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root/OpenMD/branches/development/src/math/Quaternion.hpp

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trunk/src/math/Quaternion.hpp (file contents), Revision 385 by tim, Tue Mar 1 20:10:14 2005 UTC vs.
branches/development/src/math/Quaternion.hpp (file contents), Revision 1465 by chuckv, Fri Jul 9 23:08:25 2010 UTC

#	Line 1 | Line 1
1	<	/*
1	>	/*
2		* Copyright (c) 2005 The University of Notre Dame. All Rights Reserved.
3		*
4		* The University of Notre Dame grants you ("Licensee") a
#	Line 6 | Line 6
6		* redistribute this software in source and binary code form, provided
7		* that the following conditions are met:
8		*
9	<	* 1. Acknowledgement of the program authors must be made in any
10	<	*    publication of scientific results based in part on use of the
11	<	*    program.  An acceptable form of acknowledgement is citation of
12	<	*    the article in which the program was described (Matthew
13	<	*    A. Meineke, Charles F. Vardeman II, Teng Lin, Christopher
14	<	*    J. Fennell and J. Daniel Gezelter, "OOPSE: An Object-Oriented
15	<	*    Parallel Simulation Engine for Molecular Dynamics,"
16	<	*    J. Comput. Chem. 26, pp. 252-271 (2005))
17	<	*
18	<	* 2. Redistributions of source code must retain the above copyright
9	>	* 1. Redistributions of source code must retain the above copyright
10		*    notice, this list of conditions and the following disclaimer.
11		*
12	<	* 3. Redistributions in binary form must reproduce the above copyright
12	>	* 2. Redistributions in binary form must reproduce the above copyright
13		*    notice, this list of conditions and the following disclaimer in the
14		*    documentation and/or other materials provided with the
15		*    distribution.
#	Line 37 | Line 28
28		* arising out of the use of or inability to use software, even if the
29		* University of Notre Dame has been advised of the possibility of
30		* such damages.
31	+	*
32	+	* SUPPORT OPEN SCIENCE!  If you use OpenMD or its source code in your
33	+	* research, please cite the appropriate papers when you publish your
34	+	* work.  Good starting points are:
35	+	*
36	+	* [1]  Meineke, et al., J. Comp. Chem. 26, 252-271 (2005).
37	+	* [2]  Fennell & Gezelter, J. Chem. Phys. 124, 234104 (2006).
38	+	* [3]  Sun, Lin & Gezelter, J. Chem. Phys. 128, 24107 (2008).
39	+	* [4]  Vardeman & Gezelter, in progress (2009).
40		*/
41
42		/**
#	Line 49 | Line 49
49		#ifndef MATH_QUATERNION_HPP
50		#define MATH_QUATERNION_HPP
51
52	<	#include "math/Vector.hpp"
52	>	#include "math/Vector3.hpp"
53		#include "math/SquareMatrix.hpp"
54	+	#define ISZERO(a,eps) ( (a)>-(eps) && (a)<(eps) )
55	+	const RealType tiny=1.0e-6;
56
57	<	namespace oopse{
57	>	namespace OpenMD{
58
59	<	/**
60	<	* @class Quaternion Quaternion.hpp "math/Quaternion.hpp"
61	<	* Quaternion is a sort of a higher-level complex number.
62	<	* It is defined as Q = w + x*i + y*j + z*k,
63	<	* where w, x, y, and z are numbers of type T (e.g. double), and
64	<	* i*i = -1; j*j = -1; k*k = -1;
65	<	* i*j = k; j*k = i; k*i = j;
66	<	*/
67	<	template<typename Real>
68	<	class Quaternion : public Vector<Real, 4> {
67	<	public:
68	<	Quaternion() : Vector<Real, 4>() {}
59	>	/**
60	>	* @class Quaternion Quaternion.hpp "math/Quaternion.hpp"
61	>	* Quaternion is a sort of a higher-level complex number.
62	>	* It is defined as Q = w + x*i + y*j + z*k,
63	>	* where w, x, y, and z are numbers of type T (e.g. RealType), and
64	>	* i*i = -1; j*j = -1; k*k = -1;
65	>	* i*j = k; j*k = i; k*i = j;
66	>	*/
67	>	template<typename Real>
68	>	class Quaternion : public Vector<Real, 4> {
69
70	<	/** Constructs and initializes a Quaternion from w, x, y, z values */
71	<	Quaternion(Real w, Real x, Real y, Real z) {
72	<	this->data_[0] = w;
73	<	this->data_[1] = x;
74	<	this->data_[2] = y;
75	<	this->data_[3] = z;
76	<	}
70	>	public:
71	>	Quaternion() : Vector<Real, 4>() {}
72	>
73	>	/** Constructs and initializes a Quaternion from w, x, y, z values */
74	>	Quaternion(Real w, Real x, Real y, Real z) {
75	>	this->data_[0] = w;
76	>	this->data_[1] = x;
77	>	this->data_[2] = y;
78	>	this->data_[3] = z;
79	>	}
80
81	<	/** Constructs and initializes a Quaternion from a  Vector<Real,4> */
82	<	Quaternion(const Vector<Real,4>& v)
83	<	: Vector<Real, 4>(v){
84	<	}
81	>	/** Constructs and initializes a Quaternion from a  Vector<Real,4> */
82	>	Quaternion(const Vector<Real,4>& v)
83	>	: Vector<Real, 4>(v){
84	>	}
85
86	<	/** copy assignment */
87	<	Quaternion& operator =(const Vector<Real, 4>& v){
88	<	if (this == & v)
89	<	return *this;
86	>	/** copy assignment */
87	>	Quaternion& operator =(const Vector<Real, 4>& v){
88	>	if (this == & v)
89	>	return *this;
90	>
91	>	Vector<Real, 4>::operator=(v);
92	>
93	>	return *this;
94	>	}
95	>
96	>	/**
97	>	* Returns the value of the first element of this quaternion.
98	>	* @return the value of the first element of this quaternion
99	>	*/
100	>	Real w() const {
101	>	return this->data_[0];
102	>	}
103
104	<	Vector<Real, 4>::operator=(v);
105	<
106	<	return *this;
107	<	}
104	>	/**
105	>	* Returns the reference of the first element of this quaternion.
106	>	* @return the reference of the first element of this quaternion
107	>	*/
108	>	Real& w() {
109	>	return this->data_[0];
110	>	}
111
112	<	/**
113	<	* Returns the value of the first element of this quaternion.
114	<	* @return the value of the first element of this quaternion
115	<	*/
116	<	Real w() const {
117	<	return this->data_[0];
118	<	}
112	>	/**
113	>	* Returns the value of the first element of this quaternion.
114	>	* @return the value of the first element of this quaternion
115	>	*/
116	>	Real x() const {
117	>	return this->data_[1];
118	>	}
119
120	<	/**
121	<	* Returns the reference of the first element of this quaternion.
122	<	* @return the reference of the first element of this quaternion
123	<	*/
124	<	Real& w() {
125	<	return this->data_[0];
126	<	}
120	>	/**
121	>	* Returns the reference of the second element of this quaternion.
122	>	* @return the reference of the second element of this quaternion
123	>	*/
124	>	Real& x() {
125	>	return this->data_[1];
126	>	}
127
128	<	/**
129	<	* Returns the value of the first element of this quaternion.
130	<	* @return the value of the first element of this quaternion
131	<	*/
132	<	Real x() const {
133	<	return this->data_[1];
134	<	}
128	>	/**
129	>	* Returns the value of the thirf element of this quaternion.
130	>	* @return the value of the third element of this quaternion
131	>	*/
132	>	Real y() const {
133	>	return this->data_[2];
134	>	}
135
136	<	/**
137	<	* Returns the reference of the second element of this quaternion.
138	<	* @return the reference of the second element of this quaternion
139	<	*/
140	<	Real& x() {
141	<	return this->data_[1];
142	<	}
136	>	/**
137	>	* Returns the reference of the third element of this quaternion.
138	>	* @return the reference of the third element of this quaternion
139	>	*/
140	>	Real& y() {
141	>	return this->data_[2];
142	>	}
143
144	<	/**
145	<	* Returns the value of the thirf element of this quaternion.
146	<	* @return the value of the third element of this quaternion
147	<	*/
148	<	Real y() const {
149	<	return this->data_[2];
150	<	}
144	>	/**
145	>	* Returns the value of the fourth element of this quaternion.
146	>	* @return the value of the fourth element of this quaternion
147	>	*/
148	>	Real z() const {
149	>	return this->data_[3];
150	>	}
151	>	/**
152	>	* Returns the reference of the fourth element of this quaternion.
153	>	* @return the reference of the fourth element of this quaternion
154	>	*/
155	>	Real& z() {
156	>	return this->data_[3];
157	>	}
158
159	<	/**
160	<	* Returns the reference of the third element of this quaternion.
161	<	* @return the reference of the third element of this quaternion
162	<	*/
163	<	Real& y() {
164	<	return this->data_[2];
139	<	}
159	>	/**
160	>	* Tests if this quaternion is equal to other quaternion
161	>	* @return true if equal, otherwise return false
162	>	* @param q quaternion to be compared
163	>	*/
164	>	inline bool operator ==(const Quaternion<Real>& q) {
165
166	<	/**
167	<	* Returns the value of the fourth element of this quaternion.
168	<	* @return the value of the fourth element of this quaternion
169	<	*/
170	<	Real z() const {
146	<	return this->data_[3];
147	<	}
148	<	/**
149	<	* Returns the reference of the fourth element of this quaternion.
150	<	* @return the reference of the fourth element of this quaternion
151	<	*/
152	<	Real& z() {
153	<	return this->data_[3];
154	<	}
155	<
156	<	/**
157	<	* Tests if this quaternion is equal to other quaternion
158	<	* @return true if equal, otherwise return false
159	<	* @param q quaternion to be compared
160	<	*/
161	<	inline bool operator ==(const Quaternion<Real>& q) {
162	<
163	<	for (unsigned int i = 0; i < 4; i ++) {
164	<	if (!equal(this->data_[i], q[i])) {
165	<	return false;
166	<	}
167	<	}
166	>	for (unsigned int i = 0; i < 4; i ++) {
167	>	if (!equal(this->data_[i], q[i])) {
168	>	return false;
169	>	}
170	>	}
171
172	<	return true;
173	<	}
172	>	return true;
173	>	}
174
175	<	/**
176	<	* Returns the inverse of this quaternion
177	<	* @return inverse
178	<	* @note since quaternion is a complex number, the inverse of quaternion
179	<	* q = w + xi + yj+ zk is inv_q = (w -xi - yj - zk)/(|q|^2)
180	<	*/
181	<	Quaternion<Real> inverse() {
182	<	Quaternion<Real> q;
183	<	Real d = this->lengthSquare();
175	>	/**
176	>	* Returns the inverse of this quaternion
177	>	* @return inverse
178	>	* @note since quaternion is a complex number, the inverse of quaternion
179	>	* q = w + xi + yj+ zk is inv_q = (w -xi - yj - zk)/(|q|^2)
180	>	*/
181	>	Quaternion<Real> inverse() {
182	>	Quaternion<Real> q;
183	>	Real d = this->lengthSquare();
184
185	<	q.w() = w() / d;
186	<	q.x() = -x() / d;
187	<	q.y() = -y() / d;
188	<	q.z() = -z() / d;
185	>	q.w() = w() / d;
186	>	q.x() = -x() / d;
187	>	q.y() = -y() / d;
188	>	q.z() = -z() / d;
189
190	<	return q;
191	<	}
190	>	return q;
191	>	}
192
193	<	/**
194	<	* Sets the value to the multiplication of itself and another quaternion
195	<	* @param q the other quaternion
196	<	*/
197	<	void mul(const Quaternion<Real>& q) {
198	<	Quaternion<Real> tmp(*this);
193	>	/**
194	>	* Sets the value to the multiplication of itself and another quaternion
195	>	* @param q the other quaternion
196	>	*/
197	>	void mul(const Quaternion<Real>& q) {
198	>	Quaternion<Real> tmp(*this);
199
200	<	this->data_[0] = (tmp[0]*q[0]) -(tmp[1]*q[1]) - (tmp[2]*q[2]) - (tmp[3]*q[3]);
201	<	this->data_[1] = (tmp[0]*q[1]) + (tmp[1]*q[0]) + (tmp[2]*q[3]) - (tmp[3]*q[2]);
202	<	this->data_[2] = (tmp[0]*q[2]) + (tmp[2]*q[0]) + (tmp[3]*q[1]) - (tmp[1]*q[3]);
203	<	this->data_[3] = (tmp[0]*q[3]) + (tmp[3]*q[0]) + (tmp[1]*q[2]) - (tmp[2]*q[1]);
204	<	}
200	>	this->data_[0] = (tmp[0]*q[0]) -(tmp[1]*q[1]) - (tmp[2]*q[2]) - (tmp[3]*q[3]);
201	>	this->data_[1] = (tmp[0]*q[1]) + (tmp[1]*q[0]) + (tmp[2]*q[3]) - (tmp[3]*q[2]);
202	>	this->data_[2] = (tmp[0]*q[2]) + (tmp[2]*q[0]) + (tmp[3]*q[1]) - (tmp[1]*q[3]);
203	>	this->data_[3] = (tmp[0]*q[3]) + (tmp[3]*q[0]) + (tmp[1]*q[2]) - (tmp[2]*q[1]);
204	>	}
205
206	<	void mul(const Real& s) {
207	<	this->data_[0] *= s;
208	<	this->data_[1] *= s;
209	<	this->data_[2] *= s;
210	<	this->data_[3] *= s;
211	<	}
206	>	void mul(const Real& s) {
207	>	this->data_[0] *= s;
208	>	this->data_[1] *= s;
209	>	this->data_[2] *= s;
210	>	this->data_[3] *= s;
211	>	}
212
213	<	/** Set the value of this quaternion to the division of itself by another quaternion */
214	<	void div(Quaternion<Real>& q) {
215	<	mul(q.inverse());
216	<	}
213	>	/** Set the value of this quaternion to the division of itself by another quaternion */
214	>	void div(Quaternion<Real>& q) {
215	>	mul(q.inverse());
216	>	}
217
218	<	void div(const Real& s) {
219	<	this->data_[0] /= s;
220	<	this->data_[1] /= s;
221	<	this->data_[2] /= s;
222	<	this->data_[3] /= s;
223	<	}
218	>	void div(const Real& s) {
219	>	this->data_[0] /= s;
220	>	this->data_[1] /= s;
221	>	this->data_[2] /= s;
222	>	this->data_[3] /= s;
223	>	}
224
225	<	Quaternion<Real>& operator *=(const Quaternion<Real>& q) {
226	<	mul(q);
227	<	return *this;
228	<	}
225	>	Quaternion<Real>& operator *=(const Quaternion<Real>& q) {
226	>	mul(q);
227	>	return *this;
228	>	}
229
230	<	Quaternion<Real>& operator *=(const Real& s) {
231	<	mul(s);
232	<	return *this;
233	<	}
230	>	Quaternion<Real>& operator *=(const Real& s) {
231	>	mul(s);
232	>	return *this;
233	>	}
234
235	<	Quaternion<Real>& operator /=(Quaternion<Real>& q) {
236	<	*this *= q.inverse();
237	<	return *this;
238	<	}
235	>	Quaternion<Real>& operator /=(Quaternion<Real>& q) {
236	>	*this *= q.inverse();
237	>	return *this;
238	>	}
239
240	<	Quaternion<Real>& operator /=(const Real& s) {
241	<	div(s);
242	<	return *this;
243	<	}
244	<	/**
245	<	* Returns the conjugate quaternion of this quaternion
246	<	* @return the conjugate quaternion of this quaternion
247	<	*/
248	<	Quaternion<Real> conjugate() {
249	<	return Quaternion<Real>(w(), -x(), -y(), -z());
250	<	}
240	>	Quaternion<Real>& operator /=(const Real& s) {
241	>	div(s);
242	>	return *this;
243	>	}
244	>	/**
245	>	* Returns the conjugate quaternion of this quaternion
246	>	* @return the conjugate quaternion of this quaternion
247	>	*/
248	>	Quaternion<Real> conjugate() const {
249	>	return Quaternion<Real>(w(), -x(), -y(), -z());
250	>	}
251
249	–	/**
250	–	* Returns the corresponding rotation matrix (3x3)
251	–	* @return a 3x3 rotation matrix
252	–	*/
253	–	SquareMatrix<Real, 3> toRotationMatrix3() {
254	–	SquareMatrix<Real, 3> rotMat3;
252
253	<	Real w2;
254	<	Real x2;
255	<	Real y2;
256	<	Real z2;
253	>	/**
254	>	return rotation angle from -PI to PI
255	>	*/
256	>	inline Real get_rotation_angle() const{
257	>	if( w < (Real)0.0 )
258	>	return 2.0*atan2(-sqrt( x()*x() + y()*y() + z()*z() ), -w() );
259	>	else
260	>	return 2.0*atan2( sqrt( x()*x() + y()*y() + z()*z() ),  w() );
261	>	}
262
263	<	if (!this->isNormalized())
264	<	this->normalize();
265	<
266	<	w2 = w() * w();
267	<	x2 = x() * x();
268	<	y2 = y() * y();
269	<	z2 = z() * z();
270	<
271	<	rotMat3(0, 0) = w2 + x2 - y2 - z2;
272	<	rotMat3(0, 1) = 2.0 * ( x() * y() + w() * z() );
273	<	rotMat3(0, 2) = 2.0 * ( x() * z() - w() * y() );
274	<
275	<	rotMat3(1, 0) = 2.0 * ( x() * y() - w() * z() );
276	<	rotMat3(1, 1) = w2 - x2 + y2 - z2;
277	<	rotMat3(1, 2) = 2.0 * ( y() * z() + w() * x() );
278	<
279	<	rotMat3(2, 0) = 2.0 * ( x() * z() + w() * y() );
280	<	rotMat3(2, 1) = 2.0 * ( y() * z() - w() * x() );
281	<	rotMat3(2, 2) = w2 - x2 -y2 +z2;
282	<
283	<	return rotMat3;
284	<	}
285	<
286	<	};//end Quaternion
263	>	/**
264	>	create a unit quaternion from axis angle representation
265	>	*/
266	>	Quaternion<Real> fromAxisAngle(const Vector3<Real>& axis,
267	>	const Real& angle){
268	>	Vector3<Real> v(axis);
269	>	v.normalize();
270	>	Real half_angle = angle*0.5;
271	>	Real sin_a = sin(half_angle);
272	>	*this = Quaternion<Real>(cos(half_angle),
273	>	v.x()*sin_a,
274	>	v.y()*sin_a,
275	>	v.z()*sin_a);
276	>	return *this;
277	>	}
278	>
279	>	/**
280	>	convert a quaternion to axis angle representation,
281	>	preserve the axis direction and angle from -PI to +PI
282	>	*/
283	>	void toAxisAngle(Vector3<Real>& axis, Real& angle)const {
284	>	Real vl = sqrt( x()*x() + y()*y() + z()*z() );
285	>	if( vl > tiny ) {
286	>	Real ivl = 1.0/vl;
287	>	axis.x() = x() * ivl;
288	>	axis.y() = y() * ivl;
289	>	axis.z() = z() * ivl;
290
291	+	if( w() < 0 )
292	+	angle = 2.0*atan2(-vl, -w()); //-PI,0
293	+	else
294	+	angle = 2.0*atan2( vl,  w()); //0,PI
295	+	} else {
296	+	axis = Vector3<Real>(0.0,0.0,0.0);
297	+	angle = 0.0;
298	+	}
299	+	}
300
301		/**
302	<	* Returns the vaule of scalar multiplication of this quaterion q (q * s).
303	<	* @return  the vaule of scalar multiplication of this vector
304	<	* @param q the source quaternion
305	<	* @param s the scalar value
306	<	*/
307	<	template<typename Real, unsigned int Dim>
308	<	Quaternion<Real> operator * ( const Quaternion<Real>& q, Real s) {
309	<	Quaternion<Real> result(q);
310	<	result.mul(s);
311	<	return result;
302	>	shortest arc quaternion rotate one vector to another by shortest path.
303	>	create rotation from -> to, for any length vectors.
304	>	*/
305	>	Quaternion<Real> fromShortestArc(const Vector3d& from,
306	>	const Vector3d& to ) {
307	>
308	>	Vector3d c( cross(from,to) );
309	>	*this = Quaternion<Real>(dot(from,to),
310	>	c.x(),
311	>	c.y(),
312	>	c.z());
313	>
314	>	this->normalize();    // if "from" or "to" not unit, normalize quat
315	>	w += 1.0f;            // reducing angle to halfangle
316	>	if( w <= 1e-6 ) {     // angle close to PI
317	>	if( ( from.z()*from.z() ) > ( from.x()*from.x() ) ) {
318	>	this->data_[0] =  w;
319	>	this->data_[1] =  0.0;       //cross(from , Vector3d(1,0,0))
320	>	this->data_[2] =  from.z();
321	>	this->data_[3] = -from.y();
322	>	} else {
323	>	this->data_[0] =  w;
324	>	this->data_[1] =  from.y();  //cross(from, Vector3d(0,0,1))
325	>	this->data_[2] = -from.x();
326	>	this->data_[3] =  0.0;
327	>	}
328	>	}
329	>	this->normalize();
330		}
299	–
300	–	/**
301	–	* Returns the vaule of scalar multiplication of this quaterion q (q * s).
302	–	* @return  the vaule of scalar multiplication of this vector
303	–	* @param s the scalar value
304	–	* @param q the source quaternion
305	–	*/
306	–	template<typename Real, unsigned int Dim>
307	–	Quaternion<Real> operator * ( const Real& s, const Quaternion<Real>& q ) {
308	–	Quaternion<Real> result(q);
309	–	result.mul(s);
310	–	return result;
311	–	}
331
332	<	/**
333	<	* Returns the multiplication of two quaternion
315	<	* @return the multiplication of two quaternion
316	<	* @param q1 the first quaternion
317	<	* @param q2 the second quaternion
318	<	*/
319	<	template<typename Real>
320	<	inline Quaternion<Real> operator *(const Quaternion<Real>& q1, const Quaternion<Real>& q2) {
321	<	Quaternion<Real> result(q1);
322	<	result *= q2;
323	<	return result;
332	>	Real ComputeTwist(const Quaternion& q) {
333	>	return  (Real)2.0 * atan2(q.z(), q.w());
334		}
335
336	+	void RemoveTwist(Quaternion& q) {
337	+	Real t = ComputeTwist(q);
338	+	Quaternion rt = fromAxisAngle(V3Z, t);
339	+
340	+	q *= rt.inverse();
341	+	}
342	+
343	+	void getTwistSwingAxisAngle(Real& twistAngle, Real& swingAngle,
344	+	Vector3<Real>& swingAxis) {
345	+
346	+	twistAngle = (Real)2.0 * atan2(z(), w());
347	+	Quaternion rt, rs;
348	+	rt.fromAxisAngle(V3Z, twistAngle);
349	+	rs = *this * rt.inverse();
350	+
351	+	Real vl = sqrt( rs.x()*rs.x() + rs.y()*rs.y() + rs.z()*rs.z() );
352	+	if( vl > tiny ) {
353	+	Real ivl = 1.0 / vl;
354	+	swingAxis.x() = rs.x() * ivl;
355	+	swingAxis.y() = rs.y() * ivl;
356	+	swingAxis.z() = rs.z() * ivl;
357	+
358	+	if( rs.w() < 0.0 )
359	+	swingAngle = 2.0*atan2(-vl, -rs.w()); //-PI,0
360	+	else
361	+	swingAngle = 2.0*atan2( vl,  rs.w()); //0,PI
362	+	} else {
363	+	swingAxis = Vector3<Real>(1.0,0.0,0.0);
364	+	swingAngle = 0.0;
365	+	}
366	+	}
367	+
368	+
369	+	Vector3<Real> rotate(const Vector3<Real>& v) {
370	+
371	+	Quaternion<Real> q(v.x() * w() + v.z() * y() - v.y() * z(),
372	+	v.y() * w() + v.x() * z() - v.z() * x(),
373	+	v.z() * w() + v.y() * x() - v.x() * y(),
374	+	v.x() * x() + v.y() * y() + v.z() * z());
375	+
376	+	return Vector3<Real>(w()*q.x() + x()*q.w() + y()*q.z() - z()*q.y(),
377	+	w()*q.y() + y()*q.w() + z()*q.x() - x()*q.z(),
378	+	w()*q.z() + z()*q.w() + x()*q.y() - y()*q.x())*
379	+	( 1.0/this->lengthSquare() );
380	+	}
381	+
382	+	Quaternion<Real>& align (const Vector3<Real>& V1,
383	+	const Vector3<Real>& V2) {
384	+
385	+	// If V1 and V2 are not parallel, the axis of rotation is the unit-length
386	+	// vector U = Cross(V1,V2)/Length(Cross(V1,V2)).  The angle of rotation,
387	+	// A, is the angle between V1 and V2.  The quaternion for the rotation is
388	+	// q = cos(A/2) + sin(A/2)*(ux*i+uy*j+uz*k) where U = (ux,uy,uz).
389	+	//
390	+	// (1) Rather than extract A = acos(Dot(V1,V2)), multiply by 1/2, then
391	+	//     compute sin(A/2) and cos(A/2), we reduce the computational costs
392	+	//     by computing the bisector B = (V1+V2)/Length(V1+V2), so cos(A/2) =
393	+	//     Dot(V1,B).
394	+	//
395	+	// (2) The rotation axis is U = Cross(V1,B)/Length(Cross(V1,B)), but
396	+	//     Length(Cross(V1,B)) = Length(V1)*Length(B)*sin(A/2) = sin(A/2), in
397	+	//     which case sin(A/2)*(ux*i+uy*j+uz*k) = (cx*i+cy*j+cz*k) where
398	+	//     C = Cross(V1,B).
399	+	//
400	+	// If V1 = V2, then B = V1, cos(A/2) = 1, and U = (0,0,0).  If V1 = -V2,
401	+	// then B = 0.  This can happen even if V1 is approximately -V2 using
402	+	// floating point arithmetic, since Vector3::Normalize checks for
403	+	// closeness to zero and returns the zero vector accordingly.  The test
404	+	// for exactly zero is usually not recommend for floating point
405	+	// arithmetic, but the implementation of Vector3::Normalize guarantees
406	+	// the comparison is robust.  In this case, the A = pi and any axis
407	+	// perpendicular to V1 may be used as the rotation axis.
408	+
409	+	Vector3<Real> Bisector = V1 + V2;
410	+	Bisector.normalize();
411	+
412	+	Real CosHalfAngle = dot(V1,Bisector);
413	+
414	+	this->data_[0] = CosHalfAngle;
415	+
416	+	if (CosHalfAngle != (Real)0.0) {
417	+	Vector3<Real> Cross = cross(V1, Bisector);
418	+	this->data_[1] = Cross.x();
419	+	this->data_[2] = Cross.y();
420	+	this->data_[3] = Cross.z();
421	+	} else {
422	+	Real InvLength;
423	+	if (fabs(V1[0]) >= fabs(V1[1])) {
424	+	// V1.x or V1.z is the largest magnitude component
425	+	InvLength = (Real)1.0/sqrt(V1[0]*V1[0] + V1[2]*V1[2]);
426	+
427	+	this->data_[1] = -V1[2]*InvLength;
428	+	this->data_[2] = (Real)0.0;
429	+	this->data_[3] = +V1[0]*InvLength;
430	+	} else {
431	+	// V1.y or V1.z is the largest magnitude component
432	+	InvLength = (Real)1.0 / sqrt(V1[1]*V1[1] + V1[2]*V1[2]);
433	+
434	+	this->data_[1] = (Real)0.0;
435	+	this->data_[2] = +V1[2]*InvLength;
436	+	this->data_[3] = -V1[1]*InvLength;
437	+	}
438	+	}
439	+	return *this;
440	+	}
441	+
442	+	void toTwistSwing ( Real& tw, Real& sx, Real& sy ) {
443	+
444	+	// First test if the swing is in the singularity:
445	+
446	+	if ( ISZERO(z(),tiny) && ISZERO(w(),tiny) ) { sx=sy=M_PI; tw=0; return; }
447	+
448	+	// Decompose into twist-swing by solving the equation:
449	+	//
450	+	//                       Qtwist(t*2) * Qswing(s*2) = q
451	+	//
452	+	// note: (x,y) is the normalized swing axis (x*x+y*y=1)
453	+	//
454	+	//          ( Ct 0 0 St ) * ( Cs xSs ySs 0 ) = ( qw qx qy qz )
455	+	//  ( CtCs  xSsCt-yStSs  xStSs+ySsCt  StCs ) = ( qw qx qy qz )      (1)
456	+	// From (1): CtCs / StCs = qw/qz => Ct/St = qw/qz => tan(t) = qz/qw (2)
457	+	//
458	+	// The swing rotation/2 s comes from:
459	+	//
460	+	// From (1): (CtCs)^2 + (StCs)^2 = qw^2 + qz^2 =>
461	+	//                                       Cs = sqrt ( qw^2 + qz^2 ) (3)
462	+	//
463	+	// From (1): (xSsCt-yStSs)^2 + (xStSs+ySsCt)^2 = qx^2 + qy^2 =>
464	+	//                                       Ss = sqrt ( qx^2 + qy^2 ) (4)
465	+	// From (1):  |SsCt -StSs| |x| = |qx|
466	+	//            |StSs +SsCt| |y|   |qy|                              (5)
467	+
468	+	Real qw, qx, qy, qz;
469	+
470	+	if ( w()<0 ) {
471	+	qw=-w();
472	+	qx=-x();
473	+	qy=-y();
474	+	qz=-z();
475	+	} else {
476	+	qw=w();
477	+	qx=x();
478	+	qy=y();
479	+	qz=z();
480	+	}
481	+
482	+	Real t = atan2 ( qz, qw ); // from (2)
483	+	Real s = atan2( sqrt(qx*qx+qy*qy), sqrt(qz*qz+qw*qw) ); // from (3)
484	+	// and (4)
485	+
486	+	Real x=0.0, y=0.0, sins=sin(s);
487	+
488	+	if ( !ISZERO(sins,tiny) ) {
489	+	Real sint = sin(t);
490	+	Real cost = cos(t);
491	+
492	+	// by solving the linear system in (5):
493	+	y = (-qx*sint + qy*cost)/sins;
494	+	x = ( qx*cost + qy*sint)/sins;
495	+	}
496	+
497	+	tw = (Real)2.0*t;
498	+	sx = (Real)2.0*x*s;
499	+	sy = (Real)2.0*y*s;
500	+	}
501	+
502	+	void toSwingTwist(Real& sx, Real& sy, Real& tw ) {
503	+
504	+	// Decompose q into swing-twist using a similar development as
505	+	// in function toTwistSwing
506	+
507	+	if ( ISZERO(z(),tiny) && ISZERO(w(),tiny) ) { sx=sy=M_PI; tw=0; }
508	+
509	+	Real qw, qx, qy, qz;
510	+	if ( w() < 0 ){
511	+	qw=-w();
512	+	qx=-x();
513	+	qy=-y();
514	+	qz=-z();
515	+	} else {
516	+	qw=w();
517	+	qx=x();
518	+	qy=y();
519	+	qz=z();
520	+	}
521	+
522	+	// Get the twist t:
523	+	Real t = 2.0 * atan2(qz,qw);
524	+
525	+	Real bet = atan2( sqrt(qx*qx+qy*qy), sqrt(qz*qz+qw*qw) );
526	+	Real gam = t/2.0;
527	+	Real sinc = ISZERO(bet,tiny)? 1.0 : sin(bet)/bet;
528	+	Real singam = sin(gam);
529	+	Real cosgam = cos(gam);
530	+
531	+	sx = Real( (2.0/sinc) * (cosgam*qx - singam*qy) );
532	+	sy = Real( (2.0/sinc) * (singam*qx + cosgam*qy) );
533	+	tw = Real( t );
534	+	}
535	+
536	+
537	+
538		/**
539	<	* Returns the division of two quaternion
540	<	* @param q1 divisor
329	<	* @param q2 dividen
539	>	* Returns the corresponding rotation matrix (3x3)
540	>	* @return a 3x3 rotation matrix
541		*/
542	+	SquareMatrix<Real, 3> toRotationMatrix3() {
543	+	SquareMatrix<Real, 3> rotMat3;
544	+
545	+	Real w2;
546	+	Real x2;
547	+	Real y2;
548	+	Real z2;
549
550	<	template<typename Real>
551	<	inline Quaternion<Real> operator /( Quaternion<Real>& q1,  Quaternion<Real>& q2) {
552	<	return q1 * q2.inverse();
550	>	if (!this->isNormalized())
551	>	this->normalize();
552	>
553	>	w2 = w() * w();
554	>	x2 = x() * x();
555	>	y2 = y() * y();
556	>	z2 = z() * z();
557	>
558	>	rotMat3(0, 0) = w2 + x2 - y2 - z2;
559	>	rotMat3(0, 1) = 2.0 * ( x() * y() + w() * z() );
560	>	rotMat3(0, 2) = 2.0 * ( x() * z() - w() * y() );
561	>
562	>	rotMat3(1, 0) = 2.0 * ( x() * y() - w() * z() );
563	>	rotMat3(1, 1) = w2 - x2 + y2 - z2;
564	>	rotMat3(1, 2) = 2.0 * ( y() * z() + w() * x() );
565	>
566	>	rotMat3(2, 0) = 2.0 * ( x() * z() + w() * y() );
567	>	rotMat3(2, 1) = 2.0 * ( y() * z() - w() * x() );
568	>	rotMat3(2, 2) = w2 - x2 -y2 +z2;
569	>
570	>	return rotMat3;
571		}
572
573	+	};//end Quaternion
574	+
575	+
576		/**
577	<	* Returns the value of the division of a scalar by a quaternion
578	<	* @return the value of the division of a scalar by a quaternion
579	<	* @param s scalar
580	<	* @param q quaternion
342	<	* @note for a quaternion q, 1/q = q.inverse()
577	>	* Returns the vaule of scalar multiplication of this quaterion q (q * s).
578	>	* @return  the vaule of scalar multiplication of this vector
579	>	* @param q the source quaternion
580	>	* @param s the scalar value
581		*/
582	<	template<typename Real>
583	<	Quaternion<Real> operator /(const Real& s,  Quaternion<Real>& q) {
582	>	template<typename Real, unsigned int Dim>
583	>	Quaternion<Real> operator * ( const Quaternion<Real>& q, Real s) {
584	>	Quaternion<Real> result(q);
585	>	result.mul(s);
586	>	return result;
587	>	}
588	>
589	>	/**
590	>	* Returns the vaule of scalar multiplication of this quaterion q (q * s).
591	>	* @return  the vaule of scalar multiplication of this vector
592	>	* @param s the scalar value
593	>	* @param q the source quaternion
594	>	*/
595	>	template<typename Real, unsigned int Dim>
596	>	Quaternion<Real> operator * ( const Real& s, const Quaternion<Real>& q ) {
597	>	Quaternion<Real> result(q);
598	>	result.mul(s);
599	>	return result;
600	>	}
601
602	<	Quaternion<Real> x;
603	<	x = q.inverse();
604	<	x *= s;
605	<	return x;
606	<	}
602	>	/**
603	>	* Returns the multiplication of two quaternion
604	>	* @return the multiplication of two quaternion
605	>	* @param q1 the first quaternion
606	>	* @param q2 the second quaternion
607	>	*/
608	>	template<typename Real>
609	>	inline Quaternion<Real> operator *(const Quaternion<Real>& q1, const Quaternion<Real>& q2) {
610	>	Quaternion<Real> result(q1);
611	>	result *= q2;
612	>	return result;
613	>	}
614	>
615	>	/**
616	>	* Returns the division of two quaternion
617	>	* @param q1 divisor
618	>	* @param q2 dividen
619	>	*/
620	>
621	>	template<typename Real>
622	>	inline Quaternion<Real> operator /( Quaternion<Real>& q1,  Quaternion<Real>& q2) {
623	>	return q1 * q2.inverse();
624	>	}
625	>
626	>	/**
627	>	* Returns the value of the division of a scalar by a quaternion
628	>	* @return the value of the division of a scalar by a quaternion
629	>	* @param s scalar
630	>	* @param q quaternion
631	>	* @note for a quaternion q, 1/q = q.inverse()
632	>	*/
633	>	template<typename Real>
634	>	Quaternion<Real> operator /(const Real& s,  Quaternion<Real>& q) {
635	>
636	>	Quaternion<Real> x;
637	>	x = q.inverse();
638	>	x *= s;
639	>	return x;
640	>	}
641
642	<	template <class T>
643	<	inline bool operator==(const Quaternion<T>& lhs, const Quaternion<T>& rhs) {
644	<	return equal(lhs[0] ,rhs[0]) && equal(lhs[1] , rhs[1]) && equal(lhs[2], rhs[2]) && equal(lhs[3], rhs[3]);
645	<	}
642	>	template <class T>
643	>	inline bool operator==(const Quaternion<T>& lhs, const Quaternion<T>& rhs) {
644	>	return equal(lhs[0] ,rhs[0]) && equal(lhs[1] , rhs[1]) && equal(lhs[2], rhs[2]) && equal(lhs[3], rhs[3]);
645	>	}
646
647	<	typedef Quaternion<double> Quat4d;
647	>	typedef Quaternion<RealType> Quat4d;
648		}
649		#endif //MATH_QUATERNION_HPP

Comparing:
trunk/src/math/Quaternion.hpp (property svn:keywords), Revision 385 by tim, Tue Mar 1 20:10:14 2005 UTC vs.
branches/development/src/math/Quaternion.hpp (property svn:keywords), Revision 1465 by chuckv, Fri Jul 9 23:08:25 2010 UTC

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