Check if two polygons overlap

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

• Determine if two triangles overlap
• Check if a polygon overlaps with a rectangle

C

Translation of: Wren

#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

Translation of: Wren

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

Works with: jq

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 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[0] - $vertex2[0], $vertex1[1] - $vertex2[1]] as $edge
      | . + [$edge | 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.

Phix

Translation of: Wren

-- 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

Wren

Library: Wren-vector

Library: Wren-dynamic

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