source: code/archive/tutorial/src/external/bullet/LinearMath/btTransformUtil.h

/*
Copyright (c) 2003-2006 Gino van den Bergen / Erwin Coumans  http://continuousphysics.com/Bullet/

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.
*/


#ifndef SIMD_TRANSFORM_UTIL_H
#define SIMD_TRANSFORM_UTIL_H

#include "btTransform.h"
#define ANGULAR_MOTION_THRESHOLD btScalar(0.5)*SIMD_HALF_PI



#define SIMDSQRT12 btScalar(0.7071067811865475244008443621048490)

#define btRecipSqrt(x) ((btScalar)(btScalar(1.0)/btSqrt(btScalar(x))))          /* reciprocal square root */

SIMD_FORCE_INLINE btVector3 btAabbSupport(const btVector3& halfExtents,const btVector3& supportDir)
{
        return btVector3(supportDir.x() < btScalar(0.0) ? -halfExtents.x() : halfExtents.x(),
      supportDir.y() < btScalar(0.0) ? -halfExtents.y() : halfExtents.y(),
      supportDir.z() < btScalar(0.0) ? -halfExtents.z() : halfExtents.z()); 
}


SIMD_FORCE_INLINE void btPlaneSpace1 (const btVector3& n, btVector3& p, btVector3& q)
{
  if (btFabs(n.z()) > SIMDSQRT12) {
    // choose p in y-z plane
    btScalar a = n[1]*n[1] + n[2]*n[2];
    btScalar k = btRecipSqrt (a);
    p.setValue(0,-n[2]*k,n[1]*k);
    // set q = n x p
    q.setValue(a*k,-n[0]*p[2],n[0]*p[1]);
  }
  else {
    // choose p in x-y plane
    btScalar a = n.x()*n.x() + n.y()*n.y();
    btScalar k = btRecipSqrt (a);
    p.setValue(-n.y()*k,n.x()*k,0);
    // set q = n x p
    q.setValue(-n.z()*p.y(),n.z()*p.x(),a*k);
  }
}



/// Utils related to temporal transforms
class btTransformUtil
{

public:

        static void integrateTransform(const btTransform& curTrans,const btVector3& linvel,const btVector3& angvel,btScalar timeStep,btTransform& predictedTransform)
        {
                predictedTransform.setOrigin(curTrans.getOrigin() + linvel * timeStep);
//      #define QUATERNION_DERIVATIVE
        #ifdef QUATERNION_DERIVATIVE
                btQuaternion predictedOrn = curTrans.getRotation();
                predictedOrn += (angvel * predictedOrn) * (timeStep * btScalar(0.5));
                predictedOrn.normalize();
        #else
                //Exponential map
                //google for "Practical Parameterization of Rotations Using the Exponential Map", F. Sebastian Grassia

                btVector3 axis;
                btScalar        fAngle = angvel.length(); 
                //limit the angular motion
                if (fAngle*timeStep > ANGULAR_MOTION_THRESHOLD)
                {
                        fAngle = ANGULAR_MOTION_THRESHOLD / timeStep;
                }

                if ( fAngle < btScalar(0.001) )
                {
                        // use Taylor's expansions of sync function
                        axis   = angvel*( btScalar(0.5)*timeStep-(timeStep*timeStep*timeStep)*(btScalar(0.020833333333))*fAngle*fAngle );
                }
                else
                {
                        // sync(fAngle) = sin(c*fAngle)/t
                        axis   = angvel*( btSin(btScalar(0.5)*fAngle*timeStep)/fAngle );
                }
                btQuaternion dorn (axis.x(),axis.y(),axis.z(),btCos( fAngle*timeStep*btScalar(0.5) ));
                btQuaternion orn0 = curTrans.getRotation();

                btQuaternion predictedOrn = dorn * orn0;
                predictedOrn.normalize();
        #endif
                predictedTransform.setRotation(predictedOrn);
        }

        static void     calculateVelocityQuaternion(const btVector3& pos0,const btVector3& pos1,const btQuaternion& orn0,const btQuaternion& orn1,btScalar timeStep,btVector3& linVel,btVector3& angVel)
        {
                linVel = (pos1 - pos0) / timeStep;
                btVector3 axis;
                btScalar  angle;
                if (orn0 != orn1)
                {
                        calculateDiffAxisAngleQuaternion(orn0,orn1,axis,angle);
                        angVel = axis * angle / timeStep;
                } else
                {
                        angVel.setValue(0,0,0);
                }
        }

        static void calculateDiffAxisAngleQuaternion(const btQuaternion& orn0,const btQuaternion& orn1a,btVector3& axis,btScalar& angle)
        {
                btQuaternion orn1 = orn0.farthest(orn1a);
                btQuaternion dorn = orn1 * orn0.inverse();
                ///floating point inaccuracy can lead to w component > 1..., which breaks 
                dorn.normalize();
                angle = dorn.getAngle();
                axis = btVector3(dorn.x(),dorn.y(),dorn.z());
                axis[3] = btScalar(0.);
                //check for axis length
                btScalar len = axis.length2();
                if (len < SIMD_EPSILON*SIMD_EPSILON)
                        axis = btVector3(btScalar(1.),btScalar(0.),btScalar(0.));
                else
                        axis /= btSqrt(len);
        }

        static void     calculateVelocity(const btTransform& transform0,const btTransform& transform1,btScalar timeStep,btVector3& linVel,btVector3& angVel)
        {
                linVel = (transform1.getOrigin() - transform0.getOrigin()) / timeStep;
                btVector3 axis;
                btScalar  angle;
                calculateDiffAxisAngle(transform0,transform1,axis,angle);
                angVel = axis * angle / timeStep;
        }

        static void calculateDiffAxisAngle(const btTransform& transform0,const btTransform& transform1,btVector3& axis,btScalar& angle)
        {
                btMatrix3x3 dmat = transform1.getBasis() * transform0.getBasis().inverse();
                btQuaternion dorn;
                dmat.getRotation(dorn);

                ///floating point inaccuracy can lead to w component > 1..., which breaks 
                dorn.normalize();
                
                angle = dorn.getAngle();
                axis = btVector3(dorn.x(),dorn.y(),dorn.z());
                axis[3] = btScalar(0.);
                //check for axis length
                btScalar len = axis.length2();
                if (len < SIMD_EPSILON*SIMD_EPSILON)
                        axis = btVector3(btScalar(1.),btScalar(0.),btScalar(0.));
                else
                        axis /= btSqrt(len);
        }

};


///The btConvexSeparatingDistanceUtil can help speed up convex collision detection 
///by conservatively updating a cached separating distance/vector instead of re-calculating the closest distance
class   btConvexSeparatingDistanceUtil
{
        btQuaternion    m_ornA;
        btQuaternion    m_ornB;
        btVector3       m_posA;
        btVector3       m_posB;
        
        btVector3       m_separatingNormal;

        btScalar        m_boundingRadiusA;
        btScalar        m_boundingRadiusB;
        btScalar        m_separatingDistance;

public:

        btConvexSeparatingDistanceUtil(btScalar boundingRadiusA,btScalar        boundingRadiusB)
                :m_boundingRadiusA(boundingRadiusA),
                m_boundingRadiusB(boundingRadiusB),
                m_separatingDistance(0.f)
        {
        }

        btScalar        getConservativeSeparatingDistance()
        {
                return m_separatingDistance;
        }

        void    updateSeparatingDistance(const btTransform& transA,const btTransform& transB)
        {
                const btVector3& toPosA = transA.getOrigin();
                const btVector3& toPosB = transB.getOrigin();
                btQuaternion toOrnA = transA.getRotation();
                btQuaternion toOrnB = transB.getRotation();

                if (m_separatingDistance>0.f)
                {
                        

                        btVector3 linVelA,angVelA,linVelB,angVelB;
                        btTransformUtil::calculateVelocityQuaternion(m_posA,toPosA,m_ornA,toOrnA,btScalar(1.),linVelA,angVelA);
                        btTransformUtil::calculateVelocityQuaternion(m_posB,toPosB,m_ornB,toOrnB,btScalar(1.),linVelB,angVelB);
                        btScalar maxAngularProjectedVelocity = angVelA.length() * m_boundingRadiusA + angVelB.length() * m_boundingRadiusB;
                        btVector3 relLinVel = (linVelB-linVelA);
                        btScalar relLinVelocLength = (linVelB-linVelA).dot(m_separatingNormal);
                        if (relLinVelocLength<0.f)
                        {
                                relLinVelocLength = 0.f;
                        }
        
                        btScalar        projectedMotion = maxAngularProjectedVelocity +relLinVelocLength;
                        m_separatingDistance -= projectedMotion;
                }
        
                m_posA = toPosA;
                m_posB = toPosB;
                m_ornA = toOrnA;
                m_ornB = toOrnB;
        }

        void    initSeparatingDistance(const btVector3& separatingVector,btScalar separatingDistance,const btTransform& transA,const btTransform& transB)
        {
                m_separatingNormal = separatingVector;
                m_separatingDistance = separatingDistance;
                
                const btVector3& toPosA = transA.getOrigin();
                const btVector3& toPosB = transB.getOrigin();
                btQuaternion toOrnA = transA.getRotation();
                btQuaternion toOrnB = transB.getRotation();
                m_posA = toPosA;
                m_posB = toPosB;
                m_ornA = toOrnA;
                m_ornB = toOrnB;
        }

};


#endif //SIMD_TRANSFORM_UTIL_H