0),(10.0,10.0)), Edge((10.0,10.0),(0.0,10.0)), Edge((0.0,10.0),(0.0,0.0))) result: List(true, true, false, false, true, true, false) figure: Square hole List(Edge((0.0,0.0),(10.0,0.0)), Edge((10.0,0.0),(10.0,10.0)), Edge((10.0,10.0),(0.0,10.0)), Edge((0.0,10.0),(0.0,0.0)), Edge((2.5,2.5),(7.5,2.5)), Edge((7.5,2.5),(7.5,7.5)), Edge((7.5,7.5),(2.5,7.5)), Edge((2.5,7.5),(2.5,2.5))) result: List(false, true, false, false, true, true, false) figure: Strange List(Edge((0.0,0.0),(2.5,2.5)), Edge((2.5,2.5),(0.0,10.0)), Edge((0.0,10.0),(2.5,7.5)), Edge((2.5,7.5),(7.5,7.5)), Edge((7.5,7.5),(10.0,10.0)), Edge((10.0,10.0),(10.0,0.0)), Edge((10.0,0.0),(2.5,2.5))) result: List(true, false, false, false, true, true, false) figure: Exagon List(Edge((3.0,0.0),(7.0,0.0)), Edge((7.0,0.0),(10.0,5.0)), Edge((10.0,5.0),(7.0,10.0)), Edge((7.0,10.0),(3.0,10.0)), Edge((3.0,10.0),(0.0,5.0)), Edge((0.0,5.0),(3.0,0.0))) result: List(true, true, false, false, true, true, false)  ## Smalltalk Works with: GNU Smalltalk The class Segment holds the code to test if a ray starting from a point (and going towards "right") intersects the segment (method doesIntersectRayFrom); while the class Polygon hosts the code to test if a point is inside the polygon (method pointInside). Object subclass: Segment [ |pts| Segment class >> new: points [ |a| a := super new. ^ a init: points ] init: points [ pts := points copy. ^self ] endPoints [ ^pts ] "utility methods" first [ ^ pts at: 1] second [ ^ pts at: 2] leftmostEndPoint [ ^ (self first x > self second x) ifTrue: [ self second ] ifFalse: [ self first ] ] rightmostEndPoint [ ^ (self first x > self second x) ifTrue: [ self first ] ifFalse: [ self second ] ] topmostEndPoint [ ^ (self first y > self second y) ifTrue: [ self first ] ifFalse: [ self second ] ] bottommostEndPoint [ ^ (self first y > self second y) ifTrue: [ self second ] ifFalse: [ self first ] ] slope [ (pts at: 1) x ~= (pts at: 2) x ifTrue: [ ^ ((pts at: 1) y - (pts at: 2) y) / ((pts at: 1) x - (pts at: 2) x) ] ifFalse: [ ^ FloatD infinity ] ] doesIntersectRayFrom: point [ |p A B| (point y = (pts at: 1) y) | (point y = (pts at: 2) y) ifTrue: [ p := Point x: (point x) y: (point y) + 0.00001 ] ifFalse: [ p := point copy ]. A := self bottommostEndPoint. B := self topmostEndPoint. (p y < A y) | (p y > B y) | (p x > (self rightmostEndPoint x)) ifTrue: [ ^false ] ifFalse: [ (p x < (self leftmostEndPoint x)) ifTrue: [ ^true ] ifFalse: [ |s| s := Segment new: { A . point }. (s slope) >= (self slope) ifTrue: [ ^ true ] ] ]. ^false ] ]. Object subclass: Polygon [ |polysegs| Polygon class >> new [ |a| a := super new. ^ a init. ] Polygon class >> fromSegments: segments [ |a| a := super new. ^ a initWithSegments: segments ] Polygon class >> fromPoints: pts and: indexes [ |a| a := self new. indexes do: [ :i | a addSegment: ( Segment new: { pts at: (i at: 1) . pts at: (i at: 2) } ) ]. ^ a ] initWithSegments: segments [ polysegs := segments copy. ^self ] init [ polysegs := OrderedCollection new. ^ self ] addSegment: segment [ polysegs add: segment ] pointInside: point [ |cnt| cnt := 0. polysegs do: [ :s | (s doesIntersectRayFrom: point) ifTrue: [ cnt := cnt + 1 ] ]. ^ ( cnt \\ 2 = 0 ) not ] ].  Testing |points names polys| points := { 0@0 . 10@0 . 10@10 . 0@10 . 2.5@2.5 . 7.5@2.5 . 7.5@7.5 . 2.5@7.5 . 0@5 . 10@5 . 3@0 . 7@0 . 7@10 . 3@10 }. names := { 'square' . 'square hole' . 'strange' . 'exagon' }. polys := OrderedCollection new. polys add: ( Polygon fromPoints: points and: { {1 . 2}. {2 . 3}. {3 . 4}. {4 . 1} } ) ; add: ( Polygon fromPoints: points and: { {1 . 2}. {2 . 3}. {3 . 4}. {4 . 1}. {5 . 6}. {6 . 7}. {7 . 8}. {8 . 5} } ) ; add: ( Polygon fromPoints: points and: { {1 . 5}. {5 . 4}. {4 . 8}. {8 . 7}. {7 . 3}. {3 . 2}. {2 . 5} } ) ; add: ( Polygon fromPoints: points and: { {11 . 12}. {12 . 10}. {10 . 13}. {13 . 14}. {14 . 9}. {9 . 11} } ). { 5@5 . 5@8 . -10@5 . 0@5 . 10@5 . 8@5 . 10@10 } do: [ :p | 1 to: 4 do: [ :i | ('point %1 inside %2? %3' % { p . names at: i. (polys at: i) pointInside: p }) displayNl ]. ' ' displayNl. ]  ## Tcl package require Tcl 8.5 proc point_in_polygon {point polygon} { set count 0 foreach side [sides$polygon] {
if {[ray_intersects_line $point$side]} {
incr count
}
}
expr {$count % 2} ;#-- 1 = odd = true, 0 = even = false } proc sides polygon { lassign$polygon x0 y0
foreach {x y} [lrange [lappend polygon $x0$y0] 2 end] {
lappend res [list $x0$y0 $x$y]
set x0 $x set y0$y
}
return $res } proc ray_intersects_line {point line} { lassign$point Px Py
lassign $line Ax Ay Bx By # Reverse line direction if necessary if {$By < $Ay} { lassign$line Bx By Ax Ay
}
if {$Py ==$Ay || $Py ==$By} {
set Py [expr {$Py + abs($Py)/1e6}]
}
# Bounding box checks
if {$Py <$Ay || $Py >$By || $Px > max($Ax,$Bx)} { return 0 } elseif {$Px < min($Ax,$Bx)} {
return 1
}
# Compare dot products to compare (cosines of) angles
set mRed [expr {$Ax !=$Bx ? ($By-$Ay)/($Bx-$Ax) : Inf}]
set mBlu [expr {$Ax !=$Px ? ($Py-$Ay)/($Px-$Ax) : Inf}]
return [expr {$mBlu >=$mRed}]
}

foreach {point poly} {
{0 0}	{-1 -1  -1 1  1 1  1 -1}
{2 2}	{-1 -1  -1 1  1 1  1 -1}
{0 0}	{-2 -2  -2 2  2 2  2 -2   2 -1  1 1  -1 1  -1 -1  1 -1  2 -1}
{1.5 1.5}	{-2 -2  -2 2  2 2  2 -2   2 -1  1 1  -1 1  -1 -1  1 -1  2 -1}
{5 5}	{0 0  2.5 2.5  0 10  2.5 7.5  7.5 7.5  10 10  10 0  7.5 0.1}
{5 8}	{0 0  2.5 2.5  0 10  2.5 7.5  7.5 7.5  10 10  10 0  7.5 0.1}
{2 2}	{0 0  2.5 2.5  0 10  2.5 7.5  7.5 7.5  10 10  10 0  7.5 0.1}
{0 0}	{0 0  2.5 2.5  0 10  2.5 7.5  7.5 7.5  10 10  10 0  7.5 0.1}
{10 10}	{0 0  2.5 2.5  0 10  2.5 7.5  7.5 7.5  10 10  10 0  7.5 0.1}
{2.5 2.5}	{0 0  2.5 2.5  0 10  2.5 7.5  7.5 7.5  10 10  10 0  7.5 0.1}
{-5 5}	{3 0  7 0  10 5  7 10  3 10  0 5}
} {
puts "$point in$poly = [point_in_polygon $point$poly]"
}


## Ursala

This function takes a point ${\displaystyle (x,y)}$ and a polygon ${\displaystyle \langle (x_{1},y_{1})\dots (x_{n},y_{n})\rangle }$ to a true value if the point is enclosed by the polygon and a false value if it's outside, using the algorithm described above. For points on the boundary the result is unspecified.

#import flo

in =

@lrzyCipPX ~|afatPRZaq ~&EZ+fleq~~lrPrbr2G&& ~&B+fleq~~lrPrbl2G!| -&
~&Y+ ~~lrPrbl2G fleq,
^E(fleq@lrrPX,@rl fleq\0.)^/~&lr ^(~&r,times)^/minus@llPrll2X vid+ minus~~rbbI&-

This test program tries it on a couple of examples.

#cast %bL

examples =

in* <
((0.5,0.6),<(0.,0.),(1.,2.),(1.,0.)>),
((0.5,0.6),<(0.,0.),(1.,1.),(1.,0.)>)>

output:

<true,false>

## Visual Basic .NET

Translation of: Java
Imports System.Math

Module RayCasting

Private square As Integer()() = {New Integer() {0, 0}, New Integer() {20, 0}, New Integer() {20, 20}, New Integer() {0, 20}}
Private squareHole As Integer()() = {New Integer() {0, 0}, New Integer() {20, 0}, New Integer() {20, 20}, New Integer() {0, 20}, New Integer() {5, 5}, New Integer() {15, 5}, New Integer() {15, 15}, New Integer() {5, 15}}
Private strange As Integer()() = {New Integer() {0, 0}, New Integer() {5, 5}, New Integer() {0, 20}, New Integer() {5, 15}, New Integer() {15, 15}, New Integer() {20, 20}, New Integer() {20, 0}}
Private hexagon As Integer()() = {New Integer() {6, 0}, New Integer() {14, 0}, New Integer() {20, 10}, New Integer() {14, 20}, New Integer() {6, 20}, New Integer() {0, 10}}
Private shapes As Integer()()() = {square, squareHole, strange, hexagon}

Public Sub Main()
Dim testPoints As Double()() = {New Double() {10, 10}, New Double() {10, 16}, New Double() {-20, 10}, New Double() {0, 10}, New Double() {20, 10}, New Double() {16, 10}, New Double() {20, 20}}

For Each shape As Integer()() In shapes
For Each point As Double() In testPoints
Next
Console.WriteLine()
Next
End Sub

Private Function Contains(shape As Integer()(), point As Double()) As Boolean

Dim inside As Boolean = False
Dim length As Integer = shape.Length

For i As Integer = 0 To length - 1
If Intersects(shape(i), shape((i + 1) Mod length), point) Then
inside = Not inside
End If
Next

Return inside
End Function

Private Function Intersects(a As Integer(), b As Integer(), p As Double()) As Boolean

If a(1) > b(1) Then Return Intersects(b, a, p)
If p(1) = a(1) Or p(1) = b(1) Then p(1) += 0.0001
If p(1) > b(1) Or p(1) < a(1) Or p(0) >= Max(a(0), b(0)) Then Return False
If p(0) < Min(a(0), b(0)) Then Return True
Dim red As Double = (p(1) - a(1)) / (p(0) - a(0))
Dim blue As Double = (b(1) - a(1)) / (b(0) - a(0))

Return red >= blue
End Function
End Module

Output:
   True    True   False    True   False    True   False
False    True   False   False   False    True   False
True   False   False   False   False    True   False
True    True   False   False   False    True   False


## Wren

Translation of: Java
Library: Wren-fmt
import "./fmt" for Fmt

class RayCasting {
static intersects(a, b, p) {
if (a[1] > b[1]) return intersects(b, a, p)
if (p[1] == a[1] || p[1] == b[1]) p[1] = p[1] + 0.0001
if (p[1] > b[1] || p[1] < a[1] || p[0] >= a[0].max(b[0])) return false
if (p[0] < a[0].min(b[0])) return true
var red  = (p[1] - a[1]) / (p[0] - a[0])
var blue = (b[1] - a[1]) / (b[0] - a[0])
return red >= blue
}

static contains(shape, pnt) {
var inside = false
var len = shape.count
for (i in 0...len) {
if (intersects(shape[i], shape[(i + 1) % len], pnt)) inside = !inside
}
return inside
}

static square { [[0, 0], [20, 0], [20, 20], [0, 20]] }

static squareHole { [[0, 0], [20, 0], [20, 20], [0, 20], [5, 5], [15, 5], [15, 15], [5, 15]] }

static strange { [[0, 0], [5, 5], [0, 20], [5, 15], [15, 15], [20, 20], [20, 0]] }

static hexagon { [[6, 0], [14, 0], [20, 10], [14, 20], [6, 20], [0, 10]] }

static shapes { [square, squareHole, strange, hexagon] }
}

var testPoints = [[10, 10], [10, 16], [-20, 10], [0, 10], [20, 10], [16, 10], [20, 20]]
for (shape in RayCasting.shapes) {
for (pnt in testPoints) Fmt.write("\$7s ", RayCasting.contains(shape, pnt))
System.print()
}

Output:
   true    true   false    true   false    true   false
false    true   false   false   false    true   false
true   false   false   false   false    true   false
true    true   false   false   false    true   false


## zkl

Translation of: Raku
const E = 0.0001;

fcn rayHitsSeg([(Px,Py)],[(Ax,Ay)],[(Bx,By)]){ // --> Bool
if(Py==Ay or Py==By) Py+=E;
if(Py<Ay or Py>By or Px>Ax.max(Bx)) False
else if(Px<Ax.min(Bx))              True
else try{ ( (Py - Ay)/(Px - Ax) )>=( (By - Ay)/(Bx - Ax) ) } //blue>=red
catch(MathError){ Px==Ax } // divide by zero == ∞, only blue?
}

fcn pointInPoly(point, polygon){ // --> Bool, polygon is ( (a,b),(c,d).. )
polygon.filter('wrap([(ab,cd)]){ a,b:=ab; c,d:=cd;
if(b<=d) rayHitsSeg(point,ab,cd); // left point has smallest y coordinate
else     rayHitsSeg(point,cd,ab);
})
.len().isOdd; // True if crossed an odd number of borders ie inside polygon
}
polys:=T( //(name,( ((a,b),(c,d)),((a,b),(c,d))... ), ... )==(nm,(ln,ln..) ..)
T("squared",
T(T(T( 0.0,  0.0), T(10.0,  0.0)),
T(T(10.0,  0.0), T(10.0, 10.0)),
T(T(10.0, 10.0), T( 0.0, 10.0)),
T(T( 0.0, 10.0), T( 0.0,  0.0)))),
T("squaredhole",
T(T(T( 0.0,  0.0), T(10.0,  0.0)),
T(T(10.0,  0.0), T(10.0, 10.0)),
T(T(10.0, 10.0), T( 0.0, 10.0)),
T(T( 0.0, 10.0), T( 0.0,  0.0)),
T(T( 2.5,  2.5), T( 7.5,  2.5)),
T(T( 7.5,  2.5), T( 7.5,  7.5)),
T(T( 7.5,  7.5), T( 2.5,  7.5)),
T(T( 2.5,  7.5), T( 2.5,  2.5)))),
T("strange",
T(T(T( 0.0,  0.0), T( 2.5,  2.5)),
T(T( 2.5,  2.5), T( 0.0, 10.0)),
T(T( 0.0, 10.0), T( 2.5,  7.5)),
T(T( 2.5,  7.5), T( 7.5,  7.5)),
T(T( 7.5,  7.5), T(10.0, 10.0)),
T(T(10.0, 10.0), T(10.0,  0.0)),
T(T(10.0,  0.0), T( 2.5,  2.5)),
T(T( 2.5,  2.5), T( 0.0,  0.0)))),  # conjecturally close polygon
T("exagon",
T(T(T( 3.0,  0.0), T( 7.0,  0.0)),
T(T( 7.0,  0.0), T(10.0,  5.0)),
T(T(10.0,  5.0), T( 7.0, 10.0)),
T(T( 7.0, 10.0), T( 3.0, 10.0)),
T(T( 3.0, 10.0), T( 0.0,  5.0)),
T(T( 0.0,  5.0), T( 3.0,  0.0)))),
);

testPoints:=T(
T(  5.0,  5.0),
T(  5.0,  8.0),
T(-10.0,  5.0),
T(  0.0,  5.0),
T( 10.0,  5.0),
T(  8.0,  5.0),
T( 10.0, 10.0)
);

foreach name,polywanna in (polys){
name.println();
foreach testPoint in (testPoints){
println("\t(%6.1f,%6.1f)\t".fmt(testPoint.xplode()),
pointInPoly(testPoint,polywanna) and "IN" or "OUT");
}
}
Output:
squared
(   5.0,   5.0)	IN
(   5.0,   8.0)	IN
( -10.0,   5.0)	OUT
(   0.0,   5.0)	OUT
(  10.0,   5.0)	IN
(   8.0,   5.0)	IN
(  10.0,  10.0)	OUT
squaredhole
(   5.0,   5.0)	OUT
(   5.0,   8.0)	IN
( -10.0,   5.0)	OUT
(   0.0,   5.0)	OUT
(  10.0,   5.0)	IN
(   8.0,   5.0)	IN
(  10.0,  10.0)	OUT
strange
(   5.0,   5.0)	IN
(   5.0,   8.0)	OUT
( -10.0,   5.0)	OUT
(   0.0,   5.0)	OUT
(  10.0,   5.0)	IN
(   8.0,   5.0)	IN
(  10.0,  10.0)	OUT
exagon
(   5.0,   5.0)	IN
(   5.0,   8.0)	IN
( -10.0,   5.0)	OUT
(   0.0,   5.0)	OUT
(  10.0,   5.0)	IN
(   8.0,   5.0)	IN
(  10.0,  10.0)	OUT