Check if two polygons overlap
You are encouraged to solve this task according to the task description, using any language you may know.
Self-explanatory: given two polygons (as a list of their vertices), check whether they overlap.
- Related tasks
ALGOL 68
BEGIN # test for overlapping 2D polygons #
# - based on a translation Go which is a translation of Wren #
# In the following a polygon is represented as a row of vertices #
# and a vertex ( POINT ) by a pair of x, y coordinates in the plane #
MODE POINT = STRUCT( REAL x, y );
MODE PROJECTION = STRUCT( REAL min, max );
MODE POLYGON = FLEX[ 1 : 0 ]POINT;
PRIO DOT = 3;
OP DOT = ( POINT v, other )REAL:
( x OF v * x OF other ) + ( y OF v * y OF other );
# returns the axes of the polygon defined by poly #
OP AXES = ( POLYGON poly )[]POINT:
BEGIN
[ LWB poly : UPB poly ]POINT result;
FOR i FROM LWB poly TO UPB poly DO
INT j = IF i = UPB poly THEN LWB poly ELSE i + 1 FI;
POINT vertex1 = poly[ i ];
POINT vertex2 = poly[ j ];
POINT edge = ( x OF vertex1 - x OF vertex2, y OF vertex1 - y OF vertex2 );
result[ i ] := ( - y OF edge, x OF edge )
OD;
result
END # AXES # ;
# returns the projection of poly onto axis #
PRIO PROJECTONTO = 3;
OP PROJECTONTO = ( POLYGON poly, POINT axis )PROJECTION:
BEGIN
REAL min := axis DOT poly[ LWB poly ];
REAL max := min;
FOR i FROM LWB poly + 1 TO UPB poly DO
REAL p = axis DOT poly[ i ];
IF p < min THEN
min := p
ELIF p > max THEN
max := p
FI
OD;
PROJECTION( min, max )
END # PROJECTONTO # ;
PRIO OVERLAPS = 5;
# returns TRUE if the projections proj1 and proj2 overlap, #
# FALSE otherrwise #
OP OVERLAPS = ( PROJECTION proj1, proj2 )BOOL:
IF max OF proj1 < min OF proj2 THEN FALSE
ELIF max OF proj2 < min OF proj1 THEN FALSE
ELSE TRUE
FI # OVERLAPS # ;
# returns TRUE if the ppolygons poly1 and poly2 overlap, #
# FALSE otherrwise #
OP OVERLAPS = ( POLYGON poly1, poly2 )BOOL:
BEGIN
[]POINT axes1 = AXES poly1, axes2 = AXES poly2;
BOOL does overlap := TRUE;
FOR a FROM LWB axes1 TO UPB axes1 WHILE does overlap DO
does overlap := ( poly1 PROJECTONTO axes1[ a ] )
OVERLAPS ( poly2 PROJECTONTO axes1[ a ] )
OD;
FOR a FROM LWB axes2 TO UPB axes2 WHILE does overlap DO
does overlap := ( poly1 PROJECTONTO axes2[ a ] )
OVERLAPS ( poly2 PROJECTONTO axes2[ a ] )
OD;
does overlap
END # OVERLAPS # ;
# returns x as a string without trailing 0 decoimals #
OP TOSTRING = ( REAL x )STRING:
BEGIN
STRING v := fixed( x, -14, 11 );
INT end pos := UPB v;
WHILE IF end pos < LWB v THEN FALSE ELSE v[ end pos ] = "0" FI DO
end pos -:= 1
OD;
IF end pos >= LWB v THEN
IF v[ end pos ] = "." THEN end pos -:= 1 FI
FI;
INT start pos := LWB v;
WHILE IF start pos > end pos THEN FALSE ELSE v[ start pos ] = " " FI DO
start pos +:= 1
OD;
IF end pos < start pos THEN "0" ELSE v[ start pos : end pos ] FI
END # TOSTRING # ;
# returns a string representation of the POINT p #
OP TOSTRING = ( POINT p )STRING: "( " + TOSTRING x OF p + ", " + TOSTRING y OF p + " )";
# returns a string representation of the points of p #
OP TOSTRING = ( POLYGON p )STRING:
BEGIN
STRING result := "(", separator := "";
FOR i FROM LWB p TO UPB p DO
result +:= separator + " " + TOSTRING p[ i ];
separator := ","
OD;
result + " )"
END # TOSTRING # ;
# test cases #
POLYGON poly1 = ( ( 0, 0 ), ( 0, 2 ), ( 1, 4 ), ( 2, 2 ), ( 2, 0 ) )
, poly2 = ( ( 4, 0 ), ( 4, 2 ), ( 5, 4 ), ( 6, 2 ), ( 6, 0 ) )
, poly3 = ( ( 1, 0 ), ( 1, 2 ), ( 5, 4 ), ( 9, 2 ), ( 9, 0 ) )
;
print( ( "poly1 = ", TOSTRING poly1, newline ) );
print( ( "poly2 = ", TOSTRING poly2, newline ) );
print( ( "poly3 = ", TOSTRING poly3, newline ) );
print( ( newline ) );
print( ( "poly1 and poly2 overlap? ", IF poly1 OVERLAPS poly2 THEN "yes" ELSE "no" FI, newline ) );
print( ( "poly1 and poly3 overlap? ", IF poly1 OVERLAPS poly3 THEN "yes" ELSE "no" FI, newline ) );
print( ( "poly2 and poly3 overlap? ", IF poly2 OVERLAPS poly3 THEN "yes" ELSE "no" FI, newline ) )
END- Output:
poly1 = ( ( 0, 0 ), ( 0, 2 ), ( 1, 4 ), ( 2, 2 ), ( 2, 0 ) ) poly2 = ( ( 4, 0 ), ( 4, 2 ), ( 5, 4 ), ( 6, 2 ), ( 6, 0 ) ) poly3 = ( ( 1, 0 ), ( 1, 2 ), ( 5, 4 ), ( 9, 2 ), ( 9, 0 ) ) poly1 and poly2 overlap? no poly1 and poly3 overlap? yes poly2 and poly3 overlap? yes
C
#include <stdio.h>
#include <stdbool.h>
typedef struct {
double x;
double y;
} Vector2;
typedef struct {
double min;
double max;
} Projection;
double dot(Vector2 v1, Vector2 v2) {
return v1.x * v2.x + v1.y * v2.y;
}
/* In the following a polygon is represented as an array of vertices
and a vertex by a pair of x, y coordinates in the plane. */
void getAxes(double poly[][2], size_t len, Vector2 axes[len]) {
int i, j;
Vector2 vector1, vector2, edge;
for (i = 0; i < len; ++i) {
vector1 = (Vector2){poly[i][0], poly[i][1]};
j = (i + 1 == len) ? 0 : i + 1;
vector2 = (Vector2){poly[j][0], poly[j][1]};
edge = (Vector2){vector1.x - vector2.x, vector1.y - vector2.y};
axes[i].x = -edge.y;
axes[i].y = edge.x;
}
}
Projection projectOntoAxis(double poly[][2], size_t len, Vector2 axis) {
int i;
Vector2 vector0, vector;
double min, max, p;
vector0 = (Vector2){poly[0][0], poly[0][1]};
min = dot(axis, vector0);
max = min;
for (i = 1; i < len; ++i) {
vector = (Vector2){poly[i][0], poly[i][1]};
p = dot(axis, vector);
if (p < min) {
min = p;
} else if (p > max) {
max = p;
}
}
return (Projection){min, max};
}
bool projectionsOverlap(Projection proj1, Projection proj2) {
if (proj1.max < proj2.min) return false;
if (proj2.max < proj1.min) return false;
return true;
}
bool polygonsOverlap(double poly1[][2], double poly2[][2], size_t len1, size_t len2) {
int i;
Vector2 axis, axes1[len1], axes2[len2];
Projection proj1, proj2;
getAxes(poly1, len1, axes1);
getAxes(poly2, len2, axes2);
for (i = 0; i < len1; ++i) {
axis = axes1[i];
proj1 = projectOntoAxis(poly1, len1, axis);
proj2 = projectOntoAxis(poly2, len2, axis);
if (!projectionsOverlap(proj1, proj2)) return false;
}
for (i = 0; i < len2; ++i) {
axis = axes2[i];
proj1 = projectOntoAxis(poly1, len1, axis);
proj2 = projectOntoAxis(poly2, len2, axis);
if (!projectionsOverlap(proj1, proj2)) return false;
}
return true;
}
void printPoly(double poly[][2], size_t len) {
int i, j;
printf("{ ");
for (i = 0; i < len; ++i) {
printf("{");
for (j = 0; j < 2; ++j) {
printf("%g", poly[i][j]);
if (j == 0) printf(", ");
}
printf("}");
if (i < len-1) printf(", ");
}
printf(" }\n");
}
int main() {
double poly1[][2] = { {0, 0}, {0, 2}, {1, 4}, {2, 2}, {2, 0} };
double poly2[][2] = { {4, 0}, {4, 2}, {5, 4}, {6, 2}, {6, 0} };
double poly3[][2] = { {1, 0}, {1, 2}, {5, 4}, {9, 2}, {9, 0} };
printf("poly1 = ");
printPoly(poly1, 5);
printf("poly2 = ");
printPoly(poly2, 5);
printf("poly3 = ");
printPoly(poly3, 5);
printf("\n");
printf("poly1 and poly2 overlap? %s\n", polygonsOverlap(poly1, poly2, 5, 5) ? "true" : "false");
printf("poly1 and poly3 overlap? %s\n", polygonsOverlap(poly1, poly3, 5, 5) ? "true" : "false");
printf("poly2 and poly3 overlap? %s\n", polygonsOverlap(poly2, poly3, 5, 5) ? "true" : "false");
return 0;
}
- Output:
poly1 = { {0, 0}, {0, 2}, {1, 4}, {2, 2}, {2, 0} }
poly2 = { {4, 0}, {4, 2}, {5, 4}, {6, 2}, {6, 0} }
poly3 = { {1, 0}, {1, 2}, {5, 4}, {9, 2}, {9, 0} }
poly1 and poly2 overlap? false
poly1 and poly3 overlap? true
poly2 and poly3 overlap? true
Go
package main
import "fmt"
type Vector2 struct {
x, y float64
}
type Projection struct {
min, max float64
}
func (v Vector2) dot(other Vector2) float64 {
return v.x*other.x + v.y*other.y
}
/* In the following a polygon is represented as a slice of vertices
and a vertex by a pair of x, y coordinates in the plane. */
func getAxes(poly [][2]float64) []Vector2 {
axes := make([]Vector2, len(poly))
for i := 0; i < len(poly); i++ {
vertex1 := poly[i]
j := i + 1
if i+1 == len(poly) {
j = 0
}
vertex2 := poly[j]
vector1 := Vector2{vertex1[0], vertex1[1]}
vector2 := Vector2{vertex2[0], vertex2[1]}
edge := Vector2{vector1.x - vector2.x, vector1.y - vector2.y}
axes[i] = Vector2{-edge.y, edge.x}
}
return axes
}
func projectOntoAxis(poly [][2]float64, axis Vector2) Projection {
vertex0 := poly[0]
vector0 := Vector2{vertex0[0], vertex0[1]}
min := axis.dot(vector0)
max := min
for i := 1; i < len(poly); i++ {
vertex := poly[i]
vector := Vector2{vertex[0], vertex[1]}
p := axis.dot(vector)
if p < min {
min = p
} else if p > max {
max = p
}
}
return Projection{min, max}
}
func projectionsOverlap(proj1, proj2 Projection) bool {
if proj1.max < proj2.min {
return false
}
if proj2.max < proj1.min {
return false
}
return true
}
func polygonsOverlap(poly1, poly2 [][2]float64) bool {
axes1 := getAxes(poly1)
axes2 := getAxes(poly2)
for _, axes := range [][]Vector2{axes1, axes2} {
for _, axis := range axes {
proj1 := projectOntoAxis(poly1, axis)
proj2 := projectOntoAxis(poly2, axis)
if !projectionsOverlap(proj1, proj2) {
return false
}
}
}
return true
}
func main() {
poly1 := [][2]float64{{0, 0}, {0, 2}, {1, 4}, {2, 2}, {2, 0}}
poly2 := [][2]float64{{4, 0}, {4, 2}, {5, 4}, {6, 2}, {6, 0}}
poly3 := [][2]float64{{1, 0}, {1, 2}, {5, 4}, {9, 2}, {9, 0}}
fmt.Println("poly1 = ", poly1)
fmt.Println("poly2 = ", poly2)
fmt.Println("poly3 = ", poly3)
fmt.Println()
fmt.Println("poly1 and poly2 overlap?", polygonsOverlap(poly1, poly2))
fmt.Println("poly1 and poly3 overlap?", polygonsOverlap(poly1, poly3))
fmt.Println("poly2 and poly3 overlap?", polygonsOverlap(poly2, poly3))
}
- Output:
poly1 = [[0 0] [0 2] [1 4] [2 2] [2 0]] poly2 = [[4 0] [4 2] [5 4] [6 2] [6 0]] poly3 = [[1 0] [1 2] [5 4] [9 2] [9 0]] poly1 and poly2 overlap? false poly1 and poly3 overlap? true poly2 and poly3 overlap? true
jq
Adapted from Wren (2D convex polygons)
Also works with gojq, the Go implementation of jq
In the following:
- a vertex is represented by a pair of x, y coordinates in the plane;
- a polygon is represented as a list of vertices;
- a projection is represented by a JSON object {min, max}
# Input: [$A, $B] where $A and $B are points
# Output: the vector $B - $A
def AB:
. as [$A, $B]
| [ $B[0] - $A[0], $B[1] - $A[1]];
# Input: a vector
# Output: perpendicular
def perp: [- .[1], .[0]];
# dot product of this and $v, assumed to be of the same dimension
def dot($v):
. as $this
| reduce range(0; $this|length) as $i (0; . + ($this[$i] * $v[$i] ));
def getAxes:
. as $poly
| reduce range(0; $poly|length) as $i ([];
$poly[$i] as $vertex1
| $poly[if $i+1 == ($poly|length) then 0 else $i+1 end] as $vertex2
| . + [ [$vertex1, $vertex2] | AB | perp] );
# emit {min, max}
def projectOntoAxis($axis):
. as $poly
| { max: - infinite, min: infinite }
| reduce range(0; $poly|length) as $i (.;
($axis | dot( $poly[$i] )) as $p
| if $p < .min then .min = $p else . end
| if $p > .max then .max = $p else . end ) ;
def projectionsOverlap($proj1; $proj2):
if $proj1.max < $proj2.min then false
elif $proj2.max < $proj1.min then false
else true
end;
# If there's an axis for which the projections do not overlap, then false; else true
def polygonsOverlap($poly1; $poly2):
any( $poly1, $poly2 | getAxes[];
. as $axis
| ($poly1 | projectOntoAxis($axis)) as $proj1
| ($poly2 | projectOntoAxis($axis)) as $proj2
| projectionsOverlap($proj1; $proj2) | not)
| not;
def poly1: [[0, 0], [0, 2], [1, 4], [2, 2], [2, 0]];
def poly2: [[4, 0], [4, 2], [5, 4], [6, 2], [6, 0]];
def poly3: [[1, 0], [1, 2], [5, 4], [9, 2], [9, 0]];
def task:
"poly1 = \(poly1)",
"poly2 = \(poly2)",
"poly3 = \(poly3)",
"",
"poly1 and poly2 overlap? \(polygonsOverlap(poly1; poly2))",
"poly1 and poly3 overlap? \(polygonsOverlap(poly1; poly3))",
"poly2 and poly3 overlap? \(polygonsOverlap(poly2; poly3))"
;
task- Output:
As for #Wren
Julia
using Plots
using Polyhedra
using GLPK
lib = DefaultLibrary{Float64}(GLPK.Optimizer)
const poly1 = polyhedron(vrep([
0 0
0 2
1 4
2 2
2 0
]), lib)
const poly2 = polyhedron(vrep([
4 0
4 2
5 4
6 2
6 0
]), lib)
const poly3 = polyhedron(vrep([
1 0
1 2
5 4
9 2
9 0
]), lib)
println("Polygons poly1 and poly2 intersect at ", npoints(intersect(poly1, poly2)), " points.")
println("Polygons poly1 and poly3 intersect at ", npoints(intersect(poly1, poly3)), " points.")
println("Polygons poly2 and poly3 intersect at ", npoints(intersect(poly2, poly3)), " points.")
const P1 = polyhedron(vrep([
-1.9 -1.7
-1.8 0.5
1.7 0.7
1.9 -0.3
0.9 -1.1
]), lib)
const P2 = polyhedron(vrep([
-2.5 -1.1
-0.8 0.8
0.1 0.9
1.8 -1.2
1.3 0.1
]), lib)
Pint = intersect(P1, P2)
println("Polygons P1 and P2 intersect at ", npoints(Pint), " points.")
plot(P1, color="blue", alpha=0.2)
plot!(P2, color="red", alpha=0.2)
plot!(Pint, color="yellow", alpha=0.6)
- Output:
Polygons poly1 and poly2 intersect at 0 points. Polygons poly1 and poly3 intersect at 5 points. Polygons poly2 and poly3 intersect at 5 points. Polygons P1 and P2 intersect at 8 points.
Nim
type
Vector2 = (float, float)
Projection = tuple[min, max: float]
Polygon = seq[Vector2]
func dot(v1, v2: Vector2): float =
v1[0] * v2[0] + v1[1] * v2[1]
func axes(poly: Polygon): seq[Vector2] =
result.setLen(poly.len)
for i, vertex1 in poly:
let vertex2 = poly[if i + 1 == poly.len: 0 else: i + 1]
let edge = (vertex1[0] - vertex2[0], vertex1[1] - vertex2[1])
result[i] = (-edge[1], edge[0])
func projectionOnAxis(poly: Polygon; axis: Vector2): Projection =
result.min = Inf
result.max = -Inf
for vertex in poly:
let p = axis.dot(vertex)
if p < result.min:
result.min = p
if p > result.max:
result.max = p
func projectionOverlaps(proj1, proj2: Projection): bool =
if proj1.max < proj2.min: return false
if proj2.max < proj1.min: return false
result = true
func polygonsOverlap(poly1, poly2: Polygon): bool =
for axes in [poly1.axes, poly2.axes]:
for axis in axes:
let proj1 = poly1.projectionOnAxis(axis)
let proj2 = poly2.projectionOnAxis(axis)
if not projectionOverlaps(proj1, proj2):
return false
result = true
let poly1 = @[(0.0, 0.0), (0.0, 2.0), (1.0, 4.0), (2.0, 2.0), (2.0, 0.0)]
let poly2 = @[(4.0, 0.0), (4.0, 2.0), (5.0, 4.0), (6.0, 2.0), (6.0, 0.0)]
let poly3 = @[(1.0, 0.0), (1.0, 2.0), (5.0, 4.0), (9.0, 2.0), (9.0, 0.0)]
echo "poly1 = ", poly1
echo "poly2 = ", poly2
echo "poly3 = ", poly3
echo()
echo "poly1 and poly2 overlap? ", polygonsOverlap(poly1, poly2)
echo "poly1 and poly3 overlap? ", polygonsOverlap(poly1, poly3)
echo "poly2 and poly3 overlap? ", polygonsOverlap(poly2, poly3)
- Output:
poly1 = @[(0.0, 0.0), (0.0, 2.0), (1.0, 4.0), (2.0, 2.0), (2.0, 0.0)] poly2 = @[(4.0, 0.0), (4.0, 2.0), (5.0, 4.0), (6.0, 2.0), (6.0, 0.0)] poly3 = @[(1.0, 0.0), (1.0, 2.0), (5.0, 4.0), (9.0, 2.0), (9.0, 0.0)] poly1 and poly2 overlap? false poly1 and poly3 overlap? true poly2 and poly3 overlap? true
Phix
-- demo\rosetta\Polygons_overlap.exw with javascript_semantics function getAxes(sequence poly) integer l = length(poly) sequence axes = repeat(0,l) for i=1 to l do sequence p = poly[i], n = poly[iff(i=l?1:i+1)] axes[i] = {n[2]-p[2],p[1]-n[1]} -- ie {-y,x} end for return axes end function function projectOntoAxis(sequence poly,axis) atom {ax,ay} = axis sequence p = repeat(0,length(poly)) for i,pi in poly do atom {px,py} = pi p[i] = ax*px+ay*py end for return {min(p),max(p)} end function function projectionsOverlap(sequence proj1, proj2) atom {min1,max1} = proj1, {min2,max2} = proj2 return max1>=min2 and max2>=min1 end function function polygonsOverlap(sequence poly1, poly2) sequence axes1 = getAxes(poly1), axes2 = getAxes(poly2) for axes in {axes1, axes2} do for axis in axes do sequence proj1 = projectOntoAxis(poly1, axis), proj2 = projectOntoAxis(poly2, axis) if not projectionsOverlap(proj1, proj2) then return false end if end for end for return true end function constant poly1 = {{0, 0}, {0, 2}, {1, 4}, {2, 2}, {2, 0}}, poly2 = {{4, 0}, {4, 2}, {5, 4}, {6, 2}, {6, 0}}, poly3 = {{1, 0}, {1, 2}, {5, 4}, {9, 2}, {9, 0}}, fmt = """ poly1 = %v poly2 = %v poly3 = %v poly1 and poly2 overlap? %t poly1 and poly3 overlap? %t poly2 and poly3 overlap? %t """ printf(1,fmt,{poly1,poly2,poly3,polygonsOverlap(poly1, poly2), polygonsOverlap(poly1, poly3), polygonsOverlap(poly2, poly3)})
- Output:
poly1 = {{0,0},{0,2},{1,4},{2,2},{2,0}}
poly2 = {{4,0},{4,2},{5,4},{6,2},{6,0}}
poly3 = {{1,0},{1,2},{5,4},{9,2},{9,0}}
poly1 and poly2 overlap? false
poly1 and poly3 overlap? true
poly2 and poly3 overlap? true
Python
# overlapping polygons by xing216
from math import inf
class Vector2:
def __init__(self, x: float, y: float) -> None:
self.x = x
self.y = y
def dot(self, other: 'Vector2') -> float:
return self.x * other.x + self.y * other.y
def __repr__(self) -> str:
return f'({self.x}, {self.y})'
class Projection:
min: float
max: float
def overlaps(self, proj2: 'Projection') -> bool:
if self.max < proj2.min or proj2.max < self.min: return False
return True
class Polygon:
def __init__(self, vertices: list[Vector2]) -> None:
self.vertices = vertices
self.axes = self.get_axes()
def get_axes(self) -> list[Vector2]:
axes = []
for i, vertex1 in enumerate(self.vertices):
if i + 1 == len(self.vertices): vertex2 = self.vertices[0]
else: self.vertices[i + 1]
edge = (vertex1.x - vertex2.x, vertex1.y - vertex2.y)
axes.append(Vector2(-edge[1], edge[0]))
return axes
def projection_on_axis(self, axis: Vector2) -> Projection:
projection = Projection()
projection.min = inf
projection.max = -inf
for vertex in self.vertices:
p = axis.dot(vertex)
if p < projection.min:
projection.min = p
if p > projection.max:
projection.max = p
return projection
def overlaps(self, other: 'Polygon') -> bool:
for axes in [self.axes, other.axes]:
for axis in axes:
proj1 = self.projection_on_axis(axis)
proj2 = other.projection_on_axis(axis)
if not proj1.overlaps(proj2): return False
return True
poly1 = [(0.0, 0.0), (0.0, 2.0), (1.0, 4.0), (2.0, 2.0), (2.0, 0.0)]
poly2 = [(4.0, 0.0), (4.0, 2.0), (5.0, 4.0), (6.0, 2.0), (6.0, 0.0)]
poly3 = [(1.0, 0.0), (1.0, 2.0), (5.0, 4.0), (9.0, 2.0), (9.0, 0.0)]
polygons = (poly1, poly2, poly3)
polygons = [Polygon([Vector2(*vertex) for vertex in polygon]) for polygon in polygons]
for i, polygon in enumerate(polygons):
print(f'poly{i+1} = {polygon.vertices}')
print(f'poly1 and poly2 overlap? {polygons[0].overlaps(polygons[1])}')
print(f'poly1 and poly3 overlap? {polygons[0].overlaps(polygons[2])}')
print(f'poly2 and poly3 overlap? {polygons[1].overlaps(polygons[2])}')
- Output:
poly1 = [(0.0, 0.0), (0.0, 2.0), (1.0, 4.0), (2.0, 2.0), (2.0, 0.0)] poly2 = [(4.0, 0.0), (4.0, 2.0), (5.0, 4.0), (6.0, 2.0), (6.0, 0.0)] poly3 = [(1.0, 0.0), (1.0, 2.0), (5.0, 4.0), (9.0, 2.0), (9.0, 0.0)] poly1 and poly2 overlap? False poly1 and poly3 overlap? True poly2 and poly3 overlap? True
Raku
# 20230810 Raku programming solution
class Vector2 { has ( $.x, $.y );
method dot ( \other ) { self.x * other.x + self.y * other.y }
};
class Projection { has ( $.min, $.max ) };
sub getAxes ( \poly ) {
return poly.append(poly[0]).rotor(2=>-1).map: -> (\vertex1,\vertex2) {
my \vector1 = Vector2.new: x => vertex1[0], y => vertex1[1];
my \vector2 = Vector2.new: x => vertex2[0], y => vertex2[1];
my \edge = Vector2.new: x => vector1.x - vector2.x,
y => vector1.y - vector2.y;
$_ = Vector2.new: x => -edge.y, y => edge.x
}
}
sub projectOntoAxis ( \poly, \axis ) {
my \vertex0 = poly[0];
my \vector0 = Vector2.new: x => vertex0[0], y => vertex0[1];
my $max = my $min = axis.dot: vector0;
for poly -> \vertex {
my \vector = Vector2.new: x => vertex[0], y => vertex[1];
given axis.dot: vector { when $_ < $min { $min = $_ }
when $_ > $max { $max = $_ } }
}
return Projection.new: min => $min, max => $max
}
sub projectionsOverlap ( \proj1, \proj2 ) {
return ! ( proj1.max < proj2.min or proj2.max < proj1.min )
}
sub polygonsOverlap( \poly1, \poly2 ) {
my (\axes1,\axes2) := (poly1,poly2).map: { getAxes $_ };
for (axes1, axes2) -> \axes {
for axes -> \axis {
my (\proj1,\proj2) := (poly1,poly2).map: { projectOntoAxis $_, axis }
return False unless projectionsOverlap(proj1, proj2)
}
}
return True
}
my \poly1 = [ <0 0>, <0 2>, <1 4>, <2 2>, <2 0> ];
my \poly2 = [ <4 0>, <4 2>, <5 4>, <6 2>, <6 0> ];
my \poly3 = [ <1 0>, <1 2>, <5 4>, <9 2>, <9 0> ];
say "poly1 = ", poly1;
say "poly2 = ", poly2;
say "poly3 = ", poly3;
say();
say "poly1 and poly2 overlap? ", polygonsOverlap(poly1, poly2);
say "poly1 and poly3 overlap? ", polygonsOverlap(poly1, poly3);
say "poly2 and poly3 overlap? ", polygonsOverlap(poly2, poly3);
You may Attempt This Online!
Wren
The task description doesn't say whether the polygons are 2D or 3D and whether they're convex or not. So for now, I've assumed the simplest case of 2D convex polygons though a non-convex polygon can always be decomposed into a combination of convex polygons.
The approach here is based on the Separating Axis theorem ("SAT"). See here for a simple explanation of this with pseudo-code.
import "./vector" for Vector2
import "./dynamic" for Tuple
var Projection = Tuple.create("Projection", ["min", "max"])
/* In the following a polygon is represented as a list of vertices
and a vertex by a pair of x, y coordinates in the plane. */
var getAxes = Fn.new { |poly|
var axes = List.filled(poly.count, null)
for (i in 0...poly.count) {
var vertex1 = poly[i]
var vertex2 = poly[(i+1 == poly.count) ? 0 : i+1]
var vector1 = Vector2.new(vertex1[0], vertex1[1])
var vector2 = Vector2.new(vertex2[0], vertex2[1])
var edge = vector1 - vector2
axes[i] = edge.perp
}
return axes
}
var projectOntoAxis = Fn.new { |poly, axis|
var vertex0 = poly[0]
var vector0 = Vector2.new(vertex0[0], vertex0[1])
var min = axis.dot(vector0)
var max = min
for (i in 1...poly.count) {
var vertex = poly[i]
var vector = Vector2.new(vertex[0], vertex[1])
var p = axis.dot(vector)
if (p < min) {
min = p
} else if (p > max) {
max = p
}
}
return Projection.new(min, max)
}
var projectionsOverlap = Fn.new { |proj1, proj2|
if (proj1.max < proj2.min) return false
if (proj2.max < proj1.min) return false
return true
}
var polygonsOverlap = Fn.new { |poly1, poly2|
var axes1 = getAxes.call(poly1)
var axes2 = getAxes.call(poly2)
for (axes in [axes1, axes2]) {
for (axis in axes) {
var proj1 = projectOntoAxis.call(poly1, axis)
var proj2 = projectOntoAxis.call(poly2, axis)
if (!projectionsOverlap.call(proj1, proj2)) return false
}
}
return true
}
var poly1 = [[0, 0], [0, 2], [1, 4], [2, 2], [2, 0]]
var poly2 = [[4, 0], [4, 2], [5, 4], [6, 2], [6, 0]]
var poly3 = [[1, 0], [1, 2], [5, 4], [9, 2], [9, 0]]
System.print("poly1 = %(poly1)")
System.print("poly2 = %(poly2)")
System.print("poly3 = %(poly3)")
System.print()
System.print("poly1 and poly2 overlap? %(polygonsOverlap.call(poly1, poly2))")
System.print("poly1 and poly3 overlap? %(polygonsOverlap.call(poly1, poly3))")
System.print("poly2 and poly3 overlap? %(polygonsOverlap.call(poly2, poly3))")
- Output:
poly1 = [[0, 0], [0, 2], [1, 4], [2, 2], [2, 0]] poly2 = [[4, 0], [4, 2], [5, 4], [6, 2], [6, 0]] poly3 = [[1, 0], [1, 2], [5, 4], [9, 2], [9, 0]] poly1 and poly2 overlap? false poly1 and poly3 overlap? true poly2 and poly3 overlap? true