public override void ComputeMass(out MassData massData)
{
// Polygon mass, centroid, and inertia.
// Let rho be the polygon density in mass per unit area.
// Then:
// mass = rho * int(dA)
// centroid.x = (1/mass) * rho * int(x * dA)
// centroid.y = (1/mass) * rho * int(y * dA)
// I = rho * int((x*x + y*y) * dA)
//
// We can compute these integrals by summing all the integrals
// for each triangle of the polygon. To evaluate the integral
// for a single triangle, we make a change of variables to
// the (u,v) coordinates of the triangle:
// x = x0 + e1x * u + e2x * v
// y = y0 + e1y * u + e2y * v
// where 0 <= u && 0 <= v && u + v <= 1.
//
// We integrate u from [0,1-v] and then v from [0,1].
// We also need to use the Jacobian of the transformation:
// D = cross(e1, e2)
//
// Simplification: triangle centroid = (1/3) * (p1 + p2 + p3)
//
// The rest of the derivation is handled by computer algebra.
Box2DXDebug.Assert(_vertexCount >= 3);
Vec2 center = new Vec2();
center.Set(0.0f, 0.0f);
float area = 0.0f;
float I = 0.0f;
// pRef is the reference point for forming triangles.
// It's location doesn't change the result (except for rounding error).
Vec2 pRef = new Vec2(0.0f, 0.0f);
#if O
// This code would put the reference point inside the polygon.
for (int i = 0; i < _vertexCount; ++i)
{
pRef += _vertices[i];
}
pRef *= 1.0f / count;
#endif
float k_inv3 = 1.0f / 3.0f;
for (int i = 0; i < _vertexCount; ++i)
{
// Triangle vertices.
Vec2 p1 = pRef;
Vec2 p2 = _vertices[i];
Vec2 p3 = i + 1 < _vertexCount ? _vertices[i + 1] : _vertices[0];
Vec2 e1 = p2 - p1;
Vec2 e2 = p3 - p1;
float D = Vec2.Cross(e1, e2);
float triangleArea = 0.5f * D;
area += triangleArea;
// Area weighted centroid
center += triangleArea * k_inv3 * (p1 + p2 + p3);
float px = p1.X, py = p1.Y;
float ex1 = e1.X, ey1 = e1.Y;
float ex2 = e2.X, ey2 = e2.Y;
float intx2 = k_inv3 * (0.25f * (ex1 * ex1 + ex2 * ex1 + ex2 * ex2) + (px * ex1 + px * ex2)) + 0.5f * px * px;
float inty2 = k_inv3 * (0.25f * (ey1 * ey1 + ey2 * ey1 + ey2 * ey2) + (py * ey1 + py * ey2)) + 0.5f * py * py;
I += D * (intx2 + inty2);
}
// Total mass
massData.Mass = _density * area;
// Center of mass
Box2DXDebug.Assert(area > Common.Settings.FLT_EPSILON);
center *= 1.0f / area;
massData.Center = center;
// Inertia tensor relative to the local origin.
massData.I = _density * I;
}
