text: /*
  * Copyright (c) 2006-2009 Erin Catto http://www.gphysics.com
  *
  * This software is provided 'as-is', without any express or implied
  * warranty.  In no event will the authors be held liable for any damages
  * arising from the use of this software.
  * Permission is granted to anyone to use this software for any purpose,
  * including commercial applications, and to alter it and redistribute it
  * freely, subject to the following restrictions:
  * 1. The origin of this software must not be misrepresented; you must not
  * claim that you wrote the original software. If you use this software
  * in a product, an acknowledgment in the product documentation would be
  * appreciated but is not required.
  * 2. Altered source versions must be plainly marked as such, and must not be
  * misrepresented as being the original software.
  * 3. This notice may not be removed or altered from any source distribution.
  */
  
  #include "Box2D/Dynamics/Contacts/b2ContactSolver.h"
  #include "Box2D/Dynamics/Contacts/b2Contact.h"
  #include "Box2D/Dynamics/b2Body.h"
  #include "Box2D/Dynamics/b2Fixture.h"
  #include "Box2D/Dynamics/b2World.h"
  #include "Box2D/Common/b2StackAllocator.h"
  
  #define B2_DEBUG_SOLVER 0
  
  b2ContactSolver::b2ContactSolver(b2Contact** contacts, int32 contactCount,
  								b2StackAllocator* allocator, float32 impulseRatio)
  {
  	m_allocator = allocator;
  
  	m_constraintCount = contactCount;
  	m_constraints = (b2ContactConstraint*)m_allocator->Allocate(m_constraintCount * sizeof(b2ContactConstraint));
  
  	for (int32 i = 0; i < m_constraintCount; ++i)
  	{
  		b2Contact* contact = contacts[i];
  
  		b2Fixture* fixtureA = contact->m_fixtureA;
  		b2Fixture* fixtureB = contact->m_fixtureB;
  		b2Shape* shapeA = fixtureA->GetShape();
  		b2Shape* shapeB = fixtureB->GetShape();
  		float32 radiusA = shapeA->m_radius;
  		float32 radiusB = shapeB->m_radius;
  		b2Body* bodyA = fixtureA->GetBody();
  		b2Body* bodyB = fixtureB->GetBody();
  		b2Manifold* manifold = contact->GetManifold();
  
  		float32 friction = b2MixFriction(fixtureA->GetFriction(), fixtureB->GetFriction());
  		float32 restitution = b2MixRestitution(fixtureA->GetRestitution(), fixtureB->GetRestitution());
  
  		b2Vec2 vA = bodyA->m_linearVelocity;
  		b2Vec2 vB = bodyB->m_linearVelocity;
  		float32 wA = bodyA->m_angularVelocity;
  		float32 wB = bodyB->m_angularVelocity;
  
  		b2Assert(manifold->pointCount > 0);
  
  		b2WorldManifold worldManifold;
  		worldManifold.Initialize(manifold, bodyA->m_xf, radiusA, bodyB->m_xf, radiusB);
  
  		b2ContactConstraint* cc = m_constraints + i;
  		cc->bodyA = bodyA;
  		cc->bodyB = bodyB;
  		cc->manifold = manifold;
  		cc->normal = worldManifold.normal;
  		cc->pointCount = manifold->pointCount;
  		cc->friction = friction;
  
  		cc->localNormal = manifold->localNormal;
  		cc->localPoint = manifold->localPoint;
  		cc->radius = radiusA + radiusB;
  		cc->type = manifold->type;
  
  		for (int32 j = 0; j < cc->pointCount; ++j)
  		{
  			b2ManifoldPoint* cp = manifold->points + j;
  			b2ContactConstraintPoint* ccp = cc->points + j;
  
  			ccp->normalImpulse = impulseRatio * cp->normalImpulse;
  			ccp->tangentImpulse = impulseRatio * cp->tangentImpulse;
  
  			ccp->localPoint = cp->localPoint;
  
  			ccp->rA = worldManifold.points[j] - bodyA->m_sweep.c;
  			ccp->rB = worldManifold.points[j] - bodyB->m_sweep.c;
  
  			float32 rnA = b2Cross(ccp->rA, cc->normal);
  			float32 rnB = b2Cross(ccp->rB, cc->normal);
  			rnA *= rnA;
  			rnB *= rnB;
  
  			float32 kNormal = bodyA->m_invMass + bodyB->m_invMass + bodyA->m_invI * rnA + bodyB->m_invI * rnB;
  
  			b2Assert(kNormal > b2_epsilon);
  			ccp->normalMass = 1.0f / kNormal;
  
  			b2Vec2 tangent = b2Cross(cc->normal, 1.0f);
  
  			float32 rtA = b2Cross(ccp->rA, tangent);
  			float32 rtB = b2Cross(ccp->rB, tangent);
  			rtA *= rtA;
  			rtB *= rtB;
  
  			float32 kTangent = bodyA->m_invMass + bodyB->m_invMass + bodyA->m_invI * rtA + bodyB->m_invI * rtB;
  
  			b2Assert(kTangent > b2_epsilon);
  			ccp->tangentMass = 1.0f /  kTangent;
  
  			// Setup a velocity bias for restitution.
  			ccp->velocityBias = 0.0f;
  			float32 vRel = b2Dot(cc->normal, vB + b2Cross(wB, ccp->rB) - vA - b2Cross(wA, ccp->rA));
  			if (vRel < -b2_velocityThreshold)
  			{
  				ccp->velocityBias = -restitution * vRel;
  			}
  		}
  
  		// If we have two points, then prepare the block solver.
  		if (cc->pointCount == 2)
  		{
  			b2ContactConstraintPoint* ccp1 = cc->points + 0;
  			b2ContactConstraintPoint* ccp2 = cc->points + 1;
  			
  			float32 invMassA = bodyA->m_invMass;
  			float32 invIA = bodyA->m_invI;
  			float32 invMassB = bodyB->m_invMass;
  			float32 invIB = bodyB->m_invI;
  
  			float32 rn1A = b2Cross(ccp1->rA, cc->normal);
  			float32 rn1B = b2Cross(ccp1->rB, cc->normal);
  			float32 rn2A = b2Cross(ccp2->rA, cc->normal);
  			float32 rn2B = b2Cross(ccp2->rB, cc->normal);
  
  			float32 k11 = invMassA + invMassB + invIA * rn1A * rn1A + invIB * rn1B * rn1B;
  			float32 k22 = invMassA + invMassB + invIA * rn2A * rn2A + invIB * rn2B * rn2B;
  			float32 k12 = invMassA + invMassB + invIA * rn1A * rn2A + invIB * rn1B * rn2B;
  
  			// Ensure a reasonable condition number.
  			const float32 k_maxConditionNumber = 100.0f;
  			if (k11 * k11 < k_maxConditionNumber * (k11 * k22 - k12 * k12))
  			{
  				// K is safe to invert.
  				cc->K.col1.Set(k11, k12);
  				cc->K.col2.Set(k12, k22);
  				cc->normalMass = cc->K.GetInverse();
  			}
  			else
  			{
  				// The constraints are redundant, just use one.
  				// TODO_ERIN use deepest?
  				cc->pointCount = 1;
  			}
  		}
  	}
  }
  
  b2ContactSolver::~b2ContactSolver()
  {
  	m_allocator->Free(m_constraints);
  }
  
  void b2ContactSolver::WarmStart()
  {
  	// Warm start.
  	for (int32 i = 0; i < m_constraintCount; ++i)
  	{
  		b2ContactConstraint* c = m_constraints + i;
  
  		b2Body* bodyA = c->bodyA;
  		b2Body* bodyB = c->bodyB;
  		float32 invMassA = bodyA->m_invMass;
  		float32 invIA = bodyA->m_invI;
  		float32 invMassB = bodyB->m_invMass;
  		float32 invIB = bodyB->m_invI;
  		b2Vec2 normal = c->normal;
  		b2Vec2 tangent = b2Cross(normal, 1.0f);
  
  		for (int32 j = 0; j < c->pointCount; ++j)
  		{
  			b2ContactConstraintPoint* ccp = c->points + j;
  			b2Vec2 P = ccp->normalImpulse * normal + ccp->tangentImpulse * tangent;
  			bodyA->m_angularVelocity -= invIA * b2Cross(ccp->rA, P);
  			bodyA->m_linearVelocity -= invMassA * P;
  			bodyB->m_angularVelocity += invIB * b2Cross(ccp->rB, P);
  			bodyB->m_linearVelocity += invMassB * P;
  		}
  	}
  }
  
  void b2ContactSolver::SolveVelocityConstraints()
  {
  	for (int32 i = 0; i < m_constraintCount; ++i)
  	{
  		b2ContactConstraint* c = m_constraints + i;
  		b2Body* bodyA = c->bodyA;
  		b2Body* bodyB = c->bodyB;
  		float32 wA = bodyA->m_angularVelocity;
  		float32 wB = bodyB->m_angularVelocity;
  		b2Vec2 vA = bodyA->m_linearVelocity;
  		b2Vec2 vB = bodyB->m_linearVelocity;
  		float32 invMassA = bodyA->m_invMass;
  		float32 invIA = bodyA->m_invI;
  		float32 invMassB = bodyB->m_invMass;
  		float32 invIB = bodyB->m_invI;
  		b2Vec2 normal = c->normal;
  		b2Vec2 tangent = b2Cross(normal, 1.0f);
  		float32 friction = c->friction;
  
  		b2Assert(c->pointCount == 1 || c->pointCount == 2);
  
  		// Solve tangent constraints
  		for (int32 j = 0; j < c->pointCount; ++j)
  		{
  			b2ContactConstraintPoint* ccp = c->points + j;
  
  			// Relative velocity at contact
  			b2Vec2 dv = vB + b2Cross(wB, ccp->rB) - vA - b2Cross(wA, ccp->rA);
  
  			// Compute tangent force
  			float32 vt = b2Dot(dv, tangent);
  			float32 lambda = ccp->tangentMass * (-vt);
  
  			// b2Clamp the accumulated force
  			float32 maxFriction = friction * ccp->normalImpulse;
  			float32 newImpulse = b2Clamp(ccp->tangentImpulse + lambda, -maxFriction, maxFriction);
  			lambda = newImpulse - ccp->tangentImpulse;
  
  			// Apply contact impulse
  			b2Vec2 P = lambda * tangent;
  
  			vA -= invMassA * P;
  			wA -= invIA * b2Cross(ccp->rA, P);
  
  			vB += invMassB * P;
  			wB += invIB * b2Cross(ccp->rB, P);
  
  			ccp->tangentImpulse = newImpulse;
  		}
  
  		// Solve normal constraints
  		if (c->pointCount == 1)
  		{
  			b2ContactConstraintPoint* ccp = c->points + 0;
  
  			// Relative velocity at contact
  			b2Vec2 dv = vB + b2Cross(wB, ccp->rB) - vA - b2Cross(wA, ccp->rA);
  
  			// Compute normal impulse
  			float32 vn = b2Dot(dv, normal);
  			float32 lambda = -ccp->normalMass * (vn - ccp->velocityBias);
  
  			// b2Clamp the accumulated impulse
  			float32 newImpulse = b2Max(ccp->normalImpulse + lambda, 0.0f);
  			lambda = newImpulse - ccp->normalImpulse;
  
  			// Apply contact impulse
  			b2Vec2 P = lambda * normal;
  			vA -= invMassA * P;
  			wA -= invIA * b2Cross(ccp->rA, P);
  
  			vB += invMassB * P;
  			wB += invIB * b2Cross(ccp->rB, P);
  			ccp->normalImpulse = newImpulse;
  		}
  		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, , 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 = vn_0 - 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 = x' - a
  			// 
  			// Plug into above equation:
  			//
  			// vn = A * x + b
  			//    = A * (x' - a) + b
  			//    = A * x' + b - A * a
  			//    = A * x' + b'
  			// b' = b - A * a;
  
  			b2ContactConstraintPoint* cp1 = c->points + 0;
  			b2ContactConstraintPoint* cp2 = c->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
  			float32 vn1 = b2Dot(dv1, normal);
  			float32 vn2 = b2Dot(dv2, normal);
  
  			b2Vec2 b;
  			b.x = vn1 - cp1->velocityBias;
  			b.y = vn2 - cp2->velocityBias;
  			b -= b2Mul(c->K, a);
  
  			const float32 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(c->normalMass, b);
  
  				if (x.x >= 0.0f && x.y >= 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 -= invMassA * (P1 + P2);
  					wA -= invIA * (b2Cross(cp1->rA, P1) + b2Cross(cp2->rA, P2));
  
  					vB += invMassB * (P1 + P2);
  					wB += invIB * (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 = c->K.col1.y * x.x + b.y;
  
  				if (x.x >= 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 -= invMassA * (P1 + P2);
  					wA -= invIA * (b2Cross(cp1->rA, P1) + b2Cross(cp2->rA, P2));
  
  					vB += invMassB * (P1 + P2);
  					wB += invIB * (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 = c->K.col2.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 -= invMassA * (P1 + P2);
  					wA -= invIA * (b2Cross(cp1->rA, P1) + b2Cross(cp2->rA, P2));
  
  					vB += invMassB * (P1 + P2);
  					wB += invIB * (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 -= invMassA * (P1 + P2);
  					wA -= invIA * (b2Cross(cp1->rA, P1) + b2Cross(cp2->rA, P2));
  
  					vB += invMassB * (P1 + P2);
  					wB += invIB * (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;
  			}
  		}
  
  		bodyA->m_linearVelocity = vA;
  		bodyA->m_angularVelocity = wA;
  		bodyB->m_linearVelocity = vB;
  		bodyB->m_angularVelocity = wB;
  	}
  }
  
  void b2ContactSolver::StoreImpulses()
  {
  	for (int32 i = 0; i < m_constraintCount; ++i)
  	{
  		b2ContactConstraint* c = m_constraints + i;
  		b2Manifold* m = c->manifold;
  
  		for (int32 j = 0; j < c->pointCount; ++j)
  		{
  			m->points[j].normalImpulse = c->points[j].normalImpulse;
  			m->points[j].tangentImpulse = c->points[j].tangentImpulse;
  		}
  	}
  }
  
  struct b2PositionSolverManifold
  {
  	void Initialize(b2ContactConstraint* cc, int32 index)
  	{
  		b2Assert(cc->pointCount > 0);
  
  		switch (cc->type)
  		{
  		case b2Manifold::e_circles:
  			{
  				b2Vec2 pointA = cc->bodyA->GetWorldPoint(cc->localPoint);
  				b2Vec2 pointB = cc->bodyB->GetWorldPoint(cc->points[0].localPoint);
  				if (b2DistanceSquared(pointA, pointB) > b2_epsilon * b2_epsilon)
  				{
  					normal = pointB - pointA;
  					normal.Normalize();
  				}
  				else
  				{
  					normal.Set(1.0f, 0.0f);
  				}
  
  				point = 0.5f * (pointA + pointB);
  				separation = b2Dot(pointB - pointA, normal) - cc->radius;
  			}
  			break;
  
  		case b2Manifold::e_faceA:
  			{
  				normal = cc->bodyA->GetWorldVector(cc->localNormal);
  				b2Vec2 planePoint = cc->bodyA->GetWorldPoint(cc->localPoint);
  
  				b2Vec2 clipPoint = cc->bodyB->GetWorldPoint(cc->points[index].localPoint);
  				separation = b2Dot(clipPoint - planePoint, normal) - cc->radius;
  				point = clipPoint;
  			}
  			break;
  
  		case b2Manifold::e_faceB:
  			{
  				normal = cc->bodyB->GetWorldVector(cc->localNormal);
  				b2Vec2 planePoint = cc->bodyB->GetWorldPoint(cc->localPoint);
  
  				b2Vec2 clipPoint = cc->bodyA->GetWorldPoint(cc->points[index].localPoint);
  				separation = b2Dot(clipPoint - planePoint, normal) - cc->radius;
  				point = clipPoint;
  
  				// Ensure normal points from A to B
  				normal = -normal;
  			}
  			break;
  		}
  	}
  
  	b2Vec2 normal;
  	b2Vec2 point;
  	float32 separation;
  };
  
  // Sequential solver.
  bool b2ContactSolver::SolvePositionConstraints(float32 baumgarte)
  {
  	float32 minSeparation = 0.0f;
  
  	for (int32 i = 0; i < m_constraintCount; ++i)
  	{
  		b2ContactConstraint* c = m_constraints + i;
  		b2Body* bodyA = c->bodyA;
  		b2Body* bodyB = c->bodyB;
  
  		float32 invMassA = bodyA->m_mass * bodyA->m_invMass;
  		float32 invIA = bodyA->m_mass * bodyA->m_invI;
  		float32 invMassB = bodyB->m_mass * bodyB->m_invMass;
  		float32 invIB = bodyB->m_mass * bodyB->m_invI;
  
  		// Solve normal constraints
  		for (int32 j = 0; j < c->pointCount; ++j)
  		{
  			b2PositionSolverManifold psm;
  			psm.Initialize(c, j);
  			b2Vec2 normal = psm.normal;
  
  			b2Vec2 point = psm.point;
  			float32 separation = psm.separation;
  
  			b2Vec2 rA = point - bodyA->m_sweep.c;
  			b2Vec2 rB = point - bodyB->m_sweep.c;
  
  			// Track max constraint error.
  			minSeparation = b2Min(minSeparation, separation);
  
  			// Prevent large corrections and allow slop.
  			float32 C = b2Clamp(baumgarte * (separation + b2_linearSlop), -b2_maxLinearCorrection, 0.0f);
  
  			// Compute the effective mass.
  			float32 rnA = b2Cross(rA, normal);
  			float32 rnB = b2Cross(rB, normal);
  			float32 K = invMassA + invMassB + invIA * rnA * rnA + invIB * rnB * rnB;
  
  			// Compute normal impulse
  			float32 impulse = K > 0.0f ? - C / K : 0.0f;
  
  			b2Vec2 P = impulse * normal;
  
  			bodyA->m_sweep.c -= invMassA * P;
  			bodyA->m_sweep.a -= invIA * b2Cross(rA, P);
  			bodyA->SynchronizeTransform();
  
  			bodyB->m_sweep.c += invMassB * P;
  			bodyB->m_sweep.a += invIB * b2Cross(rB, P);
  			bodyB->SynchronizeTransform();
  		}
  	}
  
  	// We can't expect minSpeparation >= -b2_linearSlop because we don't
  	// push the separation above -b2_linearSlop.
  	return minSeparation >= -1.5f * b2_linearSlop;
  }
  