summaryrefslogtreecommitdiff
path: root/indra/llmath/llquaternion.h
blob: aa0b1752f4e0a56883ff4abaa27e885311a8c760 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
/**
 * @file llquaternion.h
 * @brief LLQuaternion class header file.
 *
 * $LicenseInfo:firstyear=2000&license=viewerlgpl$
 * Second Life Viewer Source Code
 * Copyright (C) 2010, Linden Research, Inc.
 * 
 * This library is free software; you can redistribute it and/or
 * modify it under the terms of the GNU Lesser General Public
 * License as published by the Free Software Foundation;
 * version 2.1 of the License only.
 * 
 * This library 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
 * Lesser General Public License for more details.
 * 
 * You should have received a copy of the GNU Lesser General Public
 * License along with this library; if not, write to the Free Software
 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
 * 
 * Linden Research, Inc., 945 Battery Street, San Francisco, CA  94111  USA
 * $/LicenseInfo$
 */

#ifndef LLQUATERNION_H
#define LLQUATERNION_H

#include <iostream>

#ifndef LLMATH_H //enforce specific include order to avoid tangling inline dependencies
#error "Please include llmath.h first."
#endif

class LLVector4;
class LLVector3;
class LLVector3d;
class LLMatrix4;
class LLMatrix3;

//	NOTA BENE: Quaternion code is written assuming Unit Quaternions!!!!
//			   Moreover, it is written assuming that all vectors and matricies
//			   passed as arguments are normalized and unitary respectively.
//			   VERY VERY VERY VERY BAD THINGS will happen if these assumptions fail.

static const U32 LENGTHOFQUAT = 4;

class LLQuaternion
{
public:
	F32 mQ[LENGTHOFQUAT];

	static const LLQuaternion DEFAULT;

	LLQuaternion();									// Initializes Quaternion to (0,0,0,1)
	explicit LLQuaternion(const LLMatrix4 &mat);				// Initializes Quaternion from Matrix4
	explicit LLQuaternion(const LLMatrix3 &mat);				// Initializes Quaternion from Matrix3
	LLQuaternion(F32 x, F32 y, F32 z, F32 w);		// Initializes Quaternion to normalize(x, y, z, w)
	LLQuaternion(F32 angle, const LLVector4 &vec);	// Initializes Quaternion to axis_angle2quat(angle, vec)
	LLQuaternion(F32 angle, const LLVector3 &vec);	// Initializes Quaternion to axis_angle2quat(angle, vec)
	LLQuaternion(const F32 *q);						// Initializes Quaternion to normalize(x, y, z, w)
	LLQuaternion(const LLVector3 &x_axis,
				 const LLVector3 &y_axis,
				 const LLVector3 &z_axis);			// Initializes Quaternion from Matrix3 = [x_axis ; y_axis ; z_axis]

	BOOL isIdentity() const;
	BOOL isNotIdentity() const;
	BOOL isFinite() const;									// checks to see if all values of LLQuaternion are finite
	void quantize16(F32 lower, F32 upper);					// changes the vector to reflect quatization
	void quantize8(F32 lower, F32 upper);							// changes the vector to reflect quatization
	void loadIdentity();											// Loads the quaternion that represents the identity rotation

	bool isEqualEps(const LLQuaternion &quat, F32 epsilon) const;
	bool isNotEqualEps(const LLQuaternion &quat, F32 epsilon) const;

	const LLQuaternion&	set(F32 x, F32 y, F32 z, F32 w);		// Sets Quaternion to normalize(x, y, z, w)
	const LLQuaternion&	set(const LLQuaternion &quat);			// Copies Quaternion
	const LLQuaternion&	set(const F32 *q);						// Sets Quaternion to normalize(quat[VX], quat[VY], quat[VZ], quat[VW])
	const LLQuaternion&	set(const LLMatrix3 &mat);				// Sets Quaternion to mat2quat(mat)
	const LLQuaternion&	set(const LLMatrix4 &mat);				// Sets Quaternion to mat2quat(mat)

	const LLQuaternion&	setAngleAxis(F32 angle, F32 x, F32 y, F32 z);	// Sets Quaternion to axis_angle2quat(angle, x, y, z)
	const LLQuaternion&	setAngleAxis(F32 angle, const LLVector3 &vec);	// Sets Quaternion to axis_angle2quat(angle, vec)
	const LLQuaternion&	setAngleAxis(F32 angle, const LLVector4 &vec);	// Sets Quaternion to axis_angle2quat(angle, vec)
	const LLQuaternion&	setEulerAngles(F32 roll, F32 pitch, F32 yaw);	// Sets Quaternion to euler2quat(pitch, yaw, roll)

	const LLQuaternion&	setQuatInit(F32 x, F32 y, F32 z, F32 w);	// deprecated
	const LLQuaternion&	setQuat(const LLQuaternion &quat);			// deprecated
	const LLQuaternion&	setQuat(const F32 *q);						// deprecated
	const LLQuaternion&	setQuat(const LLMatrix3 &mat);				// deprecated
	const LLQuaternion&	setQuat(const LLMatrix4 &mat);				// deprecated
	const LLQuaternion&	setQuat(F32 angle, F32 x, F32 y, F32 z);	// deprecated
	const LLQuaternion&	setQuat(F32 angle, const LLVector3 &vec);	// deprecated
	const LLQuaternion&	setQuat(F32 angle, const LLVector4 &vec);	// deprecated
	const LLQuaternion&	setQuat(F32 roll, F32 pitch, F32 yaw);		// deprecated

	LLMatrix4	getMatrix4(void) const;							// Returns the Matrix4 equivalent of Quaternion
	LLMatrix3	getMatrix3(void) const;							// Returns the Matrix3 equivalent of Quaternion
	void		getAngleAxis(F32* angle, F32* x, F32* y, F32* z) const;	// returns rotation in radians about axis x,y,z
	void		getAngleAxis(F32* angle, LLVector3 &vec) const;
	void		getEulerAngles(F32 *roll, F32* pitch, F32 *yaw) const;

	F32	normalize();	// Normalizes Quaternion and returns magnitude
	F32	normQuat();		// deprecated

	const LLQuaternion&	conjugate(void);	// Conjugates Quaternion and returns result
	const LLQuaternion&	conjQuat(void);		// deprecated

	// Other useful methods
	const LLQuaternion&	transpose();		// transpose (same as conjugate)
	const LLQuaternion&	transQuat();		// deprecated

	void			shortestArc(const LLVector3 &a, const LLVector3 &b);	// shortest rotation from a to b
	const LLQuaternion& constrain(F32 radians);						// constrains rotation to a cone angle specified in radians

	// Standard operators
	friend std::ostream& operator<<(std::ostream &s, const LLQuaternion &a);					// Prints a
	friend LLQuaternion operator+(const LLQuaternion &a, const LLQuaternion &b);	// Addition
	friend LLQuaternion operator-(const LLQuaternion &a, const LLQuaternion &b);	// Subtraction
	friend LLQuaternion operator-(const LLQuaternion &a);							// Negation
	friend LLQuaternion operator*(F32 a, const LLQuaternion &q);					// Scale
	friend LLQuaternion operator*(const LLQuaternion &q, F32 b);					// Scale
	friend LLQuaternion operator*(const LLQuaternion &a, const LLQuaternion &b);	// Returns a * b
	friend LLQuaternion operator~(const LLQuaternion &a);							// Returns a* (Conjugate of a)
	bool operator==(const LLQuaternion &b) const;			// Returns a == b
	bool operator!=(const LLQuaternion &b) const;			// Returns a != b

	friend const LLQuaternion& operator*=(LLQuaternion &a, const LLQuaternion &b);	// Returns a * b

	friend LLVector4 operator*(const LLVector4 &a, const LLQuaternion &rot);		// Rotates a by rot
	friend LLVector3 operator*(const LLVector3 &a, const LLQuaternion &rot);		// Rotates a by rot
	friend LLVector3d operator*(const LLVector3d &a, const LLQuaternion &rot);		// Rotates a by rot

	// Non-standard operators
	friend F32 dot(const LLQuaternion &a, const LLQuaternion &b);
	friend LLQuaternion lerp(F32 t, const LLQuaternion &p, const LLQuaternion &q);		// linear interpolation (t = 0 to 1) from p to q
	friend LLQuaternion lerp(F32 t, const LLQuaternion &q);								// linear interpolation (t = 0 to 1) from identity to q
	friend LLQuaternion slerp(F32 t, const LLQuaternion &p, const LLQuaternion &q); 	// spherical linear interpolation from p to q
	friend LLQuaternion slerp(F32 t, const LLQuaternion &q);							// spherical linear interpolation from identity to q
	friend LLQuaternion nlerp(F32 t, const LLQuaternion &p, const LLQuaternion &q); 	// normalized linear interpolation from p to q
	friend LLQuaternion nlerp(F32 t, const LLQuaternion &q); 							// normalized linear interpolation from p to q

	LLVector3	packToVector3() const;						// Saves space by using the fact that our quaternions are normalized
	void		unpackFromVector3(const LLVector3& vec);	// Saves space by using the fact that our quaternions are normalized

	enum Order {
		XYZ = 0,
		YZX = 1,
		ZXY = 2,
		XZY = 3,
		YXZ = 4,
		ZYX = 5
	};
	// Creates a quaternions from maya's rotation representation,
	// which is 3 rotations (in DEGREES) in the specified order
	friend LLQuaternion mayaQ(F32 x, F32 y, F32 z, Order order);

	// Conversions between Order and strings like "xyz" or "ZYX"
	friend const char *OrderToString( const Order order );
	friend Order StringToOrder( const char *str );

	static BOOL parseQuat(const std::string& buf, LLQuaternion* value);

	// For debugging, only
	//static U32 mMultCount;
};

// checker
inline BOOL	LLQuaternion::isFinite() const
{
	return (llfinite(mQ[VX]) && llfinite(mQ[VY]) && llfinite(mQ[VZ]) && llfinite(mQ[VS]));
}

inline BOOL LLQuaternion::isIdentity() const
{
	return 
		( mQ[VX] == 0.f ) &&
		( mQ[VY] == 0.f ) &&
		( mQ[VZ] == 0.f ) &&
		( mQ[VS] == 1.f );
}

inline BOOL LLQuaternion::isNotIdentity() const
{
	return 
		( mQ[VX] != 0.f ) ||
		( mQ[VY] != 0.f ) ||
		( mQ[VZ] != 0.f ) ||
		( mQ[VS] != 1.f );
}



inline LLQuaternion::LLQuaternion(void)
{
	mQ[VX] = 0.f;
	mQ[VY] = 0.f;
	mQ[VZ] = 0.f;
	mQ[VS] = 1.f;
}

inline LLQuaternion::LLQuaternion(F32 x, F32 y, F32 z, F32 w)
{
	mQ[VX] = x;
	mQ[VY] = y;
	mQ[VZ] = z;
	mQ[VS] = w;

	//RN: don't normalize this case as its used mainly for temporaries during calculations
	//normalize();
	/*
	F32 mag = sqrtf(mQ[VX]*mQ[VX] + mQ[VY]*mQ[VY] + mQ[VZ]*mQ[VZ] + mQ[VS]*mQ[VS]);
	mag -= 1.f;
	mag = fabs(mag);
	llassert(mag < 10.f*FP_MAG_THRESHOLD);
	*/
}

inline LLQuaternion::LLQuaternion(const F32 *q)
{
	mQ[VX] = q[VX];
	mQ[VY] = q[VY];
	mQ[VZ] = q[VZ];
	mQ[VS] = q[VW];

	normalize();
	/*
	F32 mag = sqrtf(mQ[VX]*mQ[VX] + mQ[VY]*mQ[VY] + mQ[VZ]*mQ[VZ] + mQ[VS]*mQ[VS]);
	mag -= 1.f;
	mag = fabs(mag);
	llassert(mag < FP_MAG_THRESHOLD);
	*/
}


inline void LLQuaternion::loadIdentity()
{
	mQ[VX] = 0.0f;
	mQ[VY] = 0.0f;
	mQ[VZ] = 0.0f;
	mQ[VW] = 1.0f;
}

inline bool LLQuaternion::isEqualEps(const LLQuaternion &quat, F32 epsilon) const
{
	return ( fabs(mQ[VX] - quat.mQ[VX]) < epsilon
		&&	 fabs(mQ[VY] - quat.mQ[VY]) < epsilon
		&&	 fabs(mQ[VZ] - quat.mQ[VZ]) < epsilon
		&&	 fabs(mQ[VS] - quat.mQ[VS]) < epsilon );
}

inline bool LLQuaternion::isNotEqualEps(const LLQuaternion &quat, F32 epsilon) const
{
	return (  fabs(mQ[VX] - quat.mQ[VX]) > epsilon
		||    fabs(mQ[VY] - quat.mQ[VY]) > epsilon
		||	  fabs(mQ[VZ] - quat.mQ[VZ]) > epsilon
		||    fabs(mQ[VS] - quat.mQ[VS]) > epsilon );
}

inline const LLQuaternion&	LLQuaternion::set(F32 x, F32 y, F32 z, F32 w)
{
	mQ[VX] = x;
	mQ[VY] = y;
	mQ[VZ] = z;
	mQ[VS] = w;
	normalize();
	return (*this);
}

inline const LLQuaternion&	LLQuaternion::set(const LLQuaternion &quat)
{
	mQ[VX] = quat.mQ[VX];
	mQ[VY] = quat.mQ[VY];
	mQ[VZ] = quat.mQ[VZ];
	mQ[VW] = quat.mQ[VW];
	normalize();
	return (*this);
}

inline const LLQuaternion&	LLQuaternion::set(const F32 *q)
{
	mQ[VX] = q[VX];
	mQ[VY] = q[VY];
	mQ[VZ] = q[VZ];
	mQ[VS] = q[VW];
	normalize();
	return (*this);
}


// deprecated
inline const LLQuaternion&	LLQuaternion::setQuatInit(F32 x, F32 y, F32 z, F32 w)
{
	mQ[VX] = x;
	mQ[VY] = y;
	mQ[VZ] = z;
	mQ[VS] = w;
	normalize();
	return (*this);
}

// deprecated
inline const LLQuaternion&	LLQuaternion::setQuat(const LLQuaternion &quat)
{
	mQ[VX] = quat.mQ[VX];
	mQ[VY] = quat.mQ[VY];
	mQ[VZ] = quat.mQ[VZ];
	mQ[VW] = quat.mQ[VW];
	normalize();
	return (*this);
}

// deprecated
inline const LLQuaternion&	LLQuaternion::setQuat(const F32 *q)
{
	mQ[VX] = q[VX];
	mQ[VY] = q[VY];
	mQ[VZ] = q[VZ];
	mQ[VS] = q[VW];
	normalize();
	return (*this);
}

inline void LLQuaternion::getAngleAxis(F32* angle, F32* x, F32* y, F32* z) const
{
	F32 v = sqrtf(mQ[VX] * mQ[VX] + mQ[VY] * mQ[VY] + mQ[VZ] * mQ[VZ]); // length of the vector-component
	if (v > FP_MAG_THRESHOLD)
	{
		F32 oomag = 1.0f / v;
		F32 w = mQ[VW];
		if (w < 0.0f)
		{
			w = -w; // make VW positive
			oomag = -oomag; // invert the axis
		}
		*x = mQ[VX] * oomag; // normalize the axis
		*y = mQ[VY] * oomag;
		*z = mQ[VZ] * oomag;
		*angle = 2.0f * atan2f(v, w); // get the angle
	}
	else
	{
		*angle = 0.0f; // no rotation
		*x = 0.0f; // around some dummy axis
		*y = 0.0f;
		*z = 1.0f;
	}
}

inline const LLQuaternion& LLQuaternion::conjugate()
{
	mQ[VX] *= -1.f;
	mQ[VY] *= -1.f;
	mQ[VZ] *= -1.f;
	return (*this);
}

inline const LLQuaternion& LLQuaternion::conjQuat()
{
	mQ[VX] *= -1.f;
	mQ[VY] *= -1.f;
	mQ[VZ] *= -1.f;
	return (*this);
}

// Transpose
inline const LLQuaternion& LLQuaternion::transpose()
{
	mQ[VX] *= -1.f;
	mQ[VY] *= -1.f;
	mQ[VZ] *= -1.f;
	return (*this);
}

// deprecated
inline const LLQuaternion& LLQuaternion::transQuat()
{
	mQ[VX] *= -1.f;
	mQ[VY] *= -1.f;
	mQ[VZ] *= -1.f;
	return (*this);
}


inline LLQuaternion 	operator+(const LLQuaternion &a, const LLQuaternion &b)
{
	return LLQuaternion( 
		a.mQ[VX] + b.mQ[VX],
		a.mQ[VY] + b.mQ[VY],
		a.mQ[VZ] + b.mQ[VZ],
		a.mQ[VW] + b.mQ[VW] );
}


inline LLQuaternion 	operator-(const LLQuaternion &a, const LLQuaternion &b)
{
	return LLQuaternion( 
		a.mQ[VX] - b.mQ[VX],
		a.mQ[VY] - b.mQ[VY],
		a.mQ[VZ] - b.mQ[VZ],
		a.mQ[VW] - b.mQ[VW] );
}


inline LLQuaternion 	operator-(const LLQuaternion &a)
{
	return LLQuaternion(
		-a.mQ[VX],
		-a.mQ[VY],
		-a.mQ[VZ],
		-a.mQ[VW] );
}


inline LLQuaternion 	operator*(F32 a, const LLQuaternion &q)
{
	return LLQuaternion(
		a * q.mQ[VX],
		a * q.mQ[VY],
		a * q.mQ[VZ],
		a * q.mQ[VW] );
}


inline LLQuaternion 	operator*(const LLQuaternion &q, F32 a)
{
	return LLQuaternion(
		a * q.mQ[VX],
		a * q.mQ[VY],
		a * q.mQ[VZ],
		a * q.mQ[VW] );
}

inline LLQuaternion	operator~(const LLQuaternion &a)
{
	LLQuaternion q(a);
	q.conjQuat();
	return q;
}

inline bool	LLQuaternion::operator==(const LLQuaternion &b) const
{
	return (  (mQ[VX] == b.mQ[VX])
			&&(mQ[VY] == b.mQ[VY])
			&&(mQ[VZ] == b.mQ[VZ])
			&&(mQ[VS] == b.mQ[VS]));
}

inline bool	LLQuaternion::operator!=(const LLQuaternion &b) const
{
	return (  (mQ[VX] != b.mQ[VX])
			||(mQ[VY] != b.mQ[VY])
			||(mQ[VZ] != b.mQ[VZ])
			||(mQ[VS] != b.mQ[VS]));
}

inline const LLQuaternion&	operator*=(LLQuaternion &a, const LLQuaternion &b)
{
#if 1
	LLQuaternion q(
		b.mQ[3] * a.mQ[0] + b.mQ[0] * a.mQ[3] + b.mQ[1] * a.mQ[2] - b.mQ[2] * a.mQ[1],
		b.mQ[3] * a.mQ[1] + b.mQ[1] * a.mQ[3] + b.mQ[2] * a.mQ[0] - b.mQ[0] * a.mQ[2],
		b.mQ[3] * a.mQ[2] + b.mQ[2] * a.mQ[3] + b.mQ[0] * a.mQ[1] - b.mQ[1] * a.mQ[0],
		b.mQ[3] * a.mQ[3] - b.mQ[0] * a.mQ[0] - b.mQ[1] * a.mQ[1] - b.mQ[2] * a.mQ[2]
	);
	a = q;
#else
	a = a * b;
#endif
	return a;
}

const F32 ONE_PART_IN_A_MILLION = 0.000001f;

inline F32	LLQuaternion::normalize()
{
	F32 mag = sqrtf(mQ[VX]*mQ[VX] + mQ[VY]*mQ[VY] + mQ[VZ]*mQ[VZ] + mQ[VS]*mQ[VS]);

	if (mag > FP_MAG_THRESHOLD)
	{
		// Floating point error can prevent some quaternions from achieving
		// exact unity length.  When trying to renormalize such quaternions we
		// can oscillate between multiple quantized states.  To prevent such
		// drifts we only renomalize if the length is far enough from unity.
		if (fabs(1.f - mag) > ONE_PART_IN_A_MILLION)
		{
			F32 oomag = 1.f/mag;
			mQ[VX] *= oomag;
			mQ[VY] *= oomag;
			mQ[VZ] *= oomag;
			mQ[VS] *= oomag;
		}
	}
	else
	{
		// we were given a very bad quaternion so we set it to identity
		mQ[VX] = 0.f;
		mQ[VY] = 0.f;
		mQ[VZ] = 0.f;
		mQ[VS] = 1.f;
	}

	return mag;
}

// deprecated
inline F32	LLQuaternion::normQuat()
{
	F32 mag = sqrtf(mQ[VX]*mQ[VX] + mQ[VY]*mQ[VY] + mQ[VZ]*mQ[VZ] + mQ[VS]*mQ[VS]);

	if (mag > FP_MAG_THRESHOLD)
	{
		if (fabs(1.f - mag) > ONE_PART_IN_A_MILLION)
		{
			// only renormalize if length not close enough to 1.0 already
			F32 oomag = 1.f/mag;
			mQ[VX] *= oomag;
			mQ[VY] *= oomag;
			mQ[VZ] *= oomag;
			mQ[VS] *= oomag;
		}
	}
	else
	{
		mQ[VX] = 0.f;
		mQ[VY] = 0.f;
		mQ[VZ] = 0.f;
		mQ[VS] = 1.f;
	}

	return mag;
}

LLQuaternion::Order StringToOrder( const char *str );

// Some notes about Quaternions

// What is a Quaternion?
// ---------------------
// A quaternion is a point in 4-dimensional complex space.
// Q = { Qx, Qy, Qz, Qw }
// 
//
// Why Quaternions?
// ----------------
// The set of quaternions that make up the the 4-D unit sphere 
// can be mapped to the set of all rotations in 3-D space.  Sometimes
// it is easier to describe/manipulate rotations in quaternion space
// than rotation-matrix space.
//
//
// How Quaternions?
// ----------------
// In order to take advantage of quaternions we need to know how to
// go from rotation-matricies to quaternions and back.  We also have
// to agree what variety of rotations we're generating.
// 
// Consider the equation...   v' = v * R 
//
// There are two ways to think about rotations of vectors.
// 1) v' is the same vector in a different reference frame
// 2) v' is a new vector in the same reference frame
//
// bookmark -- which way are we using?
// 
// 
// Quaternion from Angle-Axis:
// ---------------------------
// Suppose we wanted to represent a rotation of some angle (theta) 
// about some axis ({Ax, Ay, Az})...
//
// axis of rotation = {Ax, Ay, Az} 
// angle_of_rotation = theta
//
// s = sin(0.5 * theta)
// c = cos(0.5 * theta)
// Q = { s * Ax, s * Ay, s * Az, c }
//
//
// 3x3 Matrix from Quaternion
// --------------------------
//
//     |                                                                    |
//     | 1 - 2 * (y^2 + z^2)   2 * (x * y + z * w)     2 * (y * w - x * z)  |
//     |                                                                    |
// M = | 2 * (x * y - z * w)   1 - 2 * (x^2 + z^2)     2 * (y * z + x * w)  |
//     |                                                                    |
//     | 2 * (x * z + y * w)   2 * (y * z - x * w)     1 - 2 * (x^2 + y^2)  |
//     |                                                                    |

#endif