/* LA: linear algebra C++ interface library Copyright (C) 2020 Jiri Pittner or 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 . */ //this actually should be compilable separately from LA as well as being a part of LA #ifndef _QUATERNION_H_ #define _QUATERNION_H_ #ifndef AVOID_STDSTREAM #include #endif #include #include #include #include #ifndef AVOID_LA #include "la.h" #endif #ifndef AVOID_LA namespace LA_Quaternion { template class Quaternion; //forward declaration } template class LA::LA_traits > { public: static bool is_plaindata() {return true;}; static void copyonwrite(LA_Quaternion::Quaternion& x) {}; typedef T normtype; }; #endif namespace LA_Quaternion { template class Quaternion { public: //just plain old data T q[4]; //methods Quaternion(void) {}; Quaternion(const T (&a)[4]) {memcpy(q,a,4*sizeof(T));}; Quaternion(const T x, const T u=0, const T v=0, const T w=0) {q[0]=x; q[1]=u; q[2]=v; q[3]=w;}; //quaternion from real(s) Quaternion(const std::complex &rhs) {q[0]=rhs.real(); q[1]=rhs.imag(); q[2]=0; q[3]=0;} //quaternion from complex explicit Quaternion(const T* x, const int shift=1) {q[0]=0; memcpy(q+shift,x,(4-shift)*sizeof(T));} //for shift=1 quaternion from xyz vector //compiler generates default copy constructor and assignment operator void identity() {q[0]=(T)1; q[1]=q[2]=q[3]=0;}; //formal indexing inline const T operator[](const int i) const {return q[i];}; inline T& operator[](const int i) {return q[i];}; //get pointer to data transparently inline operator const T*() const {return q;}; inline operator T*() {return q;}; //operations of quaternions with scalars void clear() {memset(q,0,4*sizeof(T));} Quaternion& operator=(const T x) {q[0]=x; memset(&q[1],0,3*sizeof(T)); return *this;}; //quaternion from real Quaternion& operator+=(const T rhs) {q[0]+=rhs; return *this;}; Quaternion& operator-=(const T rhs) {q[0]-=rhs; return *this;}; Quaternion& operator*=(const T rhs) {q[0]*=rhs; q[1]*=rhs; q[2]*=rhs; q[3]*=rhs; return *this;}; Quaternion& operator/=(const T rhs) {return *this *= ((T)1/rhs);}; const Quaternion operator+(const T rhs) const {return Quaternion(*this) += rhs;}; const Quaternion operator-(const T rhs) const {return Quaternion(*this) -= rhs;}; const Quaternion operator*(const T rhs) const {return Quaternion(*this) *= rhs;}; const Quaternion operator/(const T rhs) const {return Quaternion(*this) /= rhs;}; //quaternion algebra const Quaternion operator-() const {Quaternion r(*this); r.q[0]= -r.q[0]; r.q[1]= -r.q[1]; r.q[2]= -r.q[2]; r.q[3]= -r.q[3]; return r;}; //unary minus Quaternion& operator+=(const Quaternion &rhs) {q[0]+=rhs.q[0];q[1]+=rhs.q[1];q[2]+=rhs.q[2];q[3]+=rhs.q[3]; return *this;}; Quaternion& operator-=(const Quaternion &rhs) {q[0]-=rhs.q[0];q[1]-=rhs.q[1];q[2]-=rhs.q[2];q[3]-=rhs.q[3]; return *this;}; Quaternion operator+(const Quaternion &rhs) const {return Quaternion(*this) += rhs;}; Quaternion operator-(const Quaternion &rhs) const {return Quaternion(*this) -= rhs;}; Quaternion operator*(const Quaternion &rhs) const; //regular product Quaternion times_vec3(const T *rhs) const; //save flops for quaternions representing vectors Quaternion vec3_times(const T *rhs) const; //save flops for quaternions representing vectors Quaternion& conjugateme(void) {q[1] = -q[1]; q[2] = -q[2]; q[3] = -q[3]; return *this;} Quaternion conjugate(void) const {return Quaternion(*this).conjugateme();} T dot(const Quaternion &rhs) const {return q[0]*rhs.q[0] + q[1]*rhs.q[1] + q[2]*rhs.q[2] + q[3]*rhs.q[3];}; T normsqr(void) const {return dot(*this);}; T norm(void) const {return sqrt(normsqr());}; Quaternion& fast_normalize(bool unique_sign=false); //using quick 1/sqrt for floats Quaternion& normalize(T *getnorm=NULL, bool unique_sign=false); Quaternion inverse(void) const {return Quaternion(*this).conjugateme()/normsqr();}; const Quaternion operator/(const Quaternion &rhs) const {return *this * rhs.inverse();}; Quaternion rotateby(const Quaternion &rhs); //conjugation-rotation of this by NORMALIZED rhs void rotate(T *to, const T *from, Quaternion *grad=NULL) const; //rotate xyz vector by NORMALIZED *this Quaternion rotate_match(T *to, const T *from, const T *match) const; //gradient of quaternion rotation which should "to" = "from" transformed by *this to match a given vector by gradient descent Quaternion commutator(const Quaternion &rhs) const {return *this * rhs - rhs * *this;}; //could be made more efficient Quaternion anticommutator(const Quaternion &rhs) const {return *this * rhs + rhs * *this;}; //could be made more efficient T geodesic_distance(const Quaternion &rhs) const {T t=dot(rhs); return acos(2*t*t-1);}; //length of great arc between two quaternions on the S3 hypersphere //some conversions (for all 12 cases of euler angles go via rotation matrices), cf. also the 1977 NASA paper void normquat2eulerzyx(T *eul) const; //corresponds to [meul -r -T xyz -d -t -R] or euler2rotmat(...,"xyz",true,true,true) inline void eulerzyx2normquat(const T *eul) {euler2normquat(eul,"zyx");}; void normquat2euler(T *eul, const char *type) const; void euler2normquat(const T *eul, const char *type); void axis2normquat(const T *axis, const T &angle); void normquat2axis(T *axis, T &angle) const; void random_rotation(); //generate uniformly random unit quaternion //C-style IO int fprintf(FILE *f, const char *format) const {return ::fprintf(f,format,q[0],q[1],q[2],q[3]);}; int sprintf(char *f, const char *format) const {return ::sprintf(f,format,q[0],q[1],q[2],q[3]);}; int fscanf(FILE *f, const char *format) const {return ::fscanf(f,format,q[0],q[1],q[2],q[3]);}; int sscanf(char *f, const char *format) const {return ::sscanf(f,format,q[0],q[1],q[2],q[3]);}; }; //stream I/O ... cannot be moved to .cc, since we do e.g. Quaternion>> #ifndef AVOID_STDSTREAM template std::istream& operator>>(std::istream &s, Quaternion &x) { s >> x.q[0]; s >> x.q[1]; s >> x.q[2]; s >> x.q[3]; return s; } template std::ostream& operator<<(std::ostream &s, const Quaternion &x) { s << x.q[0]<<" "; s << x.q[1]<<" "; s << x.q[2]<<" "; s << x.q[3]; return s; } #endif //the following must be in .h due to the generic M type which is unspecified and can be any type providing [][], either plain C matrix, Mat3 class, or std::matrix or LA matrix NRMat //maybe we go via T* and recast it to T (*)[3] and move this to .cc to avoid replication of the code in multiple object files? //conversion from normalized quaternion to SU(2) matrix (+/- q yields different SU(2) element) template void normquat2su2mat(const Quaternion &q, M &a) { a[0][0] = std::complex(q[0],q[1]); a[0][1] = std::complex(q[2],q[3]); a[1][0] = std::complex(-q[2],q[3]); a[1][1] = std::complex(q[0],-q[1]); } //use transpose option to match nasa paper definition //conversion between quanternions and rotation matrices (+/- q yields the same rotation) // template void normquat2rotmat(const Quaternion &q, M &a, bool transpose=false) { //some explicit common subexpression optimizations { T q00= q[0]*q[0]; T q11= q[1]*q[1]; T q22= q[2]*q[2]; T q33= q[3]*q[3]; a[0][0] = q00+q11-q22-q33; a[1][1] = q00+q22-q11-q33; a[2][2] = q00+q33-q11-q22; } 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]; if(transpose) //effectively sign change of the temporaries { q01 = -q01; q02 = -q03; q03 = -q02; } a[0][1] = 2*(q12+q03); a[1][0] = 2*(q12-q03); a[0][2] = 2*(q13-q02); a[2][0] = 2*(q13+q02); a[1][2] = 2*(q23+q01); a[2][1] = 2*(q23-q01); } //transpose option to match nasa template void quat2rotmat(Quaternion q, M &a, bool transpose=false, const bool already_normalized=false) { if(!already_normalized) q.normalize(); normquat2rotmat(q,a,transpose); } //use transpose option to match nasa //derivative of the rotation matrix by quaternion elements template void normquat2rotmatder(const Quaternion &q, Quaternion &a, bool transpose=false) { //some explicit common subexpression optimizations T q0= q[0]+q[0]; T q1= q[1]+q[1]; T q2= q[2]+q[2]; T q3= q[3]+q[3]; a[0][0][0]= q0; a[0][0][1]= q3; a[0][0][2]= -q2; a[0][1][0]= -q3; a[0][1][1]= q0; a[0][1][2]= q1; a[0][2][0]= q2; a[0][2][1]= -q1; a[0][2][2]= q0; a[1][0][0]= q1; a[1][0][1]= q2; a[1][0][2]= q3; a[1][1][0]= q2; a[1][1][1]= -q1; a[1][1][2]= q0; a[1][2][0]= q3; a[1][2][1]= -q0; a[1][2][2]= -q1; a[2][0][0]= -q2; a[2][0][1]= q1; a[2][0][2]= -q0; a[2][1][0]= q1; a[2][1][1]= q2; a[2][1][2]= q3; a[2][2][0]= q0; a[2][2][1]= q3; a[2][2][2]= -q2; a[3][0][0]= -q3; a[3][0][1]= q0; a[3][0][2]= q1; a[3][1][0]= -q0; a[3][1][1]= -q3; a[3][1][2]= q2; a[3][2][0]= q1; a[3][2][1]= q2; a[3][2][2]= q3; if(transpose) { a[0].transposeme(); a[1].transposeme(); a[2].transposeme(); a[3].transposeme(); } } //normalized quaternion from rotation matrix //convention compatible with the paper on MEMS sensors by Sebastian O.H. Madgwick //the rotation matrix correcponds to transpose of (4) in Sarabandi and Thomas paper or the NASA paper //where the method is described template void rotmat2normquat(const M &a, Quaternion &q, bool transpose=false) { T tr= a[0][0]+a[1][1]+a[2][2]; T a12m = transpose? a[1][0]-a[0][1] : a[0][1]-a[1][0]; T a13m = transpose? a[2][0]-a[0][2] : a[0][2]-a[2][0]; T a23m = transpose? a[2][1]-a[1][2] : a[1][2]-a[2][1]; if(tr>=0) { q[0] = (T).5*sqrt((T)1. +tr); q[1] = (T).5*sqrt((T)1. +a[0][0]-a[1][1]-a[2][2]); q[2] = (T).5*sqrt((T)1. -a[0][0]+a[1][1]-a[2][2]); q[3] = (T).5*sqrt((T)1. -a[0][0]-a[1][1]+a[2][2]); } else { T a12p = a[0][1]+a[1][0]; T a13p = a[0][2]+a[2][0]; T a23p = a[1][2]+a[2][1]; q[0] = (T).5*sqrt((a23m*a23m+a13m*a13m+a12m*a12m)/((T)3.-tr)); q[1] = (T).5*sqrt((a23m*a23m+a12p*a12p+a13p*a13p)/((T)3.-a[0][0]+a[1][1]+a[2][2])); q[2] = (T).5*sqrt((a13m*a13m+a12p*a12p+a23p*a23p)/((T)3.+a[0][0]-a[1][1]+a[2][2])); q[3] = (T).5*sqrt((a12m*a12m+a13p*a13p+a23p*a23p)/((T)3.+a[0][0]+a[1][1]-a[2][2])); } if(a23m<0) q[1] = -q[1]; if(a13m>0) q[2] = -q[2]; if(a12m<0) q[3] = -q[3]; } //Quaternion Functions - cf. https://en.wikipedia.org/wiki/Quaternion template Quaternion exp(const Quaternion &x); //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 Quaternion log(const Quaternion &x); template Quaternion pow(const Quaternion &x, const T &y); } //namespace #endif /* _QUATERNION_H_ */