// GJK using Voronoi regions (Christer Ericson) and Barycentric coordinates.
void b2Distance(b2DistanceOutput* output,b2SimplexCache* cache,
const b2DistanceInput* input)
{
++b2_gjkCalls;
const b2DistanceProxy* proxyA = &input->proxyA;
const b2DistanceProxy* proxyB = &input->proxyB;
b2Transform transformA = input->transformA;
b2Transform transformB = input->transformB;
// Initialize the simplex.
b2Simplex simplex;
simplex.ReadCache(cache, proxyA, transformA, proxyB, transformB);
// Get simplex vertices as an array.
b2SimplexVertex* vertices = &simplex.m_v1;
const int32 k_maxIters = 20;
// These store the vertices of the last simplex so that we
// can check for duplicates and prevent cycling.
int32 saveA[3], saveB[3];
int32 saveCount = 0;
float32 distanceSqr1 = b2_maxFloat;
// Main iteration loop.
int32 iter = 0;
while (iter < k_maxIters)
{
// Copy simplex so we can identify duplicates.
saveCount = simplex.m_count;
for (int32 i = 0; i < saveCount; ++i)
{
saveA[i] = vertices[i].indexA;
saveB[i] = vertices[i].indexB;
}
switch (simplex.m_count)
{
case 1:
break;
case 2:
simplex.Solve2();
break;
case 3:
simplex.Solve3();
break;
default:
b2Assert(false);
}
// If we have 3 points, then the origin is in the corresponding triangle.
if (simplex.m_count == 3)
{
break;
}
// Compute closest point.
b2Vec2 p = simplex.GetClosestPoint();
float32 distanceSqr2 = p.LengthSquared();
// Ensure progress
if (distanceSqr2 >= distanceSqr1)
{
//break;
}
distanceSqr1 = distanceSqr2;
// Get search direction.
b2Vec2 d = simplex.GetSearchDirection();
// Ensure the search direction is numerically fit.
if (d.LengthSquared() < b2_epsilon * b2_epsilon)
{
// 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.
b2SimplexVertex* vertex = vertices + simplex.m_count;
vertex->indexA = proxyA->GetSupport(b2MulT(transformA.q, -d));
vertex->wA = b2Mul(transformA, proxyA->GetVertex(vertex->indexA));
b2Vec2 wBLocal;
vertex->indexB = proxyB->GetSupport(b2MulT(transformB.q, d));
vertex->wB = b2Mul(transformB, proxyB->GetVertex(vertex->indexB));
vertex->w = vertex->wB - vertex->wA;
// Iteration count is equated to the number of support point calls.
++iter;
++b2_gjkIters;
// Check for duplicate support points. This is the main termination criteria.
bool duplicate = false;
for (int32 i = 0; i < saveCount; ++i)
{
if (vertex->indexA == saveA[i] && vertex->indexB == saveB[i])
{
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.m_count;
}
b2_gjkMaxIters = b2Max(b2_gjkMaxIters, iter);
// Prepare output.
simplex.GetWitnessPoints(&output->pointA, &output->pointB);
output->distance = b2Distance(output->pointA, output->pointB);
output->iterations = iter;
// Cache the simplex.
simplex.WriteCache(cache);
// Apply radii if requested.
if (input->useRadii)
{
float32 rA = proxyA->m_radius;
float32 rB = proxyB->m_radius;
if (output->distance > rA + rB && output->distance > b2_epsilon)
{
// Shapes are still no overlapped.
// Move the witness points to the outer surface.
output->distance -= rA + rB;
b2Vec2 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.
b2Vec2 p = 0.5f * (output->pointA + output->pointB);
output->pointA = p;
output->pointB = p;
output->distance = 0.0f;
}
}
}
================================
// 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
void b2Simplex::Solve2()
{
b2Vec2 w1 = m_v1.w;
b2Vec2 w2 = m_v2.w;
b2Vec2 e12 = w2 - w1;
// w1 region
float32 d12_2 = -b2Dot(w1, e12);
if (d12_2 <= 0.0f)
{
// a2 <= 0, so we clamp it to 0
m_v1.a = 1.0f;
m_count = 1;
return;
}
// w2 region
float32 d12_1 = b2Dot(w2, e12);
if (d12_1 <= 0.0f)
{
// a1 <= 0, so we clamp it to 0
m_v2.a = 1.0f;
m_count = 1;
m_v1 = m_v2;
return;
}
// Must be in e12 region.
float32 inv_d12 = 1.0f / (d12_1 + d12_2);
m_v1.a = d12_1 * inv_d12;
m_v2.a = d12_2 * inv_d12;
m_count = 2;
}
// Possible regions:
// - points[2]
// - edge points[0]-points[2]
// - edge points[1]-points[2]
// - inside the triangle
void b2Simplex::Solve3()
{
b2Vec2 w1 = m_v1.w;
b2Vec2 w2 = m_v2.w;
b2Vec2 w3 = m_v3.w;
// Edge12
// [1 1 ][a1] = [1]
// [w1.e12 w2.e12][a2] = [0]
// a3 = 0
b2Vec2 e12 = w2 - w1;
float32 w1e12 = b2Dot(w1, e12);
float32 w2e12 = b2Dot(w2, e12);
float32 d12_1 = w2e12;
float32 d12_2 = -w1e12;
// Edge13
// [1 1 ][a1] = [1]
// [w1.e13 w3.e13][a3] = [0]
// a2 = 0
b2Vec2 e13 = w3 - w1;
float32 w1e13 = b2Dot(w1, e13);
float32 w3e13 = b2Dot(w3, e13);
float32 d13_1 = w3e13;
float32 d13_2 = -w1e13;
// Edge23
// [1 1 ][a2] = [1]
// [w2.e23 w3.e23][a3] = [0]
// a1 = 0
b2Vec2 e23 = w3 - w2;
float32 w2e23 = b2Dot(w2, e23);
float32 w3e23 = b2Dot(w3, e23);
float32 d23_1 = w3e23;
float32 d23_2 = -w2e23;
// Triangle123
float32 n123 = b2Cross(e12, e13);
float32 d123_1 = n123 * b2Cross(w2, w3);
float32 d123_2 = n123 * b2Cross(w3, w1);
float32 d123_3 = n123 * b2Cross(w1, w2);
// w1 region
if (d12_2 <= 0.0f && d13_2 <= 0.0f)
{
m_v1.a = 1.0f;
m_count = 1;
return;
}
// e12
if (d12_1 > 0.0f && d12_2 > 0.0f && d123_3 <= 0.0f)
{
float32 inv_d12 = 1.0f / (d12_1 + d12_2);
m_v1.a = d12_1 * inv_d12;
m_v2.a = d12_2 * inv_d12;
m_count = 2;
return;
}
// e13
if (d13_1 > 0.0f && d13_2 > 0.0f && d123_2 <= 0.0f)
{
float32 inv_d13 = 1.0f / (d13_1 + d13_2);
m_v1.a = d13_1 * inv_d13;
m_v3.a = d13_2 * inv_d13;
m_count = 2;
m_v2 = m_v3;
return;
}
// w2 region
if (d12_1 <= 0.0f && d23_2 <= 0.0f)
{
m_v2.a = 1.0f;
m_count = 1;
m_v1 = m_v2;
return;
}
// w3 region
if (d13_1 <= 0.0f && d23_1 <= 0.0f)
{
m_v3.a = 1.0f;
m_count = 1;
m_v1 = m_v3;
return;
}
// e23
if (d23_1 > 0.0f && d23_2 > 0.0f && d123_1 <= 0.0f)
{
float32 inv_d23 = 1.0f / (d23_1 + d23_2);
m_v2.a = d23_1 * inv_d23;
m_v3.a = d23_2 * inv_d23;
m_count = 2;
m_v1 = m_v3;
return;
}
// Must be in triangle123
float32 inv_d123 = 1.0f / (d123_1 + d123_2 + d123_3);
m_v1.a = d123_1 * inv_d123;
m_v2.a = d123_2 * inv_d123;
m_v3.a = d123_3 * inv_d123;
m_count = 3;
}
==============================
void GetWitnessPoints(b2Vec2* pA, b2Vec2* pB) const
{
switch (m_count)
{
case 0:
b2Assert(false);
break;
case 1:
*pA = m_v1.wA;
*pB = m_v1.wB;
break;
case 2:
pA = m_v1.a m_v1.wA + m_v2.a * m_v2.wA;
pB = m_v1.a m_v1.wB + m_v2.a * m_v2.wB;
break;
case 3:
pA = m_v1.a m_v1.wA + m_v2.a m_v2.wA + m_v3.a m_v3.wA;
pB = pA;
break;
default:
b2Assert(false);
break;
}
}
float32 GetMetric() const
{
switch (m_count)
{
case 0:
b2Assert(false);
return 0.0f;
case 1:
return 0.0f;
case 2:
return b2Distance(m_v1.w, m_v2.w);
case 3:
return b2Cross(m_v2.w - m_v1.w, m_v3.w - m_v1.w);
default:
b2Assert(false);
return 0.0f;
}
}
void Solve2();
void Solve3();
b2SimplexVertex m_v1, m_v2, m_v3;
int32 m_count;
};
===================================================
int32 b2_gjkCalls, b2_gjkIters, b2_gjkMaxIters;
void b2DistanceProxy::Set(const b2Shape* shape, int32 index)
{
switch (shape->GetType())
{
case b2Shape::e_circle:
{
const b2CircleShape* circle = static_cast<const b2CircleShape*>(shape);
m_vertices = &circle->m_p;
m_count = 1;
m_radius = circle->m_radius;
}
break;
case b2Shape::e_polygon:
{
const b2PolygonShape* polygon = static_cast<const b2PolygonShape*>(shape);
m_vertices = polygon->m_vertices;
m_count = polygon->m_count;
m_radius = polygon->m_radius;
}
break;
case b2Shape::e_chain:
{
const b2ChainShape* chain = static_cast<const b2ChainShape*>(shape);
b2Assert(0 <= index && index < chain->m_count);
m_buffer[0] = chain->m_vertices[index];
if (index + 1 < chain->m_count)
{
m_buffer[1] = chain->m_vertices[index + 1];
}
else
{
m_buffer[1] = chain->m_vertices[0];
}
m_vertices = m_buffer;
m_count = 2;
m_radius = chain->m_radius;
}
break;
case b2Shape::e_edge:
{
const b2EdgeShape* edge = static_cast<const b2EdgeShape*>(shape);
m_vertices = &edge->m_vertex1;
m_count = 2;
m_radius = edge->m_radius;
}
break;
default:
b2Assert(false);
}
}
struct b2SimplexVertex
{
b2Vec2 wA; // support point in proxyA
b2Vec2 wB; // support point in proxyB
b2Vec2 w; // wB - wA
float32 a; // barycentric coordinate for closest point
int32 indexA; // wA index
int32 indexB; // wB index
};
struct b2Simplex
{
void ReadCache( const b2SimplexCache* cache,
const b2DistanceProxy* proxyA, const b2Transform& transformA,
const b2DistanceProxy* proxyB, const b2Transform& transformB)
{
b2Assert(cache->count <= 3);
// Copy data from cache.
m_count = cache->count;
b2SimplexVertex* vertices = &m_v1;
for (int32 i = 0; i < m_count; ++i)
{
b2SimplexVertex* v = vertices + i;
v->indexA = cache->indexA[i];
v->indexB = cache->indexB[i];
b2Vec2 wALocal = proxyA->GetVertex(v->indexA);
b2Vec2 wBLocal = proxyB->GetVertex(v->indexB);
v->wA = b2Mul(transformA, wALocal);
v->wB = b2Mul(transformB, wBLocal);
v->w = v->wB - v->wA;
v->a = 0.0f;
}
// Compute the new simplex metric, if it is substantially different than
// old metric then flush the simplex.
if (m_count > 1)
{
float32 metric1 = cache->metric;
float32 metric2 = GetMetric();
if (metric2 < 0.5f metric1 || 2.0f metric1 < metric2 || metric2 < b2_epsilon)
{
// Reset the simplex.
m_count = 0;
}
}
// If the cache is empty or invalid ...
if (m_count == 0)
{
b2SimplexVertex* v = vertices + 0;
v->indexA = 0;
v->indexB = 0;
b2Vec2 wALocal = proxyA->GetVertex(0);
b2Vec2 wBLocal = proxyB->GetVertex(0);
v->wA = b2Mul(transformA, wALocal);
v->wB = b2Mul(transformB, wBLocal);
v->w = v->wB - v->wA;
v->a = 1.0f;
m_count = 1;
}
}
void WriteCache(b2SimplexCache* cache) const
{
cache->metric = GetMetric();
cache->count = uint16(m_count);
const b2SimplexVertex* vertices = &m_v1;
for (int32 i = 0; i < m_count; ++i)
{
cache->indexA[i] = uint8(vertices[i].indexA);
cache->indexB[i] = uint8(vertices[i].indexB);
}
}
b2Vec2 GetSearchDirection() const
{
switch (m_count)
{
case 1:
return -m_v1.w;
case 2:
{
b2Vec2 e12 = m_v2.w - m_v1.w;
float32 sgn = b2Cross(e12, -m_v1.w);
if (sgn > 0.0f)
{
// Origin is left of e12.
return b2Cross(1.0f, e12);
}
else
{
// Origin is right of e12.
return b2Cross(e12, 1.0f);
}
}
default:
b2Assert(false);
return b2Vec2_zero;
}
}
b2Vec2 GetClosestPoint() const
{
switch (m_count)
{
case 0:
b2Assert(false);
return b2Vec2_zero;
case 1:
return m_v1.w;
case 2:
return m_v1.a m_v1.w + m_v2.a m_v2.w;
case 3:
return b2Vec2_zero;
default:
b2Assert(false);
return b2Vec2_zero;
}
}
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