text: // MIT License
  
  // Copyright (c) 2019 Erin Catto
  
  // Permission is hereby granted, free of charge, to any person obtaining a copy
  // of this software and associated documentation files (the "Software"), to deal
  // in the Software without restriction, including without limitation the rights
  // to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
  // copies of the Software, and to permit persons to whom the Software is
  // furnished to do so, subject to the following conditions:
  
  // The above copyright notice and this permission notice shall be included in all
  // copies or substantial portions of the Software.
  
  // THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
  // IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
  // FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
  // AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
  // LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
  // OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
  // SOFTWARE.
  
  #include "b2_contact_solver.h"
  
  #include "box2d/b2_body.h"
  #include "box2d/b2_contact.h"
  #include "box2d/b2_fixture.h"
  #include "box2d/b2_stack_allocator.h"
  #include "box2d/b2_world.h"
  
  // Solver debugging is normally disabled because the block solver sometimes has to deal with a poorly conditioned effective mass matrix.
  #define B2_DEBUG_SOLVER 0
  
  bool g_blockSolve = true;
  
  struct b2ContactPositionConstraint
  {
  	b2Vec2 localPoints[b2_maxManifoldPoints];
  	b2Vec2 localNormal;
  	b2Vec2 localPoint;
  	int32 indexA;
  	int32 indexB;
  	float invMassA, invMassB;
  	b2Vec2 localCenterA, localCenterB;
  	float invIA, invIB;
  	b2Manifold::Type type;
  	float radiusA, radiusB;
  	int32 pointCount;
  };
  
  b2ContactSolver::b2ContactSolver(b2ContactSolverDef* def)
  {
  	m_step = def->step;
  	m_allocator = def->allocator;
  	m_count = def->count;
  	m_positionConstraints = (b2ContactPositionConstraint*)m_allocator->Allocate(m_count * sizeof(b2ContactPositionConstraint));
  	m_velocityConstraints = (b2ContactVelocityConstraint*)m_allocator->Allocate(m_count * sizeof(b2ContactVelocityConstraint));
  	m_positions = def->positions;
  	m_velocities = def->velocities;
  	m_contacts = def->contacts;
  
  	// Initialize position independent portions of the constraints.
  	for (int32 i = 0; i < m_count; ++i)
  	{
  		b2Contact* contact = m_contacts[i];
  
  		b2Fixture* fixtureA = contact->m_fixtureA;
  		b2Fixture* fixtureB = contact->m_fixtureB;
  		b2Shape* shapeA = fixtureA->GetShape();
  		b2Shape* shapeB = fixtureB->GetShape();
  		float radiusA = shapeA->m_radius;
  		float radiusB = shapeB->m_radius;
  		b2Body* bodyA = fixtureA->GetBody();
  		b2Body* bodyB = fixtureB->GetBody();
  		b2Manifold* manifold = contact->GetManifold();
  
  		int32 pointCount = manifold->pointCount;
  		b2Assert(pointCount > 0);
  
  		b2ContactVelocityConstraint* vc = m_velocityConstraints + i;
  		vc->friction = contact->m_friction;
  		vc->restitution = contact->m_restitution;
  		vc->threshold = contact->m_restitutionThreshold;
  		vc->tangentSpeed = contact->m_tangentSpeed;
  		vc->indexA = bodyA->m_islandIndex;
  		vc->indexB = bodyB->m_islandIndex;
  		vc->invMassA = bodyA->m_invMass;
  		vc->invMassB = bodyB->m_invMass;
  		vc->invIA = bodyA->m_invI;
  		vc->invIB = bodyB->m_invI;
  		vc->contactIndex = i;
  		vc->pointCount = pointCount;
  		vc->K.SetZero();
  		vc->normalMass.SetZero();
  
  		b2ContactPositionConstraint* pc = m_positionConstraints + i;
  		pc->indexA = bodyA->m_islandIndex;
  		pc->indexB = bodyB->m_islandIndex;
  		pc->invMassA = bodyA->m_invMass;
  		pc->invMassB = bodyB->m_invMass;
  		pc->localCenterA = bodyA->m_sweep.localCenter;
  		pc->localCenterB = bodyB->m_sweep.localCenter;
  		pc->invIA = bodyA->m_invI;
  		pc->invIB = bodyB->m_invI;
  		pc->localNormal = manifold->localNormal;
  		pc->localPoint = manifold->localPoint;
  		pc->pointCount = pointCount;
  		pc->radiusA = radiusA;
  		pc->radiusB = radiusB;
  		pc->type = manifold->type;
  
  		for (int32 j = 0; j < pointCount; ++j)
  		{
  			b2ManifoldPoint* cp = manifold->points + j;
  			b2VelocityConstraintPoint* vcp = vc->points + j;
  	
  			if (m_step.warmStarting)
  			{
  				vcp->normalImpulse = m_step.dtRatio * cp->normalImpulse;
  				vcp->tangentImpulse = m_step.dtRatio * cp->tangentImpulse;
  			}
  			else
  			{
  				vcp->normalImpulse = 0.0f;
  				vcp->tangentImpulse = 0.0f;
  			}
  
  			vcp->rA.SetZero();
  			vcp->rB.SetZero();
  			vcp->normalMass = 0.0f;
  			vcp->tangentMass = 0.0f;
  			vcp->velocityBias = 0.0f;
  
  			pc->localPoints[j] = cp->localPoint;
  		}
  	}
  }
  
  b2ContactSolver::~b2ContactSolver()
  {
  	m_allocator->Free(m_velocityConstraints);
  	m_allocator->Free(m_positionConstraints);
  }
  
  // Initialize position dependent portions of the velocity constraints.
  void b2ContactSolver::InitializeVelocityConstraints()
  {
  	for (int32 i = 0; i < m_count; ++i)
  	{
  		b2ContactVelocityConstraint* vc = m_velocityConstraints + i;
  		b2ContactPositionConstraint* pc = m_positionConstraints + i;
  
  		float radiusA = pc->radiusA;
  		float radiusB = pc->radiusB;
  		b2Manifold* manifold = m_contacts[vc->contactIndex]->GetManifold();
  
  		int32 indexA = vc->indexA;
  		int32 indexB = vc->indexB;
  
  		float mA = vc->invMassA;
  		float mB = vc->invMassB;
  		float iA = vc->invIA;
  		float iB = vc->invIB;
  		b2Vec2 localCenterA = pc->localCenterA;
  		b2Vec2 localCenterB = pc->localCenterB;
  
  		b2Vec2 cA = m_positions[indexA].c;
  		float aA = m_positions[indexA].a;
  		b2Vec2 vA = m_velocities[indexA].v;
  		float wA = m_velocities[indexA].w;
  
  		b2Vec2 cB = m_positions[indexB].c;
  		float aB = m_positions[indexB].a;
  		b2Vec2 vB = m_velocities[indexB].v;
  		float wB = m_velocities[indexB].w;
  
  		b2Assert(manifold->pointCount > 0);
  
  		b2Transform xfA, xfB;
  		xfA.q.Set(aA);
  		xfB.q.Set(aB);
  		xfA.p = cA - b2Mul(xfA.q, localCenterA);
  		xfB.p = cB - b2Mul(xfB.q, localCenterB);
  
  		b2WorldManifold worldManifold;
  		worldManifold.Initialize(manifold, xfA, radiusA, xfB, radiusB);
  
  		vc->normal = worldManifold.normal;
  
  		int32 pointCount = vc->pointCount;
  		for (int32 j = 0; j < pointCount; ++j)
  		{
  			b2VelocityConstraintPoint* vcp = vc->points + j;
  
  			vcp->rA = worldManifold.points[j] - cA;
  			vcp->rB = worldManifold.points[j] - cB;
  
  			float rnA = b2Cross(vcp->rA, vc->normal);
  			float rnB = b2Cross(vcp->rB, vc->normal);
  
  			float kNormal = mA + mB + iA * rnA * rnA + iB * rnB * rnB;
  
  			vcp->normalMass = kNormal > 0.0f ? 1.0f / kNormal : 0.0f;
  
  			b2Vec2 tangent = b2Cross(vc->normal, 1.0f);
  
  			float rtA = b2Cross(vcp->rA, tangent);
  			float rtB = b2Cross(vcp->rB, tangent);
  
  			float kTangent = mA + mB + iA * rtA * rtA + iB * rtB * rtB;
  
  			vcp->tangentMass = kTangent > 0.0f ? 1.0f /  kTangent : 0.0f;
  
  			// Setup a velocity bias for restitution.
  			vcp->velocityBias = 0.0f;
  			float vRel = b2Dot(vc->normal, vB + b2Cross(wB, vcp->rB) - vA - b2Cross(wA, vcp->rA));
  			if (vRel < -vc->threshold)
  			{
  				vcp->velocityBias = -vc->restitution * vRel;
  			}
  		}
  
  		// If we have two points, then prepare the block solver.
  		if (vc->pointCount == 2 && g_blockSolve)
  		{
  			b2VelocityConstraintPoint* vcp1 = vc->points + 0;
  			b2VelocityConstraintPoint* vcp2 = vc->points + 1;
  
  			float rn1A = b2Cross(vcp1->rA, vc->normal);
  			float rn1B = b2Cross(vcp1->rB, vc->normal);
  			float rn2A = b2Cross(vcp2->rA, vc->normal);
  			float rn2B = b2Cross(vcp2->rB, vc->normal);
  
  			float k11 = mA + mB + iA * rn1A * rn1A + iB * rn1B * rn1B;
  			float k22 = mA + mB + iA * rn2A * rn2A + iB * rn2B * rn2B;
  			float k12 = mA + mB + iA * rn1A * rn2A + iB * rn1B * rn2B;
  
  			// Ensure a reasonable condition number.
  			const float k_maxConditionNumber = 1000.0f;
  			if (k11 * k11 < k_maxConditionNumber * (k11 * k22 - k12 * k12))
  			{
  				// K is safe to invert.
  				vc->K.ex.Set(k11, k12);
  				vc->K.ey.Set(k12, k22);
  				vc->normalMass = vc->K.GetInverse();
  			}
  			else
  			{
  				// The constraints are redundant, just use one.
  				// TODO_ERIN use deepest?
  				vc->pointCount = 1;
  			}
  		}
  	}
  }
  
  void b2ContactSolver::WarmStart()
  {
  	// Warm start.
  	for (int32 i = 0; i < m_count; ++i)
  	{
  		b2ContactVelocityConstraint* vc = m_velocityConstraints + i;
  
  		int32 indexA = vc->indexA;
  		int32 indexB = vc->indexB;
  		float mA = vc->invMassA;
  		float iA = vc->invIA;
  		float mB = vc->invMassB;
  		float iB = vc->invIB;
  		int32 pointCount = vc->pointCount;
  
  		b2Vec2 vA = m_velocities[indexA].v;
  		float wA = m_velocities[indexA].w;
  		b2Vec2 vB = m_velocities[indexB].v;
  		float wB = m_velocities[indexB].w;
  
  		b2Vec2 normal = vc->normal;
  		b2Vec2 tangent = b2Cross(normal, 1.0f);
  
  		for (int32 j = 0; j < pointCount; ++j)
  		{
  			b2VelocityConstraintPoint* vcp = vc->points + j;
  			b2Vec2 P = vcp->normalImpulse * normal + vcp->tangentImpulse * tangent;
  			wA -= iA * b2Cross(vcp->rA, P);
  			vA -= mA * P;
  			wB += iB * b2Cross(vcp->rB, P);
  			vB += mB * P;
  		}
  
  		m_velocities[indexA].v = vA;
  		m_velocities[indexA].w = wA;
  		m_velocities[indexB].v = vB;
  		m_velocities[indexB].w = wB;
  	}
  }
  
  void b2ContactSolver::SolveVelocityConstraints()
  {
  	for (int32 i = 0; i < m_count; ++i)
  	{
  		b2ContactVelocityConstraint* vc = m_velocityConstraints + i;
  
  		int32 indexA = vc->indexA;
  		int32 indexB = vc->indexB;
  		float mA = vc->invMassA;
  		float iA = vc->invIA;
  		float mB = vc->invMassB;
  		float iB = vc->invIB;
  		int32 pointCount = vc->pointCount;
  
  		b2Vec2 vA = m_velocities[indexA].v;
  		float wA = m_velocities[indexA].w;
  		b2Vec2 vB = m_velocities[indexB].v;
  		float wB = m_velocities[indexB].w;
  
  		b2Vec2 normal = vc->normal;
  		b2Vec2 tangent = b2Cross(normal, 1.0f);
  		float friction = vc->friction;
  
  		b2Assert(pointCount == 1 || pointCount == 2);
  
  		// Solve tangent constraints first because non-penetration is more important
  		// than friction.
  		for (int32 j = 0; j < pointCount; ++j)
  		{
  			b2VelocityConstraintPoint* vcp = vc->points + j;
  
  			// Relative velocity at contact
  			b2Vec2 dv = vB + b2Cross(wB, vcp->rB) - vA - b2Cross(wA, vcp->rA);
  
  			// Compute tangent force
  			float vt = b2Dot(dv, tangent) - vc->tangentSpeed;
  			float lambda = vcp->tangentMass * (-vt);
  
  			// b2Clamp the accumulated force
  			float maxFriction = friction * vcp->normalImpulse;
  			float newImpulse = b2Clamp(vcp->tangentImpulse + lambda, -maxFriction, maxFriction);
  			lambda = newImpulse - vcp->tangentImpulse;
  			vcp->tangentImpulse = newImpulse;
  
  			// Apply contact impulse
  			b2Vec2 P = lambda * tangent;
  
  			vA -= mA * P;
  			wA -= iA * b2Cross(vcp->rA, P);
  
  			vB += mB * P;
  			wB += iB * b2Cross(vcp->rB, P);
  		}
  
  		// Solve normal constraints
  		if (pointCount == 1 || g_blockSolve == false)
  		{
  			for (int32 j = 0; j < pointCount; ++j)
  			{
  				b2VelocityConstraintPoint* vcp = vc->points + j;
  
  				// Relative velocity at contact
  				b2Vec2 dv = vB + b2Cross(wB, vcp->rB) - vA - b2Cross(wA, vcp->rA);
  
  				// Compute normal impulse
  				float vn = b2Dot(dv, normal);
  				float lambda = -vcp->normalMass * (vn - vcp->velocityBias);
  
  				// b2Clamp the accumulated impulse
  				float newImpulse = b2Max(vcp->normalImpulse + lambda, 0.0f);
  				lambda = newImpulse - vcp->normalImpulse;
  				vcp->normalImpulse = newImpulse;
  
  				// Apply contact impulse
  				b2Vec2 P = lambda * normal;
  				vA -= mA * P;
  				wA -= iA * b2Cross(vcp->rA, P);
  
  				vB += mB * P;
  				wB += iB * b2Cross(vcp->rB, P);
  			}
  		}
  		else
  		{
  			// Block solver developed in collaboration with Dirk Gregorius (back in 01/07 on Box2D_Lite).
  			// Build the mini LCP for this contact patch
  			//
  			// vn = A * x + b, vn >= 0, x >= 0 and vn_i * x_i = 0 with i = 1..2
  			//
  			// A = J * W * JT and J = ( -n, -r1 x n, n, r2 x n )
  			// b = vn0 - velocityBias
  			//
  			// The system is solved using the "Total enumeration method" (s. Murty). The complementary constraint vn_i * x_i
  			// implies that we must have in any solution either vn_i = 0 or x_i = 0. So for the 2D contact problem the cases
  			// vn1 = 0 and vn2 = 0, x1 = 0 and x2 = 0, x1 = 0 and vn2 = 0, x2 = 0 and vn1 = 0 need to be tested. The first valid
  			// solution that satisfies the problem is chosen.
  			// 
  			// In order to account of the accumulated impulse 'a' (because of the iterative nature of the solver which only requires
  			// that the accumulated impulse is clamped and not the incremental impulse) we change the impulse variable (x_i).
  			//
  			// Substitute:
  			// 
  			// x = a + d
  			// 
  			// a := old total impulse
  			// x := new total impulse
  			// d := incremental impulse 
  			//
  			// For the current iteration we extend the formula for the incremental impulse
  			// to compute the new total impulse:
  			//
  			// vn = A * d + b
  			//    = A * (x - a) + b
  			//    = A * x + b - A * a
  			//    = A * x + b'
  			// b' = b - A * a;
  
  			b2VelocityConstraintPoint* cp1 = vc->points + 0;
  			b2VelocityConstraintPoint* cp2 = vc->points + 1;
  
  			b2Vec2 a(cp1->normalImpulse, cp2->normalImpulse);
  			b2Assert(a.x >= 0.0f && a.y >= 0.0f);
  
  			// Relative velocity at contact
  			b2Vec2 dv1 = vB + b2Cross(wB, cp1->rB) - vA - b2Cross(wA, cp1->rA);
  			b2Vec2 dv2 = vB + b2Cross(wB, cp2->rB) - vA - b2Cross(wA, cp2->rA);
  
  			// Compute normal velocity
  			float vn1 = b2Dot(dv1, normal);
  			float vn2 = b2Dot(dv2, normal);
  
  			b2Vec2 b;
  			b.x = vn1 - cp1->velocityBias;
  			b.y = vn2 - cp2->velocityBias;
  
  			// Compute b'
  			b -= b2Mul(vc->K, a);
  
  			const float k_errorTol = 1e-3f;
  			B2_NOT_USED(k_errorTol);
  
  			for (;;)
  			{
  				//
  				// Case 1: vn = 0
  				//
  				// 0 = A * x + b'
  				//
  				// Solve for x:
  				//
  				// x = - inv(A) * b'
  				//
  				b2Vec2 x = - b2Mul(vc->normalMass, b);
  
  				if (x.x >= 0.0f && x.y >= 0.0f)
  				{
  					// Get the incremental impulse
  					b2Vec2 d = x - a;
  
  					// Apply incremental impulse
  					b2Vec2 P1 = d.x * normal;
  					b2Vec2 P2 = d.y * normal;
  					vA -= mA * (P1 + P2);
  					wA -= iA * (b2Cross(cp1->rA, P1) + b2Cross(cp2->rA, P2));
  
  					vB += mB * (P1 + P2);
  					wB += iB * (b2Cross(cp1->rB, P1) + b2Cross(cp2->rB, P2));
  
  					// Accumulate
  					cp1->normalImpulse = x.x;
  					cp2->normalImpulse = x.y;
  
  #if B2_DEBUG_SOLVER == 1
  					// Postconditions
  					dv1 = vB + b2Cross(wB, cp1->rB) - vA - b2Cross(wA, cp1->rA);
  					dv2 = vB + b2Cross(wB, cp2->rB) - vA - b2Cross(wA, cp2->rA);
  
  					// Compute normal velocity
  					vn1 = b2Dot(dv1, normal);
  					vn2 = b2Dot(dv2, normal);
  
  					b2Assert(b2Abs(vn1 - cp1->velocityBias) < k_errorTol);
  					b2Assert(b2Abs(vn2 - cp2->velocityBias) < k_errorTol);
  #endif
  					break;
  				}
  
  				//
  				// Case 2: vn1 = 0 and x2 = 0
  				//
  				//   0 = a11 * x1 + a12 * 0 + b1' 
  				// vn2 = a21 * x1 + a22 * 0 + b2'
  				//
  				x.x = - cp1->normalMass * b.x;
  				x.y = 0.0f;
  				vn1 = 0.0f;
  				vn2 = vc->K.ex.y * x.x + b.y;
  				if (x.x >= 0.0f && vn2 >= 0.0f)
  				{
  					// Get the incremental impulse
  					b2Vec2 d = x - a;
  
  					// Apply incremental impulse
  					b2Vec2 P1 = d.x * normal;
  					b2Vec2 P2 = d.y * normal;
  					vA -= mA * (P1 + P2);
  					wA -= iA * (b2Cross(cp1->rA, P1) + b2Cross(cp2->rA, P2));
  
  					vB += mB * (P1 + P2);
  					wB += iB * (b2Cross(cp1->rB, P1) + b2Cross(cp2->rB, P2));
  
  					// Accumulate
  					cp1->normalImpulse = x.x;
  					cp2->normalImpulse = x.y;
  
  #if B2_DEBUG_SOLVER == 1
  					// Postconditions
  					dv1 = vB + b2Cross(wB, cp1->rB) - vA - b2Cross(wA, cp1->rA);
  
  					// Compute normal velocity
  					vn1 = b2Dot(dv1, normal);
  
  					b2Assert(b2Abs(vn1 - cp1->velocityBias) < k_errorTol);
  #endif
  					break;
  				}
  
  
  				//
  				// Case 3: vn2 = 0 and x1 = 0
  				//
  				// vn1 = a11 * 0 + a12 * x2 + b1' 
  				//   0 = a21 * 0 + a22 * x2 + b2'
  				//
  				x.x = 0.0f;
  				x.y = - cp2->normalMass * b.y;
  				vn1 = vc->K.ey.x * x.y + b.x;
  				vn2 = 0.0f;
  
  				if (x.y >= 0.0f && vn1 >= 0.0f)
  				{
  					// Resubstitute for the incremental impulse
  					b2Vec2 d = x - a;
  
  					// Apply incremental impulse
  					b2Vec2 P1 = d.x * normal;
  					b2Vec2 P2 = d.y * normal;
  					vA -= mA * (P1 + P2);
  					wA -= iA * (b2Cross(cp1->rA, P1) + b2Cross(cp2->rA, P2));
  
  					vB += mB * (P1 + P2);
  					wB += iB * (b2Cross(cp1->rB, P1) + b2Cross(cp2->rB, P2));
  
  					// Accumulate
  					cp1->normalImpulse = x.x;
  					cp2->normalImpulse = x.y;
  
  #if B2_DEBUG_SOLVER == 1
  					// Postconditions
  					dv2 = vB + b2Cross(wB, cp2->rB) - vA - b2Cross(wA, cp2->rA);
  
  					// Compute normal velocity
  					vn2 = b2Dot(dv2, normal);
  
  					b2Assert(b2Abs(vn2 - cp2->velocityBias) < k_errorTol);
  #endif
  					break;
  				}
  
  				//
  				// Case 4: x1 = 0 and x2 = 0
  				// 
  				// vn1 = b1
  				// vn2 = b2;
  				x.x = 0.0f;
  				x.y = 0.0f;
  				vn1 = b.x;
  				vn2 = b.y;
  
  				if (vn1 >= 0.0f && vn2 >= 0.0f )
  				{
  					// Resubstitute for the incremental impulse
  					b2Vec2 d = x - a;
  
  					// Apply incremental impulse
  					b2Vec2 P1 = d.x * normal;
  					b2Vec2 P2 = d.y * normal;
  					vA -= mA * (P1 + P2);
  					wA -= iA * (b2Cross(cp1->rA, P1) + b2Cross(cp2->rA, P2));
  
  					vB += mB * (P1 + P2);
  					wB += iB * (b2Cross(cp1->rB, P1) + b2Cross(cp2->rB, P2));
  
  					// Accumulate
  					cp1->normalImpulse = x.x;
  					cp2->normalImpulse = x.y;
  
  					break;
  				}
  
  				// No solution, give up. This is hit sometimes, but it doesn't seem to matter.
  				break;
  			}
  		}
  
  		m_velocities[indexA].v = vA;
  		m_velocities[indexA].w = wA;
  		m_velocities[indexB].v = vB;
  		m_velocities[indexB].w = wB;
  	}
  }
  
  void b2ContactSolver::StoreImpulses()
  {
  	for (int32 i = 0; i < m_count; ++i)
  	{
  		b2ContactVelocityConstraint* vc = m_velocityConstraints + i;
  		b2Manifold* manifold = m_contacts[vc->contactIndex]->GetManifold();
  
  		for (int32 j = 0; j < vc->pointCount; ++j)
  		{
  			manifold->points[j].normalImpulse = vc->points[j].normalImpulse;
  			manifold->points[j].tangentImpulse = vc->points[j].tangentImpulse;
  		}
  	}
  }
  
  struct b2PositionSolverManifold
  {
  	void Initialize(b2ContactPositionConstraint* pc, const b2Transform& xfA, const b2Transform& xfB, int32 index)
  	{
  		b2Assert(pc->pointCount > 0);
  
  		switch (pc->type)
  		{
  		case b2Manifold::e_circles:
  			{
  				b2Vec2 pointA = b2Mul(xfA, pc->localPoint);
  				b2Vec2 pointB = b2Mul(xfB, pc->localPoints[0]);
  				normal = pointB - pointA;
  				normal.Normalize();
  				point = 0.5f * (pointA + pointB);
  				separation = b2Dot(pointB - pointA, normal) - pc->radiusA - pc->radiusB;
  			}
  			break;
  
  		case b2Manifold::e_faceA:
  			{
  				normal = b2Mul(xfA.q, pc->localNormal);
  				b2Vec2 planePoint = b2Mul(xfA, pc->localPoint);
  
  				b2Vec2 clipPoint = b2Mul(xfB, pc->localPoints[index]);
  				separation = b2Dot(clipPoint - planePoint, normal) - pc->radiusA - pc->radiusB;
  				point = clipPoint;
  			}
  			break;
  
  		case b2Manifold::e_faceB:
  			{
  				normal = b2Mul(xfB.q, pc->localNormal);
  				b2Vec2 planePoint = b2Mul(xfB, pc->localPoint);
  
  				b2Vec2 clipPoint = b2Mul(xfA, pc->localPoints[index]);
  				separation = b2Dot(clipPoint - planePoint, normal) - pc->radiusA - pc->radiusB;
  				point = clipPoint;
  
  				// Ensure normal points from A to B
  				normal = -normal;
  			}
  			break;
  		}
  	}
  
  	b2Vec2 normal;
  	b2Vec2 point;
  	float separation;
  };
  
  // Sequential solver.
  bool b2ContactSolver::SolvePositionConstraints()
  {
  	float minSeparation = 0.0f;
  
  	for (int32 i = 0; i < m_count; ++i)
  	{
  		b2ContactPositionConstraint* pc = m_positionConstraints + i;
  
  		int32 indexA = pc->indexA;
  		int32 indexB = pc->indexB;
  		b2Vec2 localCenterA = pc->localCenterA;
  		float mA = pc->invMassA;
  		float iA = pc->invIA;
  		b2Vec2 localCenterB = pc->localCenterB;
  		float mB = pc->invMassB;
  		float iB = pc->invIB;
  		int32 pointCount = pc->pointCount;
  
  		b2Vec2 cA = m_positions[indexA].c;
  		float aA = m_positions[indexA].a;
  
  		b2Vec2 cB = m_positions[indexB].c;
  		float aB = m_positions[indexB].a;
  
  		// Solve normal constraints
  		for (int32 j = 0; j < pointCount; ++j)
  		{
  			b2Transform xfA, xfB;
  			xfA.q.Set(aA);
  			xfB.q.Set(aB);
  			xfA.p = cA - b2Mul(xfA.q, localCenterA);
  			xfB.p = cB - b2Mul(xfB.q, localCenterB);
  
  			b2PositionSolverManifold psm;
  			psm.Initialize(pc, xfA, xfB, j);
  			b2Vec2 normal = psm.normal;
  
  			b2Vec2 point = psm.point;
  			float separation = psm.separation;
  
  			b2Vec2 rA = point - cA;
  			b2Vec2 rB = point - cB;
  
  			// Track max constraint error.
  			minSeparation = b2Min(minSeparation, separation);
  
  			// Prevent large corrections and allow slop.
  			float C = b2Clamp(b2_baumgarte * (separation + b2_linearSlop), -b2_maxLinearCorrection, 0.0f);
  
  			// Compute the effective mass.
  			float rnA = b2Cross(rA, normal);
  			float rnB = b2Cross(rB, normal);
  			float K = mA + mB + iA * rnA * rnA + iB * rnB * rnB;
  
  			// Compute normal impulse
  			float impulse = K > 0.0f ? - C / K : 0.0f;
  
  			b2Vec2 P = impulse * normal;
  
  			cA -= mA * P;
  			aA -= iA * b2Cross(rA, P);
  
  			cB += mB * P;
  			aB += iB * b2Cross(rB, P);
  		}
  
  		m_positions[indexA].c = cA;
  		m_positions[indexA].a = aA;
  
  		m_positions[indexB].c = cB;
  		m_positions[indexB].a = aB;
  	}
  
  	// We can't expect minSpeparation >= -b2_linearSlop because we don't
  	// push the separation above -b2_linearSlop.
  	return minSeparation >= -3.0f * b2_linearSlop;
  }
  
  // Sequential position solver for position constraints.
  bool b2ContactSolver::SolveTOIPositionConstraints(int32 toiIndexA, int32 toiIndexB)
  {
  	float minSeparation = 0.0f;
  
  	for (int32 i = 0; i < m_count; ++i)
  	{
  		b2ContactPositionConstraint* pc = m_positionConstraints + i;
  
  		int32 indexA = pc->indexA;
  		int32 indexB = pc->indexB;
  		b2Vec2 localCenterA = pc->localCenterA;
  		b2Vec2 localCenterB = pc->localCenterB;
  		int32 pointCount = pc->pointCount;
  
  		float mA = 0.0f;
  		float iA = 0.0f;
  		if (indexA == toiIndexA || indexA == toiIndexB)
  		{
  			mA = pc->invMassA;
  			iA = pc->invIA;
  		}
  
  		float mB = 0.0f;
  		float iB = 0.;
  		if (indexB == toiIndexA || indexB == toiIndexB)
  		{
  			mB = pc->invMassB;
  			iB = pc->invIB;
  		}
  
  		b2Vec2 cA = m_positions[indexA].c;
  		float aA = m_positions[indexA].a;
  
  		b2Vec2 cB = m_positions[indexB].c;
  		float aB = m_positions[indexB].a;
  
  		// Solve normal constraints
  		for (int32 j = 0; j < pointCount; ++j)
  		{
  			b2Transform xfA, xfB;
  			xfA.q.Set(aA);
  			xfB.q.Set(aB);
  			xfA.p = cA - b2Mul(xfA.q, localCenterA);
  			xfB.p = cB - b2Mul(xfB.q, localCenterB);
  
  			b2PositionSolverManifold psm;
  			psm.Initialize(pc, xfA, xfB, j);
  			b2Vec2 normal = psm.normal;
  
  			b2Vec2 point = psm.point;
  			float separation = psm.separation;
  
  			b2Vec2 rA = point - cA;
  			b2Vec2 rB = point - cB;
  
  			// Track max constraint error.
  			minSeparation = b2Min(minSeparation, separation);
  
  			// Prevent large corrections and allow slop.
  			float C = b2Clamp(b2_toiBaumgarte * (separation + b2_linearSlop), -b2_maxLinearCorrection, 0.0f);
  
  			// Compute the effective mass.
  			float rnA = b2Cross(rA, normal);
  			float rnB = b2Cross(rB, normal);
  			float K = mA + mB + iA * rnA * rnA + iB * rnB * rnB;
  
  			// Compute normal impulse
  			float impulse = K > 0.0f ? - C / K : 0.0f;
  
  			b2Vec2 P = impulse * normal;
  
  			cA -= mA * P;
  			aA -= iA * b2Cross(rA, P);
  
  			cB += mB * P;
  			aB += iB * b2Cross(rB, P);
  		}
  
  		m_positions[indexA].c = cA;
  		m_positions[indexA].a = aA;
  
  		m_positions[indexB].c = cB;
  		m_positions[indexB].a = aB;
  	}
  
  	// We can't expect minSpeparation >= -b2_linearSlop because we don't
  	// push the separation above -b2_linearSlop.
  	return minSeparation >= -1.5f * b2_linearSlop;
  }
  