How to check intersection between 2 rotated rectangles?

Question

Can someone explain how to check if one rotated rectangle intersect other rectangle?

Answer 1

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.

Answer 2

Maybe it will help someone. The same algorithm in 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;
}

Answer 3

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;
}

Answer 4

Lua implementation built in love2d framework. Collision detection function works in pure lua anyway

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

Answer 5

In Python:

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