imathquat.h

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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 _WIN32 || defined _WIN64) && defined _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;

  private:

    void                    setRotationInternal (const Vec3<T>& f0,
						 const Vec3<T>& t0,
						 Quat<T> &q);
};


//--------------------
// 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>
T
angle4D (const Quat<T> &q1, const Quat<T> &q2)
{
    //
    // Compute the angle between two quaternions,
    // interpreting the quaternions as 4D vectors.
    //

    Quat<T> d = q1 - q2;
    T lengthD = Math<T>::sqrt (d ^ d);

    Quat<T> s = q1 + q2;
    T lengthS = Math<T>::sqrt (s ^ s);

    return 2 * Math<T>::atan2 (lengthD, lengthS);
}


template<class T>
Quat<T>
slerp(const Quat<T> &q1,const Quat<T> &q2, T t)
{
    //
    // Spherical linear interpolation.
    // Assumes q1 and q2 are normalized and that q1 != -q2.
    //
    // This method does *not* interpolate along the shortest
    // arc between q1 and q2.  If you desire interpolation
    // along the shortest arc, and q1^q2 is negative, then
    // consider flipping the second quaternion explicitly.
    //
    // The implementation of squad() depends on a slerp()
    // that interpolates as is, without the automatic
    // flipping.
    //
    // Don Hatch explains the method we use here on his
    // web page, The Right Way to Calculate Stuff, at
    // http://www.plunk.org/~hatch/rightway.php
    //

    T a = angle4D (q1, q2);
    T s = 1 - t;

    Quat<T> q = sinx_over_x (s * a) / sinx_over_x (a) * s * q1 +
	        sinx_over_x (t * a) / sinx_over_x (a) * t * q2;

    return q.normalized();
}


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.