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

Question:

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.Subtract(polygonB.body->GetPosition());
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);
n.Normalize();

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 too

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 {
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;
else
*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);
else
b.body.SetPosition(b.body.GetPosition() + translation);
return true;
}
``````

Support,

``````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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