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LA_library/quaternion.cc

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/*
LA: linear algebra C++ interface library
Copyright (C) 2020 Jiri Pittner <jiri.pittner@jh-inst.cas.cz> or <jiri@pittnerovi.com>
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
#include "quaternion.h"
#include "vecmat3.h"
using namespace LA_Vecmat3;
namespace LA_Quaternion {
//do not replicate this code in each object file, therefore not in .h
//and instantize the templates for the types needed
template<typename T>
Quaternion<T>& Quaternion<T>::normalize(T *getnorm, bool unique_sign)
{
T nn=norm();
if(getnorm) *getnorm=nn;
if(unique_sign && q[0]<0) nn= -nn;
*this /= nn;
return *this;
};
template<>
Quaternion<float> & Quaternion<float>::fast_normalize(bool unique_sign)
{
float nn=fast_sqrtinv(normsqr());
if(unique_sign && q[0]<0) nn= -nn;
*this *= nn;
return *this;
}
template<typename T>
Quaternion<T>& Quaternion<T>::fast_normalize(bool unique_sign)
{
return normalize(NULL,unique_sign);
};
template<typename T>
Quaternion<T> Quaternion<T>::operator*(const Quaternion<T> &rhs) const
{
return Quaternion<T>
(
q[0]*rhs.q[0]-q[1]*rhs.q[1]-q[2]*rhs.q[2]-q[3]*rhs.q[3],
q[0]*rhs.q[1]+q[1]*rhs.q[0]+q[2]*rhs.q[3]-q[3]*rhs.q[2],
q[0]*rhs.q[2]+q[2]*rhs.q[0]+q[3]*rhs.q[1]-q[1]*rhs.q[3],
q[0]*rhs.q[3]+q[3]*rhs.q[0]+q[1]*rhs.q[2]-q[2]*rhs.q[1]
);
};
template<typename T>
Quaternion<T> Quaternion<T>::times_vec3(const T *rhs) const
{
return Quaternion<T>
(
-q[1]*rhs[0]-q[2]*rhs[1]-q[3]*rhs[2],
q[0]*rhs[0]+q[2]*rhs[2]-q[3]*rhs[1],
q[0]*rhs[1]+q[3]*rhs[0]-q[1]*rhs[2],
q[0]*rhs[2]+q[1]*rhs[1]-q[2]*rhs[0]
);
};
template<typename T>
Quaternion<T> Quaternion<T>::vec3_times(const T *lhs) const
{
return Quaternion<T>
(
-lhs[0]*q[1]-lhs[1]*q[2]-lhs[2]*q[3],
lhs[0]*q[0]+lhs[1]*q[3]-lhs[2]*q[2],
lhs[1]*q[0]+lhs[2]*q[1]-lhs[0]*q[3],
lhs[2]*q[0]+lhs[0]*q[2]-lhs[1]*q[1]
);
};
//basically the same code as in normquat2rotmat, but avoiding extra storage
template<typename T>
void Quaternion<T>::rotate(T *to, const T *from, Quaternion<T> *grad) const
{
//some subexpression eliminations
{
T q00= q[0]*q[0];
T q11= q[1]*q[1];
T q22= q[2]*q[2];
T q33= q[3]*q[3];
to[0] = (q00+q11-q22-q33) * from[0];
to[1] = (q00+q22-q11-q33) * from[1];
to[2] = (q00+q33-q11-q22) * from[2];
}
T q01= q[0]*q[1];
T q02= q[0]*q[2];
T q03= q[0]*q[3];
T q12= q[1]*q[2];
T q13= q[1]*q[3];
T q23= q[2]*q[3];
T f0=from[0]+from[0];
T f1=from[1]+from[1];
T f2=from[2]+from[2];
to[0] += (q12+q03)*f1;
to[0] += (q13-q02)*f2;
to[1] += (q12-q03)*f0;
to[1] += (q23+q01)*f2;
to[2] += (q13+q02)*f0;
to[2] += (q23-q01)*f1;
/*
to[0] = (2*q[0]*q[0]-1+2*q[1]*q[1]) * from[0] +
2*(q[1]*q[2]+q[0]*q[3]) * from[1] +
2*(q[1]*q[3]-q[0]*q[2]) * from[2];
to[1] = 2*(q[1]*q[2]-q[0]*q[3]) * from[0] +
(2*q[0]*q[0]-1+2*q[2]*q[2]) * from[1] +
2*(q[2]*q[3]+q[0]*q[1]) * from[2];
to[2] = 2*(q[1]*q[3]+q[0]*q[2]) * from[0] +
2*(q[2]*q[3]-q[0]*q[1]) * from[1] +
(2*q[0]*q[0]-1+2*q[3]*q[3]) * from[2];
*/
if(grad)
{
grad[0][0]= q[0]*f0 + q[3]*f1 - q[2]*f2;
grad[0][1]= q[1]*f0 + q[2]*f1 + q[3]*f2;
grad[0][2]= -q[2]*f0 + q[1]*f1 - q[0]*f2;
grad[0][3]= -q[3]*f0 + q[0]*f1 + q[1]*f2;
grad[1][0]= -q[3]*f0 + q[0]*f1 + q[1]*f2;
grad[1][1]= q[2]*f0 - q[1]*f1 + q[0]*f2;
grad[1][2]= q[1]*f0 + q[2]*f1 + q[3]*f2;
grad[1][3]= -q[0]*f0 - q[3]*f1 + q[2]*f2;
grad[2][0]= q[2]*f0 - q[1]*f1 + q[0]*f2;
grad[2][1]= q[3]*f0 - q[0]*f1 - q[1]*f2;
grad[2][2]= q[0]*f0 + q[3]*f1 - q[2]*f2;
grad[2][3]= q[1]*f0 + q[2]*f1 + q[3]*f2;
}
}
template<typename T>
Quaternion<T> Quaternion<T>::rotateby(const Quaternion<T> &rhs)
{
//return rhs.inverse() * *this * rhs; //inefficient reference implementation
Quaternion<T> r;
r[0]=0;
rhs.rotate(&r[1],&q[1]);
return r;
}
template<typename T>
Quaternion<T> Quaternion<T>::rotate_match(T *to, const T *from, const T *match) const
{
Quaternion<T> grad[3];
Quaternion<T>::rotate(to, from, grad);
Quaternion<T> derivative;
derivative = grad[0] * (to[0]-match[0]);
derivative += grad[1] * (to[1]-match[1]);
derivative += grad[2] * (to[2]-match[2]);
return derivative;
}
//optionally skip this for microcontrollers if not needed
//note that C++ standard headers automatically use float versions of the goniometric functions for T=float
template<typename T>
void Quaternion<T>::normquat2eulerzyx(T *e) const
{
e[0]= atan2(2*q[1]*q[2]-2*q[0]*q[3],2*q[0]*q[0]+2*q[1]*q[1]-1);
e[1]= -asin(2*q[1]*q[3]+2*q[0]*q[2]);
e[2]= atan2(2*q[2]*q[3]-2*q[0]*q[1],2*q[0]*q[0]+2*q[3]*q[3]-1);
}
template<typename T>
void Quaternion<T>::normquat2euler(T *eul, const char *type) const
{
Mat3<T> rotmat;
normquat2rotmat(*this,rotmat,false);
rotmat2euler(eul,rotmat,type,false,false,false);
}
//could be done more efficiently since not all rotmat element s are needed in each case, but this is lazy implementation
template<typename T>
void Quaternion<T>::euler2normquat(const T *e, const char *type)
{
T s1=sin(e[0]*(T)0.5);
T s2=sin(e[1]*(T)0.5);
T s3=sin(e[2]*(T)0.5);
T c1=cos(e[0]*(T)0.5);
T c2=cos(e[1]*(T)0.5);
T c3=cos(e[2]*(T)0.5);
switch(Euler_case(type[0],type[1],type[2]))
{
case Euler_case('x','z','x'):
{
q[0] = c2*(c1*c3-s1*s3);
q[1] = c2*(s1*c3+c1*s3);
q[2] = -s2*(s1*c3-c1*s3);
q[3] = s2*(c1*c3+s1*s3);
}
break;
case Euler_case('x','y','x'):
{
q[0] = c2*(c1*c3-s1*s3);
q[1] = c2*(s1*c3+c1*s3);
q[2] = s2*(c1*c3+s1*s3);
q[3] = s2*(s1*c3-c1*s3);
}
break;
case Euler_case('y','x','y'):
{
q[0] = c2*(c1*c3-s1*s3);
q[1] = s2*(c1*c3+s1*s3);
q[2] = c2*(s1*c3+c1*s3);
q[3] = -s2*(s1*c3-c1*s3);
}
break;
case Euler_case('y','z','y'):
{
q[0] = c2*(c1*c3-s1*s3);
q[1] = s2*(s1*c3-c1*s3);
q[2] = c2*(s1*c3+c1*s3);
q[3] = s2*(c1*c3+s1*s3);
}
break;
case Euler_case('z','y','z'):
{
q[0] = c2*(c1*c3-s1*s3);
q[1] = -s2*(s1*c3-c1*s3);
q[2] = s2*(c1*c3+s1*s3);
q[3] = c2*(s1*c3+c1*s3);
}
break;
case Euler_case('z','x','z'):
{
q[0] = c2*(c1*c3-s1*s3);
q[1] = s2*(c1*c3+s1*s3);
q[2] = s2*(s1*c3-c1*s3);
q[3] = c2*(s1*c3+c1*s3);
}
break;
case Euler_case('x','z','y'):
{
q[0] = s1*s2*s3 +c1*c2*c3;
q[1] = s1*c2*c3 - c1*s2*s3;
q[2] = -s1*s2*c3+c1*c2*s3;
q[3] = s1*c2*s3 +c1*s2*c3;
}
break;
case Euler_case('x','y','z'):
{
q[0] = -s1*s2*s3 + c1*c2*c3;
q[1] = s1*c2*c3 +c1*s2*s3;
q[2] = -s1*c2*s3 + c1*s2*c3;
q[3] = s1*s2*c3 + c1*c2*s3;
}
break;
case Euler_case('y','x','z'):
{
q[0] = s1*s2*s3 + c1*c2*c3;
q[1] = s1*c2*s3 + c1*s2*c3;
q[2] = s1*c2*c3 - c1*s2*s3;
q[3] = -s1*s2*c3 + c1*c2*s3;
}
break;
case Euler_case('y','z','x'):
{
q[0] = -s1*s2*s3 + c1*c2*c3;
q[1] = s1*s2*c3 + c1*c2*s3;
q[2] = s1*c2*c3 + c1*s2*s3;
q[3] = -s1*c2*s3 + c1*s2*c3;
}
break;
case Euler_case('z','y','x'):
{
q[0] = c1*c2*c3 + s1*s2*s3;
q[1] = s1*s2*c3 - c1*c2*s3;
q[2] = -s1*c2*s3 - c1*s2*c3;
q[3] = -s1*c2*c3 + c1*s2*s3;
}
break;
case Euler_case('z','x','y'):
{
q[0] = -s1*s2*s3 + c1*c2*c3;
q[1] = -s1*c2*s3 + c1*s2*c3;
q[2] = s1*s2*c3 + c1*c2*s3;
q[3] = s1*c2*c3 + c1*s2*s3;
}
break;
}//switch
}
//
template<typename T>
void Quaternion<T>::axis2normquat(const T *axis, const T &angle)
{
T a = ((T).5)*angle;
q[0]=cos(a);
T s=sin(a);
q[1]=axis[0]*s;
q[2]=axis[1]*s;
q[3]=axis[2]*s;
}
template<typename T>
void Quaternion<T>::normquat2axis(T *axis, T &angle) const
{
T s = sqrt(q[1]*q[1] + q[2]*q[2] +q[3]*q[3]);
angle = 2*atan2(s,q[0]);
s= 1/s;
axis[0]= q[1]*s;
axis[1]= q[2]*s;
axis[2]= q[3]*s;
}
template<typename T>
Quaternion<T> exp(const Quaternion<T> &x)
{
Quaternion<T> r;
T vnorm = sqrt(x[1]*x[1]+x[2]*x[2]+x[3]*x[3]);
r[0] = cos(vnorm);
vnorm = sin(vnorm)/vnorm;
r[1] = x[1] * vnorm;
r[2] = x[2] * vnorm;
r[3] = x[3] * vnorm;
r*= ::exp(x[0]);
return r;
}
//NOTE: log(exp(x)) need not be always = x ... log is not unique!
//NOTE2: log(x*y) != log(y*x) != log(x)+log(y)
template<typename T>
Quaternion<T> log(const Quaternion<T> &x)
{
Quaternion<T> r;
T vnorm = x[1]*x[1]+x[2]*x[2]+x[3]*x[3];
T xnorm = vnorm + x[0]*x[0];
vnorm = sqrt(vnorm);
xnorm = sqrt(xnorm);
r[0] = ::log(xnorm);
T tmp = acos(x[0]/xnorm)/vnorm;
r[1] = x[1] * tmp;
r[2] = x[2] * tmp;
r[3] = x[3] * tmp;
return r;
}
template<typename T>
Quaternion<T> pow(const Quaternion<T> &x, const T &y)
{
Quaternion<T> r;
T vnorm = x[1]*x[1]+x[2]*x[2]+x[3]*x[3];
T xnorm = vnorm + x[0]*x[0];
vnorm = sqrt(vnorm);
xnorm = sqrt(xnorm);
T phi = acos(x[0]/xnorm);
r[0] = cos(y*phi);
T tmp = sin(y*phi)/vnorm;
r[1] = x[1] * tmp;
r[2] = x[2] * tmp;
r[3] = x[3] * tmp;
r *= ::pow(xnorm,y);
return r;
}
#ifndef QUAT_NO_RANDOM
//method to efficiently generate a uniformly random point on a unit 4-sphere by G. Marsaglia, Ann. Math. Stat. 43, 645 (1972)
template<typename T>
void Quaternion<T>::random_rotation()
{
T s1,s2,s;
do
{
q[0]=2.*random()/(1. + RAND_MAX) - 1.;
q[1]=2.*random()/(1. + RAND_MAX) - 1.;
s1 = q[0]*q[0] + q[1]*q[1];
}
while(s1>1);
do
{
q[2]=2.*random()/(1. + RAND_MAX) - 1.;
q[3]=2.*random()/(1. + RAND_MAX) - 1.;
s2 = q[2]*q[2] + q[3]*q[3];
}
while(s2>1);
s = sqrt((1-s1)/s2);
q[2] *= s;
q[3] *= s;
}
#endif
//force instantization
#define INSTANTIZE(T) \
template class Quaternion<T>; \
template Quaternion<T> pow(const Quaternion<T> &x, const T &y); \
template Quaternion<T> log(const Quaternion<T> &x); \
template Quaternion<T> exp(const Quaternion<T> &x); \
INSTANTIZE(float)
#ifndef QUAT_NO_DOUBLE
INSTANTIZE(double)
#endif
} //namespace
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