How to check intersection between 2 rotated rectangles?

  1. For each edge in both polygons, check if it can be used as a separating line. If so, you are done: No intersection.
  2. If no separation line was found, you have an 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;
}

For more information, see this article: 2D Polygon Collision Detection - Code Project

NB: The algorithm only works for convex polygons, specified in either clockwise, or counterclockwise order.


In javascript, the exact same algorithm is (for convenience):

/**
 * 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;
};

Hope this helps someone.


Check out the method designed by Oren Becker to detect intersection of rotated rectangles with form:

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;
};

And calling the following function will return whether two rotated rectangles intersect or not:

// 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;
}

Here's the same algorithm in Java if anybody is interested.

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;
}