Comment vérifier l'intersection entre 2 rectangles pivotés?
Quelqu'un peut-il expliquer comment vérifier si un rectangle pivoté croise un autre rectangle ?
- Vérifiez si elle peut être utilisée comme ligne de séparation pour chaque bord dans les deux polygones. Si c'est le cas, vous avez terminé: pas d'intersection.
- Si aucune ligne de séparation n'a été trouvée, vous avez une intersection.
/// 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;
}
Pour plus d'informations, consultez cet article: Détection de collision de polygones 2D - Projet de code
NB: L'algorithme ne fonctionne que pour les polygones convexes, spécifiés dans le sens horaire ou dans le sens antihoraire.
En javascript, le même algorithme est (par commodité):
/**
* 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;
};
J'espère que ça aide quelqu'un.
Voici le même algorithme en Java si quelqu'un est intéressé.
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;
}
Découvrez la méthode conçue par Oren Becker pour détecter l'intersection de rectangles pivotés avec la forme:
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;
};
Et appeler la fonction suivante retournera si deux rectangles pivotés se croisent ou non:
// 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;
}
Cela aidera peut-être quelqu'un. Le même algorithme en PHP:
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;
}
Vous pouvez également utiliser Rect.IntersectsWith () .
Par exemple, dans WPF, si vous avez deux UIElements, avec RenderTransform et placés sur un canevas, et que vous voulez savoir s'ils se croisent, vous pouvez utiliser quelque chose de similaire:
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);
}
Une implémentation de script de type (Java) avec une bascule pour (ex) inclure les situations "tactiles":
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;
}