-=[ Mr. Bumblebee ]=-
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///////////////////////////////////////////////////////////////////////////
//
// Copyright (c) 2002, Industrial Light & Magic, a division of Lucas
// Digital Ltd. LLC
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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
Copyright © 2017 || Recoded By Mr.Bumblebee