Source/OpenEXR/Imath/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.
    // 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 = 1;
    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 = 1;
    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)
{
    //
    // Create a quaternion that rotates vector from into vector to,
    // such that the rotation is around an axis that is the cross
    // product of from and to.
    //
    // This function calls function setRotationInternal(), which is
    // numerically accurate only for rotation angles that are not much
    // greater than pi/2.  In order to achieve good accuracy for angles
    // greater than pi/2, we split large angles in half, and rotate in
    // two steps.
    //

    //
    // Normalize from and to, yielding f0 and t0.
    //

    Vec3<T> f0 = from.normalized();
    Vec3<T> t0 = to.normalized();

    if ((f0 ^ t0) >= 0)
    {
	//
	// The rotation angle is less than or equal to pi/2.
	//

	setRotationInternal (f0, t0, *this);
    }
    else
    {
	//
	// The angle is greater than pi/2.  After computing h0,
	// which is halfway between f0 and t0, we rotate first
	// from f0 to h0, then from h0 to t0.
	//

	Vec3<T> h0 = (f0 + t0).normalized();

	if ((h0 ^ h0) != 0)
	{
	    setRotationInternal (f0, h0, *this);

	    Quat<T> q;
	    setRotationInternal (h0, t0, q);

	    *this *= q;
	}
	else
	{
	    //
	    // f0 and t0 point in exactly opposite directions.
	    // Pick an arbitrary axis that is orthogonal to f0,
	    // and rotate by pi.
	    //

	    r = T (0);

	    Vec3<T> f02 = f0 * f0;

	    if (f02.x <= f02.y && f02.x <= f02.z)
		v = (f0 % Vec3<T> (1, 0, 0)).normalized();
	    else if (f02.y <= f02.z)
		v = (f0 % Vec3<T> (0, 1, 0)).normalized();
	    else
		v = (f0 % Vec3<T> (0, 0, 1)).normalized();
	}
    }

    return *this;
}


template <class T>
void
Quat<T>::setRotationInternal (const Vec3<T>& f0, const Vec3<T>& t0, Quat<T> &q)
{
    //
    // The following is equivalent to setAxisAngle(n,2*phi),
    // where the rotation axis, is orthogonal to the f0 and
    // t0 vectors, and 2*phi is the angle between f0 and t0.
    //
    // This function is called by setRotation(), above; it assumes
    // that f0 and t0 are normalized and that the angle between
    // them is not much greater than pi/2.  This function becomes
    // numerically inaccurate if f0 and t0 point into nearly
    // opposite directions.
    //

    //
    // Find a normalized vector, h0, that is half way between f0 and t0.
    // The angle between f0 and h0 is phi.
    //

    Vec3<T> h0 = (f0 + t0).normalized();

    //
    // Store the rotation axis and rotation angle.
    //

    q.r = f0 ^ h0;	//  f0 ^ h0 == cos (phi)
    q.v = f0 % h0;	// (f0 % h0).length() == sin (phi)
}


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 );
}

template<class T>
inline Vec3<T> operator* (const Vec3<T>& v, const Quat<T>& q)
{
    Vec3<T> a = q.v % v;
    Vec3<T> b = q.v % a;
    return v + T (2) * (q.r * a + b);
}

#if (defined _WIN32 || defined _WIN64) && defined _MSC_VER
#pragma warning(default:4244)
#endif

} // namespace Imath

#endif