// Solve a line segment using barycentric coordinates.
//
// p = a1 * w1 + a2 * w2
// a1 + a2 = 1
//
// The vector from the origin to the closest point on the line is
// perpendicular to the line.
// e12 = w2 - w1
// dot(p, e) = 0
// a1 * dot(w1, e) + a2 * dot(w2, e) = 0
//
// 2-by-2 linear system
// [1 1 ][a1] = [1]
// [w1.e12 w2.e12][a2] = [0]
//
// Define
// d12_1 = dot(w2, e12)
// d12_2 = -dot(w1, e12)
// d12 = d12_1 + d12_2
//
// Solution
// a1 = d12_1 / d12
// a2 = d12_2 / d12
internal void Solve2()
{
Vector2 w1 = _v1.w;
Vector2 w2 = _v2.w;
Vector2 e12 = w2 - w1;
// w1 region
float d12_2 = -Vector2.Dot(w1, e12);
if (d12_2 <= 0.0f)
{
// a2 <= 0, so we clamp it to 0
_v1.a = 1.0f;
_count = 1;
return;
}
// w2 region
float d12_1 = Vector2.Dot(w2, e12);
if (d12_1 <= 0.0f)
{
// a1 <= 0, so we clamp it to 0
_v2.a = 1.0f;
_count = 1;
_v1 = _v2;
return;
}
// Must be in e12 region.
float inv_d12 = 1.0f / (d12_1 + d12_2);
_v1.a = d12_1 * inv_d12;
_v2.a = d12_2 * inv_d12;
_count = 2;
}
static void Distance(out DistanceOutput output, ref SimplexCache cache, ref DistanceInput input, Shape shapeA, Shape shapeB)
{
output = new DistanceOutput();
Transform transformA = input.TransformA;
Transform transformB = input.TransformB;
// Initialize the simplex.
Simplex simplex = new Simplex();
#if ALLOWUNSAFE
fixed(SimplexCache *sPtr = &cache)
{
simplex.ReadCache(sPtr, shapeA, transformA, shapeB, transformB);
}
#else
simplex.ReadCache(cache, shapeA, transformA, shapeB, transformB);
#endif
// Get simplex vertices as an array.
#if ALLOWUNSAFE
SimplexVertex *vertices = &simplex._v1;
#else
SimplexVertex[] vertices = new SimplexVertex[] { simplex._v1, simplex._v2, simplex._v3 };
#endif
// These store the vertices of the last simplex so that we
// can check for duplicates and prevent cycling.
#if ALLOWUNSAFE
int *lastA = stackalloc int[4], lastB = stackalloc int[4];
#else
int[] lastA = new int[4];
int[] lastB = new int[4];
#endif // ALLOWUNSAFE
int lastCount;
// Main iteration loop.
int iter = 0;
const int k_maxIterationCount = 20;
while (iter < k_maxIterationCount)
{
// Copy simplex so we can identify duplicates.
lastCount = simplex._count;
int i;
for (i = 0; i < lastCount; ++i)
{
lastA[i] = vertices[i].indexA;
lastB[i] = vertices[i].indexB;
}
switch (simplex._count)
{
case 1:
break;
case 2:
simplex.Solve2();
break;
case 3:
simplex.Solve3();
break;
default:
break;
}
// If we have 3 points, then the origin is in the corresponding triangle.
if (simplex._count == 3)
{
break;
}
// Compute closest point.
Vector2 p = simplex.GetClosestPoint();
float distanceSqr = p.LengthSquared();
// Ensure the search direction is numerically fit.
if (distanceSqr < Common.Settings.FLT_EPSILON_SQUARED)
{
// The origin is probably contained by a line segment
// or triangle. Thus the shapes are overlapped.
// We can't return zero here even though there may be overlap.
// In case the simplex is a point, segment, or triangle it is difficult
// to determine if the origin is contained in the CSO or very close to it.
break;
}
// Compute a tentative new simplex vertex using support points.
#if ALLOWUNSAFE
SimplexVertex *vertex = vertices + simplex._count;
vertex->indexA = shapeA.GetSupport(transformA.InverseTransformDirection(p));
vertex->wA = transformA.TransformPoint(shapeA.GetVertex(vertex->indexA));
//Vec2 wBLocal;
vertex->indexB = shapeB.GetSupport(transformB.InverseTransformDirection(-p));
vertex->wB = transformB.TransformPoint(shapeB.GetVertex(vertex->indexB));
vertex->w = vertex->wB - vertex->wA;
#else
SimplexVertex vertex = vertices[simplex._count - 1];
vertex.indexA = shapeA.GetSupport(transformA.InverseTransformDirection(p));
vertex.wA = transformA.TransformPoint(shapeA.GetVertex(vertex.indexA));
//Vec2 wBLocal;
vertex.indexB = shapeB.GetSupport(transformB.InverseTransformDirection(-p));
vertex.wB = transformB.TransformPoint(shapeB.GetVertex(vertex.indexB));
vertex.w = vertex.wB - vertex.wA;
#endif // ALLOWUNSAFE
// Iteration count is equated to the number of support point calls.
++iter;
// Check for convergence.
#if ALLOWUNSAFE
float lowerBound = Vector2.Dot(p, vertex->w);
#else
float lowerBound = Vector2.Dot(p, vertex.w);
#endif
float upperBound = distanceSqr;
const float k_relativeTolSqr = 0.01f * 0.01f; // 1:100
if (upperBound - lowerBound <= k_relativeTolSqr * upperBound)
{
// Converged!
break;
}
// Check for duplicate support points.
bool duplicate = false;
for (i = 0; i < lastCount; ++i)
{
#if ALLOWUNSAFE
if (vertex->indexA == lastA[i] && vertex->indexB == lastB[i])
#else
if (vertex.indexA == lastA[i] && vertex.indexB == lastB[i])
#endif
{
duplicate = true;
break;
}
}
// If we found a duplicate support point we must exit to avoid cycling.
if (duplicate)
{
break;
}
// New vertex is ok and needed.
++simplex._count;
}
#if ALLOWUNSAFE
fixed(DistanceOutput *doPtr = &output)
{
// Prepare output.
simplex.GetWitnessPoints(&doPtr->PointA, &doPtr->PointB);
doPtr->Distance = Vector2.Distance(doPtr->PointA, doPtr->PointB);
doPtr->Iterations = iter;
}
fixed(SimplexCache *sPtr = &cache)
{
// Cache the simplex.
simplex.WriteCache(sPtr);
}
#else
// Prepare output.
simplex.GetWitnessPoints(ref output.PointA, ref output.PointB);
output.Distance = Box2DNet.Common.Math.Distance(output.PointA, output.PointB);
output.Iterations = iter;
// Cache the simplex.
simplex.WriteCache(cache);
#endif
// Apply radii if requested.
if (input.UseRadii)
{
float rA = shapeA._radius;
float rB = shapeB._radius;
if (output.Distance > rA + rB && output.Distance > Common.Settings.FLT_EPSILON)
{
// Shapes are still no overlapped.
// Move the witness points to the outer surface.
output.Distance -= rA + rB;
Vector2 normal = output.PointB - output.PointA;
normal.Normalize();
output.PointA += rA * normal;
output.PointB -= rB * normal;
}
else
{
// Shapes are overlapped when radii are considered.
// Move the witness points to the middle.
Vector2 p = 0.5f * (output.PointA + output.PointB);
output.PointA = p;
output.PointB = p;
output.Distance = 0.0f;
}
}
}
// Possible regions:
// - points[2]
// - edge points[0]-points[2]
// - edge points[1]-points[2]
// - inside the triangle
internal void Solve3()
{
Vector2 w1 = _v1.w;
Vector2 w2 = _v2.w;
Vector2 w3 = _v3.w;
// Edge12
// [1 1 ][a1] = [1]
// [w1.e12 w2.e12][a2] = [0]
// a3 = 0
Vector2 e12 = w2 - w1;
float w1e12 = Vector2.Dot(w1, e12);
float w2e12 = Vector2.Dot(w2, e12);
float d12_1 = w2e12;
float d12_2 = -w1e12;
// Edge13
// [1 1 ][a1] = [1]
// [w1.e13 w3.e13][a3] = [0]
// a2 = 0
Vector2 e13 = w3 - w1;
float w1e13 = Vector2.Dot(w1, e13);
float w3e13 = Vector2.Dot(w3, e13);
float d13_1 = w3e13;
float d13_2 = -w1e13;
// Edge23
// [1 1 ][a2] = [1]
// [w2.e23 w3.e23][a3] = [0]
// a1 = 0
Vector2 e23 = w3 - w2;
float w2e23 = Vector2.Dot(w2, e23);
float w3e23 = Vector2.Dot(w3, e23);
float d23_1 = w3e23;
float d23_2 = -w2e23;
// Triangle123
float n123 = e12.Cross(e13);
float d123_1 = n123 * w2.Cross(w3);
float d123_2 = n123 * w3.Cross(w1);
float d123_3 = n123 * w1.Cross(w2);
// w1 region
if (d12_2 <= 0.0f && d13_2 <= 0.0f)
{
_v1.a = 1.0f;
_count = 1;
return;
}
// e12
if (d12_1 > 0.0f && d12_2 > 0.0f && d123_3 <= 0.0f)
{
float inv_d12 = 1.0f / (d12_1 + d12_2);
_v1.a = d12_1 * inv_d12;
_v2.a = d12_1 * inv_d12;
_count = 2;
return;
}
// e13
if (d13_1 > 0.0f && d13_2 > 0.0f && d123_2 <= 0.0f)
{
float inv_d13 = 1.0f / (d13_1 + d13_2);
_v1.a = d13_1 * inv_d13;
_v3.a = d13_2 * inv_d13;
_count = 2;
_v2 = _v3;
return;
}
// w2 region
if (d12_1 <= 0.0f && d23_2 <= 0.0f)
{
_v2.a = 1.0f;
_count = 1;
_v1 = _v2;
return;
}
// w3 region
if (d13_1 <= 0.0f && d23_1 <= 0.0f)
{
_v3.a = 1.0f;
_count = 1;
_v1 = _v3;
return;
}
// e23
if (d23_1 > 0.0f && d23_2 > 0.0f && d123_1 <= 0.0f)
{
float inv_d23 = 1.0f / (d23_1 + d23_2);
_v2.a = d23_1 * inv_d23;
_v3.a = d23_2 * inv_d23;
_count = 2;
_v1 = _v3;
return;
}
// Must be in triangle123
float inv_d123 = 1.0f / (d123_1 + d123_2 + d123_3);
_v1.a = d123_1 * inv_d123;
_v2.a = d123_2 * inv_d123;
_v3.a = d123_3 * inv_d123;
_count = 3;
}