- 両方のポリゴンの各エッジについて、それを分離線として使用できるかどうかを確認します。もしそうなら、あなたは終わった：交差点なし。
- 分割線が見つからなかった場合は、交差点があります。
/// Checks if the two polygons are intersecting.
bool IsPolygonsIntersecting(Polygon a, Polygon b)
{
foreach (var polygon in new[] { a, b })
{
for (int i1 = 0; i1 < polygon.Points.Count; i1++)
{
int i2 = (i1 + 1) % polygon.Points.Count;
var p1 = polygon.Points[i1];
var p2 = polygon.Points[i2];
var normal = new Point(p2.Y - p1.Y, p1.X - p2.X);
double? minA = null, maxA = null;
foreach (var p in a.Points)
{
var projected = normal.X * p.X + normal.Y * p.Y;
if (minA == null || projected < minA)
minA = projected;
if (maxA == null || projected > maxA)
maxA = projected;
}
double? minB = null, maxB = null;
foreach (var p in b.Points)
{
var projected = normal.X * p.X + normal.Y * p.Y;
if (minB == null || projected < minB)
minB = projected;
if (maxB == null || projected > maxB)
maxB = projected;
}
if (maxA < minB || maxB < minA)
return false;
}
}
return true;
}
詳細については、次の記事を参照してください。 2Dポリゴン衝突検出-コードプロジェクト
NB：アルゴリズムは、時計回りまたは反時計回りのいずれかの順序で指定された凸多角形に対してのみ機能します。
JavaScriptでは、まったく同じアルゴリズムが（便宜上）あります。
/**
* Helper function to determine whether there is an intersection between the two polygons described
* by the lists of vertices. Uses the Separating Axis Theorem
*
* @param a an array of connected points [{x:, y:}, {x:, y:},...] that form a closed polygon
* @param b an array of connected points [{x:, y:}, {x:, y:},...] that form a closed polygon
* @return true if there is any intersection between the 2 polygons, false otherwise
*/
function doPolygonsIntersect (a, b) {
var polygons = [a, b];
var minA, maxA, projected, i, i1, j, minB, maxB;
for (i = 0; i < polygons.length; i++) {
// for each polygon, look at each Edge of the polygon, and determine if it separates
// the two shapes
var polygon = polygons[i];
for (i1 = 0; i1 < polygon.length; i1++) {
// grab 2 vertices to create an Edge
var i2 = (i1 + 1) % polygon.length;
var p1 = polygon[i1];
var p2 = polygon[i2];
// find the line perpendicular to this Edge
var normal = { x: p2.y - p1.y, y: p1.x - p2.x };
minA = maxA = undefined;
// for each vertex in the first shape, project it onto the line perpendicular to the Edge
// and keep track of the min and max of these values
for (j = 0; j < a.length; j++) {
projected = normal.x * a[j].x + normal.y * a[j].y;
if (isUndefined(minA) || projected < minA) {
minA = projected;
}
if (isUndefined(maxA) || projected > maxA) {
maxA = projected;
}
}
// for each vertex in the second shape, project it onto the line perpendicular to the Edge
// and keep track of the min and max of these values
minB = maxB = undefined;
for (j = 0; j < b.length; j++) {
projected = normal.x * b[j].x + normal.y * b[j].y;
if (isUndefined(minB) || projected < minB) {
minB = projected;
}
if (isUndefined(maxB) || projected > maxB) {
maxB = projected;
}
}
// if there is no overlap between the projects, the Edge we are looking at separates the two
// polygons, and we know there is no overlap
if (maxA < minB || maxB < minA) {
CONSOLE("polygons don't intersect!");
return false;
}
}
}
return true;
};
これが誰かを助けることを願っています。
これは、Java=の同じアルゴリズムで、もし誰かが興味があれば。
boolean isPolygonsIntersecting(Polygon a, Polygon b)
{
for (int x=0; x<2; x++)
{
Polygon polygon = (x==0) ? a : b;
for (int i1=0; i1<polygon.getPoints().length; i1++)
{
int i2 = (i1 + 1) % polygon.getPoints().length;
Point p1 = polygon.getPoints()[i1];
Point p2 = polygon.getPoints()[i2];
Point normal = new Point(p2.y - p1.y, p1.x - p2.x);
double minA = Double.POSITIVE_INFINITY;
double maxA = Double.NEGATIVE_INFINITY;
for (Point p : a.getPoints())
{
double projected = normal.x * p.x + normal.y * p.y;
if (projected < minA)
minA = projected;
if (projected > maxA)
maxA = projected;
}
double minB = Double.POSITIVE_INFINITY;
double maxB = Double.NEGATIVE_INFINITY;
for (Point p : b.getPoints())
{
double projected = normal.x * p.x + normal.y * p.y;
if (projected < minB)
minB = projected;
if (projected > maxB)
maxB = projected;
}
if (maxA < minB || maxB < minA)
return false;
}
}
return true;
}
Oren Beckerが設計した、回転した長方形とフォームの交差を検出する方法を確認してください。
struct _Vector2D
{
float x, y;
};
// C:center; S: size (w,h); ang: in radians,
// rotate the plane by [-ang] to make the second rectangle axis in C aligned (vertical)
struct _RotRect
{
_Vector2D C;
_Vector2D S;
float ang;
};
また、次の関数を呼び出すと、2つの回転した長方形が交差するかどうかが返されます。
// Rotated Rectangles Collision Detection, Oren Becker, 2001
bool check_two_rotated_rects_intersect(_RotRect * rr1, _RotRect * rr2)
{
_Vector2D A, B, // vertices of the rotated rr2
C, // center of rr2
BL, TR; // vertices of rr2 (bottom-left, top-right)
float ang = rr1->ang - rr2->ang, // orientation of rotated rr1
cosa = cos(ang), // precalculated trigonometic -
sina = sin(ang); // - values for repeated use
float t, x, a; // temporary variables for various uses
float dx; // deltaX for linear equations
float ext1, ext2; // min/max vertical values
// move rr2 to make rr1 cannonic
C = rr2->C;
SubVectors2D(&C, &rr1->C);
// rotate rr2 clockwise by rr2->ang to make rr2 axis-aligned
RotateVector2DClockwise(&C, rr2->ang);
// calculate vertices of (moved and axis-aligned := 'ma') rr2
BL = TR = C;
/*SubVectors2D(&BL, &rr2->S);
AddVectors2D(&TR, &rr2->S);*/
//-----------------------------------
BL.x -= rr2->S.x/2; BL.y -= rr2->S.y/2;
TR.x += rr2->S.x/2; TR.y += rr2->S.y/2;
// calculate vertices of (rotated := 'r') rr1
A.x = -(rr1->S.y/2)*sina; B.x = A.x; t = (rr1->S.x/2)*cosa; A.x += t; B.x -= t;
A.y = (rr1->S.y/2)*cosa; B.y = A.y; t = (rr1->S.x/2)*sina; A.y += t; B.y -= t;
//---------------------------------------
//// calculate vertices of (rotated := 'r') rr1
//A.x = -rr1->S.y*sina; B.x = A.x; t = rr1->S.x*cosa; A.x += t; B.x -= t;
//A.y = rr1->S.y*cosa; B.y = A.y; t = rr1->S.x*sina; A.y += t; B.y -= t;
t = sina*cosa;
// verify that A is vertical min/max, B is horizontal min/max
if (t < 0)
{
t = A.x; A.x = B.x; B.x = t;
t = A.y; A.y = B.y; B.y = t;
}
// verify that B is horizontal minimum (leftest-vertex)
if (sina < 0) { B.x = -B.x; B.y = -B.y; }
// if rr2(ma) isn't in the horizontal range of
// colliding with rr1(r), collision is impossible
if (B.x > TR.x || B.x > -BL.x) return 0;
// if rr1(r) is axis-aligned, vertical min/max are easy to get
if (t == 0) {ext1 = A.y; ext2 = -ext1; }
// else, find vertical min/max in the range [BL.x, TR.x]
else
{
x = BL.x-A.x; a = TR.x-A.x;
ext1 = A.y;
// if the first vertical min/max isn't in (BL.x, TR.x), then
// find the vertical min/max on BL.x or on TR.x
if (a*x > 0)
{
dx = A.x;
if (x < 0) { dx -= B.x; ext1 -= B.y; x = a; }
else { dx += B.x; ext1 += B.y; }
ext1 *= x; ext1 /= dx; ext1 += A.y;
}
x = BL.x+A.x; a = TR.x+A.x;
ext2 = -A.y;
// if the second vertical min/max isn't in (BL.x, TR.x), then
// find the local vertical min/max on BL.x or on TR.x
if (a*x > 0)
{
dx = -A.x;
if (x < 0) { dx -= B.x; ext2 -= B.y; x = a; }
else { dx += B.x; ext2 += B.y; }
ext2 *= x; ext2 /= dx; ext2 -= A.y;
}
}
// check whether rr2(ma) is in the vertical range of colliding with rr1(r)
// (for the horizontal range of rr2)
return !((ext1 < BL.y && ext2 < BL.y) ||
(ext1 > TR.y && ext2 > TR.y));
}
inline void AddVectors2D(_Vector2D * v1, _Vector2D * v2)
{
v1->x += v2->x; v1->y += v2->y;
}
inline void SubVectors2D(_Vector2D * v1, _Vector2D * v2)
{
v1->x -= v2->x; v1->y -= v2->y;
}
inline void RotateVector2DClockwise(_Vector2D * v, float ang)
{
float t, cosa = cos(ang), sina = sin(ang);
t = v->x;
v->x = t*cosa + v->y*sina;
v->y = -t*sina + v->y*cosa;
}
多分それは誰かを助けるでしょう。 PHPの同じアルゴリズム：
function isPolygonsIntersecting($a, $b) {
$polygons = array($a, $b);
for ($i = 0; $i < count($polygons); $i++) {
$polygon = $polygons[$i];
for ($i1 = 0; $i1 < count($polygon); $i1++) {
$i2 = ($i1 + 1) % count($polygon);
$p1 = $polygon[$i1];
$p2 = $polygon[$i2];
$normal = array(
"x" => $p2["y"] - $p1["y"],
"y" => $p1["x"] - $p2["x"]
);
$minA = NULL; $maxA = NULL;
for ($j = 0; $j < count($a); $j++) {
$projected = $normal["x"] * $a[$j]["x"] + $normal["y"] * $a[$j]["y"];
if (!isset($minA) || $projected < $minA) {
$minA = $projected;
}
if (!isset($maxA) || $projected > $maxA) {
$maxA = $projected;
}
}
$minB = NULL; $maxB = NULL;
for ($j = 0; $j < count($b); $j++) {
$projected = $normal["x"] * $b[$j]["x"] + $normal["y"] * $b[$j]["y"];
if (!isset($minB) || $projected < $minB) {
$minB = $projected;
}
if (!isset($maxB) || $projected > $maxB) {
$maxB = $projected;
}
}
if ($maxA < $minB || $maxB < $minA) {
return false;
}
}
}
return true;
}
Rect.IntersectsWith（） を使用することもできます。
たとえば、WPFでは、RenderTransformを使用してCanvasに配置された2つのUIElementがあり、それらが交差するかどうかを確認したい場合は、次のようなものを使用できます。
bool IsIntersecting(UIElement element1, UIElement element2)
{
Rect area1 = new Rect(
(double)element1.GetValue(Canvas.TopProperty),
(double)element1.GetValue(Canvas.LeftProperty),
(double)element1.GetValue(Canvas.WidthProperty),
(double)element1.GetValue(Canvas.HeightProperty));
Rect area2 = new Rect(
(double)element2.GetValue(Canvas.TopProperty),
(double)element2.GetValue(Canvas.LeftProperty),
(double)element2.GetValue(Canvas.WidthProperty),
(double)element2.GetValue(Canvas.HeightProperty));
Transform transform1 = element1.RenderTransform as Transform;
Transform transform2 = element2.RenderTransform as Transform;
if (transform1 != null)
{
area1.Transform(transform1.Value);
}
if (transform2 != null)
{
area2.Transform(transform2.Value);
}
return area1.IntersectsWith(area2);
}
Pythonの場合：
def do_polygons_intersect(a, b):
"""
* Helper function to determine whether there is an intersection between the two polygons described
* by the lists of vertices. Uses the Separating Axis Theorem
*
* @param a an ndarray of connected points [[x_1, y_1], [x_2, y_2],...] that form a closed polygon
* @param b an ndarray of connected points [[x_1, y_1], [x_2, y_2],...] that form a closed polygon
* @return true if there is any intersection between the 2 polygons, false otherwise
"""
polygons = [a, b];
minA, maxA, projected, i, i1, j, minB, maxB = None, None, None, None, None, None, None, None
for i in range(len(polygons)):
# for each polygon, look at each Edge of the polygon, and determine if it separates
# the two shapes
polygon = polygons[i];
for i1 in range(len(polygon)):
# grab 2 vertices to create an Edge
i2 = (i1 + 1) % len(polygon);
p1 = polygon[i1];
p2 = polygon[i2];
# find the line perpendicular to this Edge
normal = { 'x': p2[1] - p1[1], 'y': p1[0] - p2[0] };
minA, maxA = None, None
# for each vertex in the first shape, project it onto the line perpendicular to the Edge
# and keep track of the min and max of these values
for j in range(len(a)):
projected = normal['x'] * a[j][0] + normal['y'] * a[j][1];
if (minA is None) or (projected < minA):
minA = projected
if (maxA is None) or (projected > maxA):
maxA = projected
# for each vertex in the second shape, project it onto the line perpendicular to the Edge
# and keep track of the min and max of these values
minB, maxB = None, None
for j in range(len(b)):
projected = normal['x'] * b[j][0] + normal['y'] * b[j][1]
if (minB is None) or (projected < minB):
minB = projected
if (maxB is None) or (projected > maxB):
maxB = projected
# if there is no overlap between the projects, the Edge we are looking at separates the two
# polygons, and we know there is no overlap
if (maxA < minB) or (maxB < minA):
print("polygons don't intersect!")
return False;
return True
Type（Java）Script実装で、（ex）includeに「タッチ」シチュエーションへの切り替えが含まれている場合：
class Position {
private _x: number;
private _y: number;
public constructor(x: number = null, y: number = null) {
this._x = x;
this._y = y;
}
public get x() { return this._x; }
public set x(value: number) { this._x = value; }
public get y() { return this._y; }
public set y(value: number) { this._y = value; }
}
class Polygon {
private _positions: Array<Position>;
public constructor(positions: Array<Position> = null) {
this._positions = positions;
}
public addPosition(position: Position) {
if (!position) {
return;
}
if (!this._positions) {
this._positions = new Array<Position>();
}
this._positions.Push(position);
}
public get positions(): ReadonlyArray<Position> { return this._positions; }
/**
* https://stackoverflow.com/a/12414951/468910
*
* Helper function to determine whether there is an intersection between the two polygons described
* by the lists of vertices. Uses the Separating Axis Theorem
*
* @param polygonToCompare a polygon to compare with
* @param allowTouch consider it an intersection when polygons only "touch"
* @return true if there is any intersection between the 2 polygons, false otherwise
*/
public isIntersecting(polygonToCompare: Polygon, allowTouch: boolean = true): boolean {
const polygons: Array<ReadonlyArray<Position>> = [this.positions, polygonToCompare.positions]
const firstPolygonPositions: ReadonlyArray<Position> = polygons[0];
const secondPolygonPositions: ReadonlyArray<Position> = polygons[1];
let minA, maxA, projected, i, i1, j, minB, maxB;
for (i = 0; i < polygons.length; i++) {
// for each polygon, look at each Edge of the polygon, and determine if it separates
// the two shapes
const polygon = polygons[i];
for (i1 = 0; i1 < polygon.length; i1++) {
// grab 2 vertices to create an Edge
const i2 = (i1 + 1) % polygon.length;
const p1 = polygon[i1];
const p2 = polygon[i2];
// find the line perpendicular to this Edge
const normal = {
x: p2.y - p1.y,
y: p1.x - p2.x
};
minA = maxA = undefined;
// for each vertex in the first shape, project it onto the line perpendicular to the Edge
// and keep track of the min and max of these values
for (j = 0; j < firstPolygonPositions.length; j++) {
projected = normal.x * firstPolygonPositions[j].x + normal.y * firstPolygonPositions[j].y;
if (!minA || projected < minA || (!allowTouch && projected === minA)) {
minA = projected;
}
if (!maxA || projected > maxA || (!allowTouch && projected === maxA)) {
maxA = projected;
}
}
// for each vertex in the second shape, project it onto the line perpendicular to the Edge
// and keep track of the min and max of these values
minB = maxB = undefined;
for (j = 0; j < secondPolygonPositions.length; j++) {
projected = normal.x * secondPolygonPositions[j].x + normal.y * secondPolygonPositions[j].y;
if (!minB || projected < minB || (!allowTouch && projected === minB)) {
minB = projected;
}
if (!maxB || projected > maxB || (!allowTouch && projected === maxB)) {
maxB = projected;
}
}
// if there is no overlap between the projects, the Edge we are looking at separates the two
// polygons, and we know there is no overlap
if (maxA < minB || (!allowTouch && maxA === minB) || maxB < minA || (!allowTouch && maxB === minA)) {
return false;
}
}
}
return true;
}
これはLUAにあります。誰かがそれを必要とするときに役立つことを願っています。
function doPolygonsIntersect(a, b)
local polygons = { a, b };
local minA, maxA, projected, i, i1, j, minB, maxB;
for i = 1, #polygons do
--// for each polygon, look at each Edge of the polygon, and determine if it separates
--// the two shapes
local polygon = polygons[i];
for i1 = 0, (#polygon-1) do
--// grab 2 vertices to create an Edge
local i2 = (i1 + 1) % (#polygon);
local p1 = polygon[i1+1];
local p2 = polygon[i2+1];
--// find the line perpendicular to this Edge
local normal = { x = p2.y - p1.y, y = p1.x - p2.x };
minA = nil;
maxA = nil;
--// for each vertex in the first shape, project it onto the line perpendicular to the Edge
--// and keep track of the min and max of these values
for j = 1, #a do
projected = normal.x * a[j].x + normal.y * a[j].y;
if (minA == nil or projected < minA) then
minA = projected;
end
if (maxA == nil or projected > maxA) then
maxA = projected;
end
end
--// for each vertex in the second shape, project it onto the line perpendicular to the Edge
--// and keep track of the min and max of these values
minB = nil;
maxB = nil;
for j = 1, #b do
projected = normal.x * b[j].x + normal.y * b[j].y;
if (minB == nil or projected < minB) then
minB = projected;
end
if (maxB == nil or projected > maxB) then
maxB = projected;
end
end
if (maxA < minB or maxB < minA) then
return false;
end
end
end
return true;
end
SriのJavaScriptを取得し、Phaser 3ポリゴンで動作するようにしました。
/// Checks if the two Phaser 3 polygons are intersecting.
gameScene.doPolygonsIntersect=function(a, b) {
// https://stackoverflow.com/questions/10962379/how-to-check-intersection-between-2-rotated-rectangles#10965077
/**
* Helper function to determine whether there is an intersection between the two polygons described
* by the lists of vertices. Uses the Separating Axis Theorem
*
* @param a an array of connected points [{x:, y:}, {x:, y:},...] that form a closed polygon
* @param b an array of connected points [{x:, y:}, {x:, y:},...] that form a closed polygon
* @return true if there is any intersection between the 2 polygons, false otherwise
*/
var polygons = [a, b];
var minA, maxA, projected, i, i1, j, minB, maxB;
for (i = 0; i < polygons.length; i++) {
// for each polygon, look at each Edge of the polygon, and determine if it separates
// the two shapes
var polygon = polygons[i];
for (i1 = 0; i1 < polygon.points.length; i1++) {
// grab 2 vertices to create an Edge
var i2 = (i1 + 1) % polygon.points.length;
var p1 = polygon.points[i1];
var p2 = polygon.points[i2];
// find the line perpendicular to this Edge
var normal = { x: p2.y - p1.y, y: p1.x - p2.x };
minA = maxA = undefined;
// for each vertex in the first shape, project it onto the line perpendicular to the Edge
// and keep track of the min and max of these values
for (j = 0; j < a.points.length; j++) {
projected = normal.x * a.points[j].x + normal.y * a.points[j].y;
if (!isDef(minA) || projected < minA) {
minA = projected;
}
if (!isDef(maxA) || projected > maxA) {
maxA = projected;
}
}
// for each vertex in the second shape, project it onto the line perpendicular to the Edge
// and keep track of the min and max of these values
minB = maxB = undefined;
for (j = 0; j < b.points.length; j++) {
projected = normal.x * b.points[j].x + normal.y * b.points[j].y;
if (!isDef(minB) || projected < minB) {
minB = projected;
}
if (!isDef(maxB) || projected > maxB) {
maxB = projected;
}
}
// if there is no overlap between the projects, the Edge we are looking at separates the two
// polygons, and we know there is no overlap
if (maxA < minB || maxB < minA) {
console.log("polygons don't intersect!");
return false;
}
}
}
return true;
};
Love2dフレームワークに組み込まれたLua実装。衝突検出機能はとにかく純粋なルアで動作します
math.inf = 1e309
function love.load()
pol = {{0, 0}, {30, 2}, {8, 30}}
pol2 = {{60, 60}, {90, 61}, {98, 100}, {80, 100}}
end
function love.draw()
for k,v in ipairs(pol) do
love.graphics.line(pol[k][1], pol[k][2], pol[k % #pol + 1][1], pol[k % #pol + 1][2])
end
for k,v in ipairs(pol2) do
love.graphics.line(pol2[k][1], pol2[k][2], pol2[k % #pol2 + 1][1], pol2[k % #pol2 + 1][2])
end
end
function love.update(dt)
pol[1][1] = love.mouse.getX()
pol[1][2] = love.mouse.getY()
pol[2][1] = pol[1][1] + 30
pol[2][2] = pol[1][2] + 2
pol[3][1] = pol[1][1] + 8
pol[3][2] = pol[1][2] + 30
--lazy way to see that's function works
print(doPolygonsIntersect(pol, pol2))
end
-------------------------------------------------------------------------
function doPolygonsIntersect(a,b)
polygons = {a,b}
for i=1, #polygons do
polygon = polygons[i]
for i1=1, #polygon do
i2 = i1 % #polygon + 1
p1 = polygon[i1]
p2 = polygon[i2]
nx,ny = p2[2] - p1[2], p1[1] - p2[1]
minA = math.inf
maxA = -math.inf
for j=1, #a do
projected = nx * a[j][1] + ny * a[j][2]
if projected < minA then minA = projected end
if projected > maxA then maxA = projected end
end
minB = math.inf
maxB = -math.inf
for j=1, #b do
projected = nx * b[j][1] + ny * b[j][2]
if projected < minB then minB = projected end
if projected > maxB then maxB = projected end
end
if maxA < minB or maxB < minA then return false end
end
end
return true
end