c++,math,rotation,rotational-matrices,separating-axis-theorem , Separating axis theorem: rotation around center of mass

Separating axis theorem: rotation around center of mass


Tag: c++,math,rotation,rotational-matrices,separating-axis-theorem

The problem is in Polygon::FindAxisLeastPenetration:

double Polygon::FindAxisLeastPenetration(unsigned int *faceIndex, const Polygon &polygonA, const Polygon &polygonB) const {
  double bestDistance = -std::numeric_limits<double>::infinity();
  unsigned int bestIndex;

  for (unsigned int i = 0; i < polygonA.points.size(); i++) {
    Vector2D n = polygonA.normals[i];
    Vector2D nw = polygonA.rotationMatrix * n; //ROTATION
    Matrix22 buT = polygonB.rotationMatrix.Transposed();
    n = buT * nw; //ROTATION

    Vector2D support = polygonB.points[polygonB.GetSupport(-n)];

    Vector2D vertex = polygonA.points[i];
    vertex = polygonA.rotationMatrix * vertex; //ROTATION
    vertex = buT * vertex; // ROTATION
    double distance = n.DotProduct(support - vertex);
    if (distance > bestDistance) {
      bestDistance = distance;
      bestIndex = i;
  *faceIndex = bestIndex;

  return bestDistance;

unsigned int Polygon::GetSupport(const Vector2D &dir) const {
  double bestProjection = -std::numeric_limits<double>::infinity();
  unsigned int bestIndex = 0;

  for (unsigned int i = 0; i < points.size(); i++) {
    Vector2D vertex = points[i];
    double projection = vertex.DotProduct(dir);

    if (projection > bestProjection) {
      bestProjection = projection;
      bestIndex = i;

  return bestIndex;

Manifold Polygon::CheckCollision(const Polygon &polygonA, const Polygon &polygonB) const {
  Manifold result;
  result.objectA = polygonA.body;
  result.objectB = polygonB.body;
  unsigned int indexA;
  double penetrationA = Polygon::FindAxisLeastPenetration(&indexA, polygonA, polygonB);
  if (penetrationA >= 0.0) {
    result.intersects = false;
    return result;

  unsigned int indexB;
  double penetrationB = Polygon::FindAxisLeastPenetration(&indexB, polygonB, polygonA);

  if (penetrationB >= 0.0) {
    result.intersects = false;
    return result;

  result.intersects = true;
  return result;

Rectangle::Rectangle(double width, double height) : Polygon() {
  double hw = width / 2.0;
  double hh = height / 2.0;
  points.push_back(Vector2D(-hw, -hh));
  points.push_back(Vector2D(hw, -hh));
  points.push_back(Vector2D(hw, hh));
  points.push_back(Vector2D(-hw, hh));

  //  points.push_back(Vector2D(0, 0));
  //  points.push_back(Vector2D(width, 0));
  //  points.push_back(Vector2D(width, height));
  //  points.push_back(Vector2D(0, height));

  normals.push_back(Vector2D(0.0, -1.0));
  normals.push_back(Vector2D(1.0, 0.0));
  normals.push_back(Vector2D(0.0, 1.0));
  normals.push_back(Vector2D(-1.0, 0.0));

  center.x = 0;
  center.y = 0;


polygon.rotationMatrix is an object of type Matrix22 which is a 2x2 matrix.
polygon.points is std::vector<Vector2D> filled with vectors.
polygon.body is a pointer to an Object instance. In this case it's only used to get position.
polygon.body->position is an instance of Vector2D containing X and Y coordinates.
Vector2D polygon.body->GetPosition() returns position vector of a body.

It works fine, except that the rotation is done around the [0, 0] point but it's supposed to rotate around the center of mass.

I know that rotation around a point can be done like this:

rotationMatrix * (vertex - point) + point

And it works fine when rendering polygons. But not in collision detection.

How do I rotate vectors around a certain point in this case?

EDIT: Here's what I have so far

double Polygon::FindAxisLeastPenetration(unsigned int *faceIndex, const Polygon &polygonA, const Polygon &polygonB) const {
  double bestDistance = -std::numeric_limits<double>::infinity();
  unsigned int bestIndex;

  for (unsigned int i = 0; i < polygonA.points.size(); i++) {
    // Calculate normal
    unsigned int j = i == points.size() ? 0 : i + 1;
    Vector2D n;
    // Rotate points
    Vector2D p1 = polygonA.rotationMatrix * (polygonA.points[i] - polygonA.Center()) + polygonA.Center();
    Vector2D p2 = polygonA.rotationMatrix * (polygonA.points[j] - polygonA.Center()) + polygonA.Center();
    n.x = p2.y - p1.y;
    n.y = -(p2.x - p1.x);

    Vector2D support = polygonB.points[polygonB.GetSupport(-n)];
    support = polygonB.rotationMatrix * (support - polygonB.Center()) + polygonB.Center();

    Vector2D vertex = polygonA.points[i];
    vertex = polygonA.rotationMatrix * (vertex - polygonA.Center()) + polygonA.Center(); //ROTATION

    double distance = n.DotProduct(support - vertex);
    if (distance > bestDistance) {
      bestDistance = distance;
      bestIndex = i;
  *faceIndex = bestIndex;

  return bestDistance;

unsigned int Polygon::GetSupport(const Vector2D &dir) const {
  double bestProjection = -std::numeric_limits<double>::infinity();
  unsigned int bestIndex = 0;

  for (unsigned int i = 0; i < points.size(); i++) {
    Vector2D vertex = rotationMatrix * (points[i] - center) + center;
    double projection = vertex.DotProduct(dir);

    if (projection > bestProjection) {
      bestProjection = projection;
      bestIndex = i;

  return bestIndex;

Right now I don't really care about optimizations. There's sill the same problem. When rotating around the center collisions aren't being detected properly. However, if the center is [0, 0] or it's not used, then collision detection works properly, but again rotation is being performed wrong.

Edit: Even when rotating before collision detection, I get the same problem. The best approach so far was to translate polygon so that its center would be at [0, 0], but at some angles collisions aren't being detected. Have no idea what to do now.

Edit: Screenshots (polygons are being translated so that their centers of mass are always at [0, 0], polygons are rectangles in this case) Collision detection didn't work well here Collision detection didn't work well here

Collision detection didn't work well here too Collision detection didn't work well here too

Collision detection worked well here Collision detection worked well here

Edit: I added the Rectangle class.


This should work whether or not polygon origin is aligned to center of gravity. I'll start with the most important stuff, and end with supporting methods that have changed.

Edit: Revised implementation.

struct Response {
            : overlap(std::numeric_limits<double>::max()) {}
        Vector2D axis;
        double overlap;

bool FindAxisLeastPenetration(const Polygon& a, const Polygon& b,
        Response* response)
        for ( unsigned long i = 0; i < a.points.size(); i++ )
            Vector2D axis = a.normals[i];
            Vector2D support = b.GetSupport(-axis);

            double overlap = axis.DotProduct(a.points[i] - support);
            if (overlap <= 0.0)
                return false;
            if (overlap < response->overlap)
                response->overlap = overlap;
                response->axis = axis;
        return true;

bool CheckCollisionLocal(const Polygon& a, const Polygon& b,
        Vector2D* min_translation)
// @note assumes untransformed polygons.
        Polygon worldA = a.ToWorld();
        Polygon worldB = b.ToWorld();

        Response responseA;
        Response responseB;

        if (!FindAxisLeastPenetration(worldA, worldB, &responseA))
            return false;
        if (!FindAxisLeastPenetration(worldB, worldA, &responseB))
            return false;

        if (responseA.overlap <= responseB.overlap)
            *min_translation =  responseA.axis * responseA.overlap;
            *min_translation = -responseB.axis * responseB.overlap;
        return true;

Use case,

bool HandleCollisionLocal(Polygon& a, Polygon& b)
        Vector2D translation;
        if (!CheckCollisionLocal(a, b, &translation))
            return false;
        if (MOVE_POLYGON_A)
            a.body.SetPosition(a.body.GetPosition() - translation);
            b.body.SetPosition(b.body.GetPosition() + translation);
        return true;


Polygon Polygon::ToWorld() const
        Polygon result = *this;
        for ( auto& point : result.points )
            point = result.rotationMatrix * point;
        for ( auto& normal : result.normals )
            normal = result.rotationMatrix * normal;
        return result;

Vector2D Polygon::GetSupport(const Vector2D& direction) const
        double best_projection = -std::numeric_limits<double>::max();
        Vector2D best_point;

        for ( auto point : points )
            double projection = point.DotProduct(direction);
            if (projection > best_projection)
                best_projection = projection;
                best_point = point;
        return best_point;

Note: This version inverts the translation vector to conform with - what seems to be - the standard.


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