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690 lines
17 KiB
690 lines
17 KiB
/////////////////////////////////////////////////////////////////////////// |
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// |
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// Copyright (c) 2002, Industrial Light & Magic, a division of Lucas |
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// Digital Ltd. LLC |
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// |
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// All rights reserved. |
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// |
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// Redistribution and use in source and binary forms, with or without |
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// modification, are permitted provided that the following conditions are |
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// met: |
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// * Redistributions of source code must retain the above copyright |
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// notice, this list of conditions and the following disclaimer. |
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// * Redistributions in binary form must reproduce the above |
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// copyright notice, this list of conditions and the following disclaimer |
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// in the documentation and/or other materials provided with the |
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// distribution. |
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// * Neither the name of Industrial Light & Magic nor the names of |
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// its contributors may be used to endorse or promote products derived |
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// from this software without specific prior written permission. |
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// |
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// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS |
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// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT |
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// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR |
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// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT |
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// OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, |
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// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT |
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// LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, |
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// DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY |
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// THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT |
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// (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE |
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// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. |
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// |
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/////////////////////////////////////////////////////////////////////////// |
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#ifndef INCLUDED_IMATHQUAT_H |
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#define INCLUDED_IMATHQUAT_H |
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//---------------------------------------------------------------------- |
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// |
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// template class Quat<T> |
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// |
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// "Quaternions came from Hamilton ... and have been an unmixed |
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// evil to those who have touched them in any way. Vector is a |
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// useless survival ... and has never been of the slightest use |
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// to any creature." |
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// |
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// - Lord Kelvin |
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// |
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// This class implements the quaternion numerical type -- you |
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// will probably want to use this class to represent orientations |
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// in R3 and to convert between various euler angle reps. You |
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// should probably use Imath::Euler<> for that. |
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// |
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//---------------------------------------------------------------------- |
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#include "ImathExc.h" |
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#include "ImathMatrix.h" |
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#include <iostream> |
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namespace Imath { |
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#if (defined _WIN32 || defined _WIN64) && defined _MSC_VER |
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// Disable MS VC++ warnings about conversion from double to float |
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#pragma warning(disable:4244) |
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#endif |
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template <class T> |
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class Quat; |
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template<class T> |
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Quat<T> slerp (const Quat<T> &q1,const Quat<T> &q2, T t); |
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template<class T> |
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Quat<T> squad (const Quat<T> &q1,const Quat<T> &q2, |
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const Quat<T> &qa,const Quat<T> &qb, T t); |
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template<class T> |
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void intermediate (const Quat<T> &q0, const Quat<T> &q1, |
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const Quat<T> &q2, const Quat<T> &q3, |
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Quat<T> &qa, Quat<T> &qb); |
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template <class T> |
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class Quat |
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{ |
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public: |
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T r; // real part |
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Vec3<T> v; // imaginary vector |
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//----------------------------------------------------- |
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// Constructors - default constructor is identity quat |
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//----------------------------------------------------- |
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Quat() : r(1), v(0,0,0) {} |
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template <class S> |
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Quat( const Quat<S>& q) : r(q.r), v(q.v) {} |
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Quat( T s, T i, T j, T k ) : r(s), v(i,j,k) {} |
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Quat( T s, Vec3<T> d ) : r(s), v(d) {} |
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static Quat<T> identity() { return Quat<T>(); } |
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//------------------------------------------------ |
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// Basic Algebra - Operators and Methods |
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// The operator return values are *NOT* normalized |
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// |
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// operator^ is 4D dot product |
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// operator/ uses the inverse() quaternion |
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// operator~ is conjugate -- if (S+V) is quat then |
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// the conjugate (S+V)* == (S-V) |
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// |
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// some operators (*,/,*=,/=) treat the quat as |
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// a 4D vector when one of the operands is scalar |
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//------------------------------------------------ |
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const Quat<T>& operator= (const Quat<T>&); |
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const Quat<T>& operator*= (const Quat<T>&); |
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const Quat<T>& operator*= (T); |
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const Quat<T>& operator/= (const Quat<T>&); |
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const Quat<T>& operator/= (T); |
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const Quat<T>& operator+= (const Quat<T>&); |
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const Quat<T>& operator-= (const Quat<T>&); |
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T& operator[] (int index); // as 4D vector |
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T operator[] (int index) const; |
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template <class S> bool operator == (const Quat<S> &q) const; |
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template <class S> bool operator != (const Quat<S> &q) const; |
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Quat<T>& invert(); // this -> 1 / this |
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Quat<T> inverse() const; |
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Quat<T>& normalize(); // returns this |
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Quat<T> normalized() const; |
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T length() const; // in R4 |
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//----------------------- |
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// Rotation conversion |
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//----------------------- |
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Quat<T>& setAxisAngle(const Vec3<T>& axis, T radians); |
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Quat<T>& setRotation(const Vec3<T>& fromDirection, |
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const Vec3<T>& toDirection); |
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T angle() const; |
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Vec3<T> axis() const; |
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Matrix33<T> toMatrix33() const; |
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Matrix44<T> toMatrix44() const; |
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Quat<T> log() const; |
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Quat<T> exp() const; |
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}; |
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//-------------------- |
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// Convenient typedefs |
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//-------------------- |
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typedef Quat<float> Quatf; |
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typedef Quat<double> Quatd; |
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//--------------- |
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// Implementation |
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//--------------- |
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template<class T> |
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inline const Quat<T>& Quat<T>::operator= (const Quat<T>& q) |
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{ |
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r = q.r; |
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v = q.v; |
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return *this; |
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} |
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template<class T> |
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inline const Quat<T>& Quat<T>::operator*= (const Quat<T>& q) |
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{ |
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T rtmp = r * q.r - (v ^ q.v); |
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v = r * q.v + v * q.r + v % q.v; |
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r = rtmp; |
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return *this; |
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} |
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template<class T> |
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inline const Quat<T>& Quat<T>::operator*= (T t) |
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{ |
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r *= t; |
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v *= t; |
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return *this; |
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} |
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template<class T> |
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inline const Quat<T>& Quat<T>::operator/= (const Quat<T>& q) |
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{ |
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*this = *this * q.inverse(); |
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return *this; |
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} |
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template<class T> |
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inline const Quat<T>& Quat<T>::operator/= (T t) |
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{ |
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r /= t; |
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v /= t; |
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return *this; |
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} |
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template<class T> |
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inline const Quat<T>& Quat<T>::operator+= (const Quat<T>& q) |
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{ |
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r += q.r; |
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v += q.v; |
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return *this; |
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} |
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template<class T> |
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inline const Quat<T>& Quat<T>::operator-= (const Quat<T>& q) |
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{ |
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r -= q.r; |
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v -= q.v; |
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return *this; |
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} |
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template<class T> |
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inline T& Quat<T>::operator[] (int index) |
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{ |
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return index ? v[index-1] : r; |
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} |
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template<class T> |
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inline T Quat<T>::operator[] (int index) const |
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{ |
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return index ? v[index-1] : r; |
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} |
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template <class T> |
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template <class S> |
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inline bool |
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Quat<T>::operator == (const Quat<S> &q) const |
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{ |
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return r == q.r && v == q.v; |
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} |
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template <class T> |
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template <class S> |
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inline bool |
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Quat<T>::operator != (const Quat<S> &q) const |
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{ |
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return r != q.r || v != q.v; |
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} |
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template<class T> |
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inline T operator^ (const Quat<T>& q1,const Quat<T>& q2) |
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{ |
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return q1.r * q2.r + (q1.v ^ q2.v); |
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} |
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template <class T> |
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inline T Quat<T>::length() const |
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{ |
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return Math<T>::sqrt( r * r + (v ^ v) ); |
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} |
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template <class T> |
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inline Quat<T>& Quat<T>::normalize() |
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{ |
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if ( T l = length() ) { r /= l; v /= l; } |
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else { r = 1; v = Vec3<T>(0); } |
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return *this; |
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} |
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template <class T> |
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inline Quat<T> Quat<T>::normalized() const |
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{ |
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if ( T l = length() ) return Quat( r / l, v / l ); |
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return Quat(); |
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} |
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template<class T> |
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inline Quat<T> Quat<T>::inverse() const |
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{ |
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// 1 Q* |
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// - = ---- where Q* is conjugate (operator~) |
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// Q Q* Q and (Q* Q) == Q ^ Q (4D dot) |
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T qdot = *this ^ *this; |
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return Quat( r / qdot, -v / qdot ); |
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} |
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template<class T> |
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inline Quat<T>& Quat<T>::invert() |
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{ |
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T qdot = (*this) ^ (*this); |
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r /= qdot; |
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v = -v / qdot; |
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return *this; |
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} |
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template<class T> |
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Quat<T> |
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slerp(const Quat<T> &q1,const Quat<T> &q2, T t) |
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{ |
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// |
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// Spherical linear interpolation. |
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// |
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// NOTE: Assumes q1 and q2 are normalized and that 0 <= t <= 1. |
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// |
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// This method does *not* interpolate along the shortest arc |
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// between q1 and q2. If you desire interpolation along the |
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// shortest arc, then consider flipping the second quaternion |
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// explicitly before calling slerp. The implementation of squad() |
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// depends on a slerp() that interpolates as is, without the |
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// automatic flipping. |
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// |
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T cosomega = q1 ^ q2; |
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if (cosomega >= (T) 1.0) |
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{ |
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// |
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// Special case: q1 and q2 are the same, so just return one of them. |
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// This also catches the case where cosomega is very slightly > 1.0 |
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// |
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return q1; |
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} |
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T sinomega = Math<T>::sqrt (1 - cosomega * cosomega); |
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Quat<T> result; |
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if (sinomega * limits<T>::max() > 1) |
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{ |
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T omega = Math<T>::acos (cosomega); |
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T s1 = Math<T>::sin ((1.0 - t) * omega) / sinomega; |
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T s2 = Math<T>::sin (t * omega) / sinomega; |
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result = s1 * q1 + s2 * q2; |
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} |
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else if (cosomega > 0) |
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{ |
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// |
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// omega == 0 |
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// |
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T s1 = 1.0 - t; |
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T s2 = t; |
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result = s1 * q1 + s2 * q2; |
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} |
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else |
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{ |
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// |
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// omega == -pi |
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// |
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result.v.x = - q1.v.y; |
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result.v.y = q1.v.x; |
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result.v.z = - q1.r; |
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result.r = q1.v.z; |
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T s1 = Math<T>::sin ((0.5 - t) * M_PI); |
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T s2 = Math<T>::sin (t * M_PI); |
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result = s1 * q1 + s2 * result; |
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} |
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return result; |
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} |
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template<class T> |
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Quat<T> spline(const Quat<T> &q0, const Quat<T> &q1, |
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const Quat<T> &q2, const Quat<T> &q3, |
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T t) |
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{ |
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// Spherical Cubic Spline Interpolation - |
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// from Advanced Animation and Rendering |
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// Techniques by Watt and Watt, Page 366: |
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// A spherical curve is constructed using three |
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// spherical linear interpolations of a quadrangle |
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// of unit quaternions: q1, qa, qb, q2. |
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// Given a set of quaternion keys: q0, q1, q2, q3, |
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// this routine does the interpolation between |
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// q1 and q2 by constructing two intermediate |
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// quaternions: qa and qb. The qa and qb are |
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// computed by the intermediate function to |
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// guarantee the continuity of tangents across |
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// adjacent cubic segments. The qa represents in-tangent |
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// for q1 and the qb represents the out-tangent for q2. |
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// |
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// The q1 q2 is the cubic segment being interpolated. |
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// The q0 is from the previous adjacent segment and q3 is |
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// from the next adjacent segment. The q0 and q3 are used |
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// in computing qa and qb. |
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// |
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Quat<T> qa = intermediate (q0, q1, q2); |
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Quat<T> qb = intermediate (q1, q2, q3); |
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Quat<T> result = squad(q1, qa, qb, q2, t); |
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return result; |
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} |
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template<class T> |
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Quat<T> squad(const Quat<T> &q1, const Quat<T> &qa, |
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const Quat<T> &qb, const Quat<T> &q2, |
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T t) |
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{ |
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// Spherical Quadrangle Interpolation - |
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// from Advanced Animation and Rendering |
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// Techniques by Watt and Watt, Page 366: |
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// It constructs a spherical cubic interpolation as |
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// a series of three spherical linear interpolations |
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// of a quadrangle of unit quaternions. |
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// |
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Quat<T> r1 = slerp(q1, q2, t); |
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Quat<T> r2 = slerp(qa, qb, t); |
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Quat<T> result = slerp(r1, r2, 2*t*(1-t)); |
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return result; |
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} |
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template<class T> |
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Quat<T> intermediate(const Quat<T> &q0, const Quat<T> &q1, const Quat<T> &q2) |
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{ |
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// From advanced Animation and Rendering |
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// Techniques by Watt and Watt, Page 366: |
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// computing the inner quadrangle |
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// points (qa and qb) to guarantee tangent |
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// continuity. |
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// |
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Quat<T> q1inv = q1.inverse(); |
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Quat<T> c1 = q1inv*q2; |
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Quat<T> c2 = q1inv*q0; |
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Quat<T> c3 = (T) (-0.25) * (c2.log() + c1.log()); |
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Quat<T> qa = q1 * c3.exp(); |
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qa.normalize(); |
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return qa; |
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} |
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template <class T> |
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inline Quat<T> Quat<T>::log() const |
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{ |
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// |
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// For unit quaternion, from Advanced Animation and |
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// Rendering Techniques by Watt and Watt, Page 366: |
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// |
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T theta = Math<T>::acos (std::min (r, (T) 1.0)); |
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if (theta == 0) |
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return Quat<T> (0, v); |
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T sintheta = Math<T>::sin (theta); |
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T k; |
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if (abs (sintheta) < 1 && abs (theta) >= limits<T>::max() * abs (sintheta)) |
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k = 0; |
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else |
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k = theta / sintheta; |
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return Quat<T> ((T) 0, v.x * k, v.y * k, v.z * k); |
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} |
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template <class T> |
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inline Quat<T> Quat<T>::exp() const |
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{ |
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// |
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// For pure quaternion (zero scalar part): |
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// from Advanced Animation and Rendering |
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// Techniques by Watt and Watt, Page 366: |
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// |
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T theta = v.length(); |
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T sintheta = Math<T>::sin (theta); |
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T k; |
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if (abs (theta) < 1 && abs (sintheta) >= limits<T>::max() * abs (theta)) |
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k = 0; |
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else |
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k = sintheta / theta; |
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T costheta = Math<T>::cos (theta); |
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return Quat<T> (costheta, v.x * k, v.y * k, v.z * k); |
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} |
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template <class T> |
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inline T Quat<T>::angle() const |
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{ |
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return 2.0*Math<T>::acos(r); |
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} |
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template <class T> |
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inline Vec3<T> Quat<T>::axis() const |
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{ |
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return v.normalized(); |
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} |
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template <class T> |
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inline Quat<T>& Quat<T>::setAxisAngle(const Vec3<T>& axis, T radians) |
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{ |
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r = Math<T>::cos(radians/2); |
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v = axis.normalized() * Math<T>::sin(radians/2); |
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return *this; |
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} |
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template <class T> |
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Quat<T>& |
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Quat<T>::setRotation(const Vec3<T>& from, const Vec3<T>& to) |
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{ |
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// |
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// Ported from SbRotation |
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// |
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T cost = from.dot(to) / Math<T>::sqrt(from.dot(from) * to.dot(to)); |
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// check for degeneracies |
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if (cost > 0.99999) |
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{ |
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// |
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// Vectors are parallel. |
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// |
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r = 1.0; |
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v = Vec3<T>(0); |
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} |
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else if (cost < -0.99999) |
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{ |
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// |
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// Vectors are opposite. Find an axis to rotate around, |
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// which should be perpendicular to the original axis. |
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// |
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Vec3<T> frm = from.normalized(); |
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v = frm.cross(Vec3<T>(1, 0, 0)); |
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if (v.length() < 0.00001) |
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v = frm.cross(Vec3<T>(0, 1, 0)); |
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r = 0; |
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v.normalize(); |
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} |
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else |
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{ |
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// |
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// Use half-angle formulae: |
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// cos^2 t = ( 1 + cos (2t) ) / 2 |
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// w part is cosine of half the rotation angle |
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// |
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r = Math<T>::sqrt(0.5 * (1.0 + cost)); |
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// |
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// sin^2 t = ( 1 - cos (2t) ) / 2 |
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// Do the normalization of the axis vector at the same time so |
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// we only call sqrt once. |
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// |
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v = from.cross(to); |
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v *= Math<T>::sqrt((0.5 * (1.0 - cost))/(v.dot(v))); |
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} |
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return *this; |
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} |
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template<class T> |
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Matrix33<T> Quat<T>::toMatrix33() const |
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{ |
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return Matrix33<T>(1. - 2.0 * (v.y * v.y + v.z * v.z), |
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2.0 * (v.x * v.y + v.z * r), |
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2.0 * (v.z * v.x - v.y * r), |
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2.0 * (v.x * v.y - v.z * r), |
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1. - 2.0 * (v.z * v.z + v.x * v.x), |
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2.0 * (v.y * v.z + v.x * r), |
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2.0 * (v.z * v.x + v.y * r), |
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2.0 * (v.y * v.z - v.x * r), |
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1. - 2.0 * (v.y * v.y + v.x * v.x)); |
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} |
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template<class T> |
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Matrix44<T> Quat<T>::toMatrix44() const |
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{ |
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return Matrix44<T>(1. - 2.0 * (v.y * v.y + v.z * v.z), |
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2.0 * (v.x * v.y + v.z * r), |
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2.0 * (v.z * v.x - v.y * r), |
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0., |
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2.0 * (v.x * v.y - v.z * r), |
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1. - 2.0 * (v.z * v.z + v.x * v.x), |
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2.0 * (v.y * v.z + v.x * r), |
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0., |
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2.0 * (v.z * v.x + v.y * r), |
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2.0 * (v.y * v.z - v.x * r), |
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1. - 2.0 * (v.y * v.y + v.x * v.x), |
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0., |
|
0., |
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0., |
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0., |
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1.0 ); |
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} |
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|
|
|
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template<class T> |
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inline Matrix33<T> operator* (const Matrix33<T> &M, const Quat<T> &q) |
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{ |
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return M * q.toMatrix33(); |
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} |
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|
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template<class T> |
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inline Matrix33<T> operator* (const Quat<T> &q, const Matrix33<T> &M) |
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{ |
|
return q.toMatrix33() * M; |
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} |
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|
|
template<class T> |
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std::ostream& operator<< (std::ostream &o, const Quat<T> &q) |
|
{ |
|
return o << "(" << q.r |
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<< " " << q.v.x |
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<< " " << 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 ); |
|
} |
|
|
|
#if (defined _WIN32 || defined _WIN64) && defined _MSC_VER |
|
#pragma warning(default:4244) |
|
#endif |
|
|
|
} // namespace Imath |
|
|
|
#endif
|
|
|