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ImathQuat.h

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//
// Copyright (c) 2002, Industrial Light & Magic, a division of Lucas
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#ifndef INCLUDED_IMATHQUAT_H
#define INCLUDED_IMATHQUAT_H

//----------------------------------------------------------------------
//
//    template class Quat<T>
//
//    "Quaternions came from Hamilton ... and have been an unmixed
//    evil to those who have touched them in any way. Vector is a
//    useless survival ... and has never been of the slightest use
//    to any creature."
//
//        - Lord Kelvin
//
//    This class implements the quaternion numerical type -- you
//      will probably want to use this class to represent orientations
//    in R3 and to convert between various euler angle reps. You
//    should probably use Imath::Euler<> for that.
//
//----------------------------------------------------------------------

#include <ImathExc.h>
#include <ImathMatrix.h>

#include <iostream>

namespace Imath {

#if defined PLATFORM_WINDOWS && _MSC_VER
// Disable MS VC++ warnings about conversion from double to float
#pragma warning(disable:4244)
#endif

template <class T>
class Quat;

template<class T>
Quat<T> slerp (const Quat<T> &q1,const Quat<T> &q2, T t);

template<class T>
Quat<T> squad (const Quat<T> &q1,const Quat<T> &q2, 
             const Quat<T> &qa,const Quat<T> &qb, T t);

template<class T>
void intermediate (const Quat<T> &q0, const Quat<T> &q1, 
               const Quat<T> &q2, const Quat<T> &q3,
               Quat<T> &qa, Quat<T> &qb);

template <class T>
class Quat
{
  public:

    T                 r;          // real part
    Vec3<T>           v;          // imaginary vector

    //-----------------------------------------------------
    //      Constructors - default constructor is identity quat
    //-----------------------------------------------------

    Quat()                  : r(1), v(0,0,0) {}

    template <class S>
    Quat( const Quat<S>& q)       : r(q.r), v(q.v) {}

    Quat( T s, T i, T j, T k )          : r(s), v(i,j,k) {}

    Quat( T s, Vec3<T> d )        : r(s), v(d) {}

    static Quat<T> identity() { return Quat<T>(); }

    //------------------------------------------------
    //      Basic Algebra - Operators and Methods
    //  The operator return values are *NOT* normalized
    //
    //  operator^ is 4D dot product
    //  operator/ uses the inverse() quaternion
    //      operator~ is conjugate -- if (S+V) is quat then
    //              the conjugate (S+V)* == (S-V)
    //
    //  some operators (*,/,*=,/=) treat the quat as
    //      a 4D vector when one of the operands is scalar
    //------------------------------------------------

    const Quat<T>&          operator=     (const Quat<T>&);
    const Quat<T>&          operator*=    (const Quat<T>&);
    const Quat<T>&          operator*=    (T);
    const Quat<T>&          operator/=    (const Quat<T>&);
    const Quat<T>&          operator/=    (T);
    const Quat<T>&          operator+=    (const Quat<T>&);
    const Quat<T>&          operator-=    (const Quat<T>&);
    T&                      operator[]    (int index);      // as 4D vector
    T                 operator[]    (int index) const;

    template <class S> bool operator == (const Quat<S> &q) const;
    template <class S> bool operator != (const Quat<S> &q) const;

    Quat<T>&                invert();               // this -> 1 / this
    Quat<T>           inverse() const;
    Quat<T>&                normalize();      // returns this
    Quat<T>           normalized() const;
    T                 length() const;         // in R4

    //-----------------------
    //      Rotation conversion
    //-----------------------

    Quat<T>&                setAxisAngle(const Vec3<T>& axis, T radians);
    Quat<T>&                setRotation(const Vec3<T>& fromDirection,
                              const Vec3<T>& toDirection);

    T                   angle() const;
    Vec3<T>                 axis() const;

    Matrix33<T>             toMatrix33() const;
    Matrix44<T>             toMatrix44() const;

    Quat<T>                 log() const;
    Quat<T>                 exp() const;
};


//--------------------
// Convenient typedefs
//--------------------

typedef Quat<float>     Quatf;
typedef Quat<double>    Quatd;


//---------------
// Implementation
//---------------

template<class T>
inline const Quat<T>& Quat<T>::operator= (const Quat<T>& q)
{
    r = q.r;
    v = q.v;
    return *this;
}

template<class T>
inline const Quat<T>& Quat<T>::operator*= (const Quat<T>& q)
{
    T rtmp = r * q.r - (v ^ q.v);
    v = r * q.v + v * q.r + v % q.v;
    r = rtmp;
    return *this;
}

template<class T>
inline const Quat<T>& Quat<T>::operator*= (T t)
{
    r *= t;
    v *= t;
    return *this;
}

template<class T>
inline const Quat<T>& Quat<T>::operator/= (const Quat<T>& q)
{
    *this = *this * q.inverse();
    return *this;
}

template<class T>
inline const Quat<T>& Quat<T>::operator/= (T t)
{
    r /= t;
    v /= t;
    return *this;
}

template<class T>
inline const Quat<T>& Quat<T>::operator+= (const Quat<T>& q)
{
    r += q.r;
    v += q.v;
    return *this;
}

template<class T>
inline const Quat<T>& Quat<T>::operator-= (const Quat<T>& q)
{
    r -= q.r;
    v -= q.v;
    return *this;
}
template<class T>
inline T& Quat<T>::operator[] (int index)
{
    return index ? v[index-1] : r;
}

template<class T>
inline T Quat<T>::operator[] (int index) const
{
    return index ? v[index-1] : r;
}

template <class T>
template <class S>
inline bool
Quat<T>::operator == (const Quat<S> &q) const
{
    return r == q.r && v == q.v;
}

template <class T>
template <class S>
inline bool
Quat<T>::operator != (const Quat<S> &q) const
{
    return r != q.r || v != q.v;
}

template<class T>
inline T operator^ (const Quat<T>& q1,const Quat<T>& q2)
{
    return q1.r * q2.r + (q1.v ^ q2.v);
}

template <class T>
inline T Quat<T>::length() const
{
    return Math<T>::sqrt( r * r + (v ^ v) );
}

template <class T>
inline Quat<T>& Quat<T>::normalize()
{
    if ( T l = length() ) { r /= l; v /= l; }
    else { r = 1; v = Vec3<T>(0); }
    return *this;
}

template <class T>
inline Quat<T> Quat<T>::normalized() const
{
    if ( T l = length() ) return Quat( r / l, v / l );
    return Quat();
}

template<class T>
inline Quat<T> Quat<T>::inverse() const
{
    // 1    Q*
    // - = ----   where Q* is conjugate (operator~)
    // Q   Q* Q   and (Q* Q) == Q ^ Q (4D dot)

    T qdot = *this ^ *this;
    return Quat( r / qdot, -v / qdot );
}

template<class T>
inline Quat<T>& Quat<T>::invert()
{
    T qdot = (*this) ^ (*this);
    r /= qdot;
    v = -v / qdot;
    return *this;
}

template<class T>
Quat<T>
slerp(const Quat<T> &q1,const Quat<T> &q2, T t)
{
    //
    // Spherical linear interpolation.
    //
    // NOTE: Assumes q1 and q2 are normalized and that 0 <= t <= 1.
    //
    // This method does *not* interpolate along the shortest arc
    // between q1 and q2. If you desire interpolation along the
    // shortest arc, then consider flipping the second quaternion
    // explicitly before calling slerp. The implementation of squad()
    // depends on a slerp() that interpolates as is, without the
    // automatic flipping.
    //

    T cosomega = q1 ^ q2;
    if (cosomega >= (T) 1.0)
    {
      //
      // Special case: q1 and q2 are the same, so just return one of them.
      // This also catches the case where cosomega is very slightly > 1.0
      //

      return q1;
    }
    
    T sinomega = Math<T>::sqrt (1 - cosomega * cosomega);

    Quat<T> result;

    if (sinomega * limits<T>::max() > 1)
    {
      T omega = Math<T>::acos (cosomega);
      T s1 = Math<T>::sin ((1.0 - t) * omega) / sinomega;
      T s2 = Math<T>::sin (t * omega) / sinomega;

      result      = s1 * q1 + s2 * q2;
    }
    else if (cosomega > 0)
    {
      //
      // omega == 0
      //

      T s1 = 1.0 - t;
      T s2 = t;

      result = s1 * q1 + s2 * q2;
    }
    else
    {
      //
      // omega == -pi
      //

      result.v.x  = - q1.v.y;
      result.v.y  =   q1.v.x;
      result.v.z  = - q1.r;
      result.r    =   q1.v.z;

      T s1 = Math<T>::sin ((0.5 - t) * M_PI);
      T s2 = Math<T>::sin (t * M_PI);

      result  = s1 * q1 + s2 * result;
    }

    return result;
}

template<class T>
Quat<T> spline(const Quat<T> &q0, const Quat<T> &q1,
             const Quat<T> &q2, const Quat<T> &q3,
             T t)
{
    // Spherical Cubic Spline Interpolation -
    // from Advanced Animation and Rendering
    // Techniques by Watt and Watt, Page 366:
    // A spherical curve is constructed using three
    // spherical linear interpolations of a quadrangle
    // of unit quaternions: q1, qa, qb, q2.
    // Given a set of quaternion keys: q0, q1, q2, q3,
    // this routine does the interpolation between
    // q1 and q2 by constructing two intermediate
    // quaternions: qa and qb. The qa and qb are 
    // computed by the intermediate function to 
    // guarantee the continuity of tangents across
    // adjacent cubic segments. The qa represents in-tangent
    // for q1 and the qb represents the out-tangent for q2.
    // 
    // The q1 q2 is the cubic segment being interpolated. 
    // The q0 is from the previous adjacent segment and q3 is 
    // from the next adjacent segment. The q0 and q3 are used
    // in computing qa and qb.
    // 

    Quat<T> qa = intermediate (q0, q1, q2);
    Quat<T> qb = intermediate (q1, q2, q3);
    Quat<T> result = squad(q1, qa, qb, q2, t);

    return result;
}

template<class T>
Quat<T> squad(const Quat<T> &q1, const Quat<T> &qa,
            const Quat<T> &qb, const Quat<T> &q2,
            T t)
{
    // Spherical Quadrangle Interpolation -
    // from Advanced Animation and Rendering
    // Techniques by Watt and Watt, Page 366:
    // It constructs a spherical cubic interpolation as 
    // a series of three spherical linear interpolations 
    // of a quadrangle of unit quaternions. 
    //     
  
    Quat<T> r1 = slerp(q1, q2, t);
    Quat<T> r2 = slerp(qa, qb, t);
    Quat<T> result = slerp(r1, r2, 2*t*(1-t));

    return result;
}

template<class T>
Quat<T> intermediate(const Quat<T> &q0, const Quat<T> &q1, const Quat<T> &q2)
{
    // From advanced Animation and Rendering
    // Techniques by Watt and Watt, Page 366:
    // computing the inner quadrangle 
    // points (qa and qb) to guarantee tangent
    // continuity.
    // 
    Quat<T> q1inv = q1.inverse();
    Quat<T> c1 = q1inv*q2;
    Quat<T> c2 = q1inv*q0;
    Quat<T> c3 = (T) (-0.25) * (c2.log() + c1.log());
    Quat<T> qa = q1 * c3.exp();
    qa.normalize();
    return qa;
}

template <class T>
inline Quat<T> Quat<T>::log() const
{
    //
    // For unit quaternion, from Advanced Animation and 
    // Rendering Techniques by Watt and Watt, Page 366:
    //

    T theta = Math<T>::acos (std::min (r, (T) 1.0));
    if (theta == 0)
      return Quat<T> (0, v);
    
    T sintheta = Math<T>::sin (theta);
    
    T k;
    if (abs (sintheta) < 1 && abs (theta) >= limits<T>::max() * abs (sintheta))
      k = 0;
    else
      k = theta / sintheta;

    return Quat<T> ((T) 0, v.x * k, v.y * k, v.z * k);
}

template <class T>
inline Quat<T> Quat<T>::exp() const
{
    //
    // For pure quaternion (zero scalar part):
    // from Advanced Animation and Rendering
    // Techniques by Watt and Watt, Page 366:
    //

    T theta = v.length();
    T sintheta = Math<T>::sin (theta);
    
    T k;
    if (abs (theta) < 1 && abs (sintheta) >= limits<T>::max() * abs (theta))
      k = 0;
    else
      k = sintheta / theta;

    T costheta = Math<T>::cos (theta);

    return Quat<T> (costheta, v.x * k, v.y * k, v.z * k);
}

template <class T>
inline T Quat<T>::angle() const
{
    return 2.0*Math<T>::acos(r);
}

template <class T>
inline Vec3<T> Quat<T>::axis() const
{
    return v.normalized();
}

template <class T>
inline Quat<T>& Quat<T>::setAxisAngle(const Vec3<T>& axis, T radians)
{
    r = Math<T>::cos(radians/2);
    v = axis.normalized() * Math<T>::sin(radians/2);
    return *this;
}


template <class T>
Quat<T>&
Quat<T>::setRotation(const Vec3<T>& from, const Vec3<T>& to)
{
    //
    // Ported from SbRotation
    //

    T cost = from.dot(to) / Math<T>::sqrt(from.dot(from) * to.dot(to));

    // check for degeneracies
    if (cost > 0.99999)
    {
      //
      // Vectors are parallel.
      //

      r = 1.0;
      v = Vec3<T>(0);
    }
    else if (cost < -0.99999)
    {
      //
      // Vectors are opposite. Find an axis to rotate around,
      // which should be perpendicular to the original axis.
      //

      Vec3<T> frm = from.normalized();
      v = frm.cross(Vec3<T>(1, 0, 0));
      if (v.length() < 0.00001)
          v   = frm.cross(Vec3<T>(0, 1, 0));
      r = 0;
      v.normalize();
    }
    else
    {
      //
      // Use half-angle formulae:
      // cos^2 t = ( 1 + cos (2t) ) / 2
      // w part is cosine of half the rotation angle
      //

      r = Math<T>::sqrt(0.5 * (1.0 + cost));

      //
      // sin^2 t = ( 1 - cos (2t) ) / 2
      // Do the normalization of the axis vector at the same time so
      // we only call sqrt once.
      //

      v = from.cross(to);
      v *= Math<T>::sqrt((0.5 * (1.0 - cost))/(v.dot(v)));
    }

    return *this;
}

template<class T>
Matrix33<T> Quat<T>::toMatrix33() const
{
    return Matrix33<T>(1. - 2.0 * (v.y * v.y + v.z * v.z),
                      2.0 * (v.x * v.y + v.z * r),
                      2.0 * (v.z * v.x - v.y * r),

                      2.0 * (v.x * v.y - v.z * r),
                   1. - 2.0 * (v.z * v.z + v.x * v.x),
                      2.0 * (v.y * v.z + v.x * r),

                      2.0 * (v.z * v.x + v.y * r),
                      2.0 * (v.y * v.z - v.x * r),
                   1. - 2.0 * (v.y * v.y + v.x * v.x));
}

template<class T>
Matrix44<T> Quat<T>::toMatrix44() const
{
    return Matrix44<T>(1. - 2.0 * (v.y * v.y + v.z * v.z),
                      2.0 * (v.x * v.y + v.z * r),
                      2.0 * (v.z * v.x - v.y * r),
                      0.,
                      2.0 * (v.x * v.y - v.z * r),
                   1. - 2.0 * (v.z * v.z + v.x * v.x),
                      2.0 * (v.y * v.z + v.x * r),
                      0.,
                      2.0 * (v.z * v.x + v.y * r),
                      2.0 * (v.y * v.z - v.x * r),
                   1. - 2.0 * (v.y * v.y + v.x * v.x),
                      0.,
                      0.,
                      0.,
                      0.,
                      1.0 );
}


template<class T>
inline Matrix33<T> operator* (const Matrix33<T> &M, const Quat<T> &q)
{
    return M * q.toMatrix33();
}

template<class T>
inline Matrix33<T> operator* (const Quat<T> &q, const Matrix33<T> &M)
{
    return q.toMatrix33() * M;
}

template<class T>
std::ostream& operator<< (std::ostream &o, const Quat<T> &q)
{
    return o << "(" << q.r
           << " " << q.v.x
           << " " << q.v.y
           << " " << q.v.z
           << ")";

}

template<class T>
inline Quat<T> operator* (const Quat<T>& q1, const Quat<T>& q2)
{
    // (S1+V1) (S2+V2) = S1 S2 - V1.V2 + S1 V2 + V1 S2 + V1 x V2
    return Quat<T>( q1.r * q2.r - (q1.v ^ q2.v),
                q1.r * q2.v + q1.v * q2.r + q1.v % q2.v );
}

template<class T>
inline Quat<T> operator/ (const Quat<T>& q1, const Quat<T>& q2)
{
    return q1 * q2.inverse();
}

template<class T>
inline Quat<T> operator/ (const Quat<T>& q,T t)
{
    return Quat<T>(q.r/t,q.v/t);
}

template<class T>
inline Quat<T> operator* (const Quat<T>& q,T t)
{
    return Quat<T>(q.r*t,q.v*t);
}

template<class T>
inline Quat<T> operator* (T t, const Quat<T>& q)
{
    return Quat<T>(q.r*t,q.v*t);
}

template<class T>
inline Quat<T> operator+ (const Quat<T>& q1, const Quat<T>& q2)
{
    return Quat<T>( q1.r + q2.r, q1.v + q2.v );
}

template<class T>
inline Quat<T> operator- (const Quat<T>& q1, const Quat<T>& q2)
{
    return Quat<T>( q1.r - q2.r, q1.v - q2.v );
}

template<class T>
inline Quat<T> operator~ (const Quat<T>& q)
{
    return Quat<T>( q.r, -q.v );    // conjugate: (S+V)* = S-V
}

template<class T>
inline Quat<T> operator- (const Quat<T>& q)
{
    return Quat<T>( -q.r, -q.v );
}

#if defined PLATFORM_WINDOWS && _MSC_VER
#pragma warning(default:4244)
#endif

} // namespace Imath

#endif

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