OpenVDB: openvdb/math/Mat.h Source File

OpenVDB 9.0.0

Mat.h
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1// Copyright Contributors to the OpenVDB Project
2// SPDX-License-Identifier: MPL-2.0
3//
4/// @file Mat.h
5/// @author Joshua Schpok
6
7#ifndef OPENVDB_MATH_MAT_HAS_BEEN_INCLUDED
8#define OPENVDB_MATH_MAT_HAS_BEEN_INCLUDED
9
10#include "Math.h"
11#include <openvdb/Exceptions.h>
12#include <algorithm> // for std::max()
13#include <cmath>
14#include <iostream>
15#include <string>
16
17
18namespace openvdb {
19 OPENVDB_USE_VERSION_NAMESPACE
20namespace OPENVDB_VERSION_NAME {
21namespace math {
22
23/// @class Mat "Mat.h"
24/// A base class for square matrices.
25template<unsigned SIZE, typename T>
26 class Mat
27{
28public:
29 using value_type = T;
30 using ValueType = T;
31 enum SIZE_ { size = SIZE };
32
33#if OPENVDB_ABI_VERSION_NUMBER >= 8
34 /// Trivial constructor, the matrix is NOT initialized
35 /// @note destructor, copy constructor, assignment operator and
36 /// move constructor are left to be defined by the compiler (default)
37 Mat() = default;
38#else
39 /// Default ctor. Does nothing. Required because declaring a copy (or
40 /// other) constructor means the default constructor gets left out.
41 Mat() { }
42
43 /// Copy constructor. Used when the class signature matches exactly.
44 Mat(Mat const &src) {
45 for (unsigned i(0); i < numElements(); ++i) {
46 mm[i] = src.mm[i];
47 }
48 }
49
50 Mat& operator=(Mat const& src) {
51 if (&src != this) {
52 for (unsigned i = 0; i < numElements(); ++i) {
53 mm[i] = src.mm[i];
54 }
55 }
56 return *this;
57 }
58#endif
59
60 // Number of cols, rows, elements
61 static unsigned numRows() { return SIZE; }
62 static unsigned numColumns() { return SIZE; }
63 static unsigned numElements() { return SIZE*SIZE; }
64
65 /// @return string representation of matrix
66 /// Since output is multiline, optional indentation argument prefixes
67 /// each newline with that much white space. It does not indent
68 /// the first line, since you might be calling this inline:
69 ///
70 /// cout << "matrix: " << mat.str(7)
71 ///
72 /// matrix: [[1 2]
73 /// [3 4]]
74 std::string
75 str(unsigned indentation = 0) const {
76
77 std::string ret;
78 std::string indent;
79
80 // We add +1 since we're indenting one for the first '['
81 indent.append(indentation+1, ' ');
82
83 ret.append("[");
84
85 // For each row,
86 for (unsigned i(0); i < SIZE; i++) {
87
88 ret.append("[");
89
90 // For each column
91 for (unsigned j(0); j < SIZE; j++) {
92
93 // Put a comma after everything except the last
94 if (j) ret.append(", ");
95 ret.append(std::to_string(mm[(i*SIZE)+j]));
96 }
97
98 ret.append("]");
99
100 // At the end of every row (except the last)...
101 if (i < SIZE - 1) {
102 // ...suffix the row bracket with a comma, newline, and advance indentation.
103 ret.append(",\n");
104 ret.append(indent);
105 }
106 }
107
108 ret.append("]");
109
110 return ret;
111 }
112
113 /// Write a Mat to an output stream
114 friend std::ostream& operator<<(
115 std::ostream& ostr,
116 const Mat<SIZE, T>& m)
117 {
118 ostr << m.str();
119 return ostr;
120 }
121
122 /// Direct access to the internal data
123 T* asPointer() { return mm; }
124 const T* asPointer() const { return mm; }
125
126 //@{
127 /// Array style reference to ith row
128 T* operator[](int i) { return &(mm[i*SIZE]); }
129 const T* operator[](int i) const { return &(mm[i*SIZE]); }
130 //@}
131
132 void write(std::ostream& os) const {
133 os.write(reinterpret_cast<const char*>(&mm), sizeof(T)*SIZE*SIZE);
134 }
135
136 void read(std::istream& is) {
137 is.read(reinterpret_cast<char*>(&mm), sizeof(T)*SIZE*SIZE);
138 }
139
140 /// Return the maximum of the absolute of all elements in this matrix
141 T absMax() const {
142 T x = static_cast<T>(std::fabs(mm[0]));
143 for (unsigned i = 1; i < numElements(); ++i) {
144 x = std::max(x, static_cast<T>(std::fabs(mm[i])));
145 }
146 return x;
147 }
148
149 /// True if a Nan is present in this matrix
150 bool isNan() const {
151 for (unsigned i = 0; i < numElements(); ++i) {
152 if (math::isNan(mm[i])) return true;
153 }
154 return false;
155 }
156
157 /// True if an Inf is present in this matrix
158 bool isInfinite() const {
159 for (unsigned i = 0; i < numElements(); ++i) {
160 if (math::isInfinite(mm[i])) return true;
161 }
162 return false;
163 }
164
165 /// True if no Nan or Inf values are present
166 bool isFinite() const {
167 for (unsigned i = 0; i < numElements(); ++i) {
168 if (!math::isFinite(mm[i])) return false;
169 }
170 return true;
171 }
172
173 /// True if all elements are exactly zero
174 bool isZero() const {
175 for (unsigned i = 0; i < numElements(); ++i) {
176 if (!math::isZero(mm[i])) return false;
177 }
178 return true;
179 }
180
181protected:
182 T mm[SIZE*SIZE];
183};
184
185
186template<typename T> class Quat;
187template<typename T> class Vec3;
188
189/// @brief Return the rotation matrix specified by the given quaternion.
190/// @details The quaternion is normalized and used to construct the matrix.
191/// Note that the matrix is transposed to match post-multiplication semantics.
192template<class MatType>
193MatType
194 rotation(const Quat<typename MatType::value_type> &q,
195 typename MatType::value_type eps = static_cast<typename MatType::value_type>(1.0e-8))
196{
197 using T = typename MatType::value_type;
198
199 T qdot(q.dot(q));
200 T s(0);
201
202 if (!isApproxEqual(qdot, T(0.0),eps)) {
203 s = T(2.0 / qdot);
204 }
205
206 T x = s*q.x();
207 T y = s*q.y();
208 T z = s*q.z();
209 T wx = x*q.w();
210 T wy = y*q.w();
211 T wz = z*q.w();
212 T xx = x*q.x();
213 T xy = y*q.x();
214 T xz = z*q.x();
215 T yy = y*q.y();
216 T yz = z*q.y();
217 T zz = z*q.z();
218
219 MatType r;
220 r[0][0]=T(1) - (yy+zz); r[0][1]=xy + wz; r[0][2]=xz - wy;
221 r[1][0]=xy - wz; r[1][1]=T(1) - (xx+zz); r[1][2]=yz + wx;
222 r[2][0]=xz + wy; r[2][1]=yz - wx; r[2][2]=T(1) - (xx+yy);
223
224 if(MatType::numColumns() == 4) padMat4(r);
225 return r;
226}
227
228
229
230/// @brief Return a matrix for rotation by @a angle radians about the given @a axis.
231/// @param axis The axis (one of X, Y, Z) to rotate about.
232/// @param angle The rotation angle, in radians.
233template<class MatType>
234MatType
235 rotation(Axis axis, typename MatType::value_type angle)
236{
237 using T = typename MatType::value_type;
238 T c = static_cast<T>(cos(angle));
239 T s = static_cast<T>(sin(angle));
240
241 MatType result;
242 result.setIdentity();
243
244 switch (axis) {
245 case X_AXIS:
246 result[1][1] = c;
247 result[1][2] = s;
248 result[2][1] = -s;
249 result[2][2] = c;
250 return result;
251 case Y_AXIS:
252 result[0][0] = c;
253 result[0][2] = -s;
254 result[2][0] = s;
255 result[2][2] = c;
256 return result;
257 case Z_AXIS:
258 result[0][0] = c;
259 result[0][1] = s;
260 result[1][0] = -s;
261 result[1][1] = c;
262 return result;
263 default:
264 throw ValueError("Unrecognized rotation axis");
265 }
266}
267
268
269/// @brief Return a matrix for rotation by @a angle radians about the given @a axis.
270/// @note The axis must be a unit vector.
271template<class MatType>
272MatType
273 rotation(const Vec3<typename MatType::value_type> &_axis, typename MatType::value_type angle)
274{
275 using T = typename MatType::value_type;
276 T txy, txz, tyz, sx, sy, sz;
277
278 Vec3<T> axis(_axis.unit());
279
280 // compute trig properties of angle:
281 T c(cos(double(angle)));
282 T s(sin(double(angle)));
283 T t(1 - c);
284
285 MatType result;
286 // handle diagonal elements
287 result[0][0] = axis[0]*axis[0] * t + c;
288 result[1][1] = axis[1]*axis[1] * t + c;
289 result[2][2] = axis[2]*axis[2] * t + c;
290
291 txy = axis[0]*axis[1] * t;
292 sz = axis[2] * s;
293
294 txz = axis[0]*axis[2] * t;
295 sy = axis[1] * s;
296
297 tyz = axis[1]*axis[2] * t;
298 sx = axis[0] * s;
299
300 // right handed space
301 // Contribution from rotation about 'z'
302 result[0][1] = txy + sz;
303 result[1][0] = txy - sz;
304 // Contribution from rotation about 'y'
305 result[0][2] = txz - sy;
306 result[2][0] = txz + sy;
307 // Contribution from rotation about 'x'
308 result[1][2] = tyz + sx;
309 result[2][1] = tyz - sx;
310
311 if(MatType::numColumns() == 4) padMat4(result);
312 return MatType(result);
313}
314
315
316/// @brief Return the Euler angles composing the given rotation matrix.
317/// @details Optional axes arguments describe in what order elementary rotations
318/// are applied. Note that in our convention, XYZ means Rz * Ry * Rx.
319/// Because we are using rows rather than columns to represent the
320/// local axes of a coordinate frame, the interpretation from a local
321/// reference point of view is to first rotate about the x axis, then
322/// about the newly rotated y axis, and finally by the new local z axis.
323/// From a fixed reference point of view, the interpretation is to
324/// rotate about the stationary world z, y, and x axes respectively.
325///
326/// Irrespective of the Euler angle convention, in the case of distinct
327/// axes, eulerAngles() returns the x, y, and z angles in the corresponding
328/// x, y, z components of the returned Vec3. For the XZX convention, the
329/// left X value is returned in Vec3.x, and the right X value in Vec3.y.
330/// For the ZXZ convention the left Z value is returned in Vec3.z and
331/// the right Z value in Vec3.y
332///
333/// Examples of reconstructing r from its Euler angle decomposition
334///
335/// v = eulerAngles(r, ZYX_ROTATION);
336/// rx.setToRotation(Vec3d(1,0,0), v[0]);
337/// ry.setToRotation(Vec3d(0,1,0), v[1]);
338/// rz.setToRotation(Vec3d(0,0,1), v[2]);
339/// r = rx * ry * rz;
340///
341/// v = eulerAngles(r, ZXZ_ROTATION);
342/// rz1.setToRotation(Vec3d(0,0,1), v[2]);
343/// rx.setToRotation (Vec3d(1,0,0), v[0]);
344/// rz2.setToRotation(Vec3d(0,0,1), v[1]);
345/// r = rz2 * rx * rz1;
346///
347/// v = eulerAngles(r, XZX_ROTATION);
348/// rx1.setToRotation (Vec3d(1,0,0), v[0]);
349/// rx2.setToRotation (Vec3d(1,0,0), v[1]);
350/// rz.setToRotation (Vec3d(0,0,1), v[2]);
351/// r = rx2 * rz * rx1;
352///
353template<class MatType>
354 Vec3<typename MatType::value_type>
355eulerAngles(
356 const MatType& mat,
357 RotationOrder rotationOrder,
358 typename MatType::value_type eps = static_cast<typename MatType::value_type>(1.0e-8))
359{
360 using ValueType = typename MatType::value_type;
361 using V = Vec3<ValueType>;
362 ValueType phi, theta, psi;
363
364 switch(rotationOrder)
365 {
366 case XYZ_ROTATION:
367 if (isApproxEqual(mat[2][0], ValueType(1.0), eps)) {
368 theta = ValueType(M_PI_2);
369 phi = ValueType(0.5 * atan2(mat[1][2], mat[1][1]));
370 psi = phi;
371 } else if (isApproxEqual(mat[2][0], ValueType(-1.0), eps)) {
372 theta = ValueType(-M_PI_2);
373 phi = ValueType(0.5 * atan2(mat[1][2], mat[1][1]));
374 psi = -phi;
375 } else {
376 psi = ValueType(atan2(-mat[1][0],mat[0][0]));
377 phi = ValueType(atan2(-mat[2][1],mat[2][2]));
378 theta = ValueType(atan2(mat[2][0],
379 sqrt( mat[2][1]*mat[2][1] +
380 mat[2][2]*mat[2][2])));
381 }
382 return V(phi, theta, psi);
383 case ZXY_ROTATION:
384 if (isApproxEqual(mat[1][2], ValueType(1.0), eps)) {
385 theta = ValueType(M_PI_2);
386 phi = ValueType(0.5 * atan2(mat[0][1], mat[0][0]));
387 psi = phi;
388 } else if (isApproxEqual(mat[1][2], ValueType(-1.0), eps)) {
389 theta = ValueType(-M_PI/2);
390 phi = ValueType(0.5 * atan2(mat[0][1],mat[2][1]));
391 psi = -phi;
392 } else {
393 psi = ValueType(atan2(-mat[0][2], mat[2][2]));
394 phi = ValueType(atan2(-mat[1][0], mat[1][1]));
395 theta = ValueType(atan2(mat[1][2],
396 sqrt(mat[0][2] * mat[0][2] +
397 mat[2][2] * mat[2][2])));
398 }
399 return V(theta, psi, phi);
400
401 case YZX_ROTATION:
402 if (isApproxEqual(mat[0][1], ValueType(1.0), eps)) {
403 theta = ValueType(M_PI_2);
404 phi = ValueType(0.5 * atan2(mat[2][0], mat[2][2]));
405 psi = phi;
406 } else if (isApproxEqual(mat[0][1], ValueType(-1.0), eps)) {
407 theta = ValueType(-M_PI/2);
408 phi = ValueType(0.5 * atan2(mat[2][0], mat[1][0]));
409 psi = -phi;
410 } else {
411 psi = ValueType(atan2(-mat[2][1], mat[1][1]));
412 phi = ValueType(atan2(-mat[0][2], mat[0][0]));
413 theta = ValueType(atan2(mat[0][1],
414 sqrt(mat[0][0] * mat[0][0] +
415 mat[0][2] * mat[0][2])));
416 }
417 return V(psi, phi, theta);
418
419 case XZX_ROTATION:
420
421 if (isApproxEqual(mat[0][0], ValueType(1.0), eps)) {
422 theta = ValueType(0.0);
423 phi = ValueType(0.5 * atan2(mat[1][2], mat[1][1]));
424 psi = phi;
425 } else if (isApproxEqual(mat[0][0], ValueType(-1.0), eps)) {
426 theta = ValueType(M_PI);
427 psi = ValueType(0.5 * atan2(mat[2][1], -mat[1][1]));
428 phi = - psi;
429 } else {
430 psi = ValueType(atan2(mat[2][0], -mat[1][0]));
431 phi = ValueType(atan2(mat[0][2], mat[0][1]));
432 theta = ValueType(atan2(sqrt(mat[0][1] * mat[0][1] +
433 mat[0][2] * mat[0][2]),
434 mat[0][0]));
435 }
436 return V(phi, psi, theta);
437
438 case ZXZ_ROTATION:
439
440 if (isApproxEqual(mat[2][2], ValueType(1.0), eps)) {
441 theta = ValueType(0.0);
442 phi = ValueType(0.5 * atan2(mat[0][1], mat[0][0]));
443 psi = phi;
444 } else if (isApproxEqual(mat[2][2], ValueType(-1.0), eps)) {
445 theta = ValueType(M_PI);
446 phi = ValueType(0.5 * atan2(mat[0][1], mat[0][0]));
447 psi = -phi;
448 } else {
449 psi = ValueType(atan2(mat[0][2], mat[1][2]));
450 phi = ValueType(atan2(mat[2][0], -mat[2][1]));
451 theta = ValueType(atan2(sqrt(mat[0][2] * mat[0][2] +
452 mat[1][2] * mat[1][2]),
453 mat[2][2]));
454 }
455 return V(theta, psi, phi);
456
457 case YXZ_ROTATION:
458
459 if (isApproxEqual(mat[2][1], ValueType(1.0), eps)) {
460 theta = ValueType(-M_PI_2);
461 phi = ValueType(0.5 * atan2(-mat[1][0], mat[0][0]));
462 psi = phi;
463 } else if (isApproxEqual(mat[2][1], ValueType(-1.0), eps)) {
464 theta = ValueType(M_PI_2);
465 phi = ValueType(0.5 * atan2(mat[1][0], mat[0][0]));
466 psi = -phi;
467 } else {
468 psi = ValueType(atan2(mat[0][1], mat[1][1]));
469 phi = ValueType(atan2(mat[2][0], mat[2][2]));
470 theta = ValueType(atan2(-mat[2][1],
471 sqrt(mat[0][1] * mat[0][1] +
472 mat[1][1] * mat[1][1])));
473 }
474 return V(theta, phi, psi);
475
476 case ZYX_ROTATION:
477
478 if (isApproxEqual(mat[0][2], ValueType(1.0), eps)) {
479 theta = ValueType(-M_PI_2);
480 phi = ValueType(0.5 * atan2(-mat[1][0], mat[1][1]));
481 psi = phi;
482 } else if (isApproxEqual(mat[0][2], ValueType(-1.0), eps)) {
483 theta = ValueType(M_PI_2);
484 phi = ValueType(0.5 * atan2(mat[2][1], mat[2][0]));
485 psi = -phi;
486 } else {
487 psi = ValueType(atan2(mat[1][2], mat[2][2]));
488 phi = ValueType(atan2(mat[0][1], mat[0][0]));
489 theta = ValueType(atan2(-mat[0][2],
490 sqrt(mat[0][1] * mat[0][1] +
491 mat[0][0] * mat[0][0])));
492 }
493 return V(psi, theta, phi);
494
495 case XZY_ROTATION:
496
497 if (isApproxEqual(mat[1][0], ValueType(-1.0), eps)) {
498 theta = ValueType(M_PI_2);
499 psi = ValueType(0.5 * atan2(mat[2][1], mat[2][2]));
500 phi = -psi;
501 } else if (isApproxEqual(mat[1][0], ValueType(1.0), eps)) {
502 theta = ValueType(-M_PI_2);
503 psi = ValueType(0.5 * atan2(- mat[2][1], mat[2][2]));
504 phi = psi;
505 } else {
506 psi = ValueType(atan2(mat[2][0], mat[0][0]));
507 phi = ValueType(atan2(mat[1][2], mat[1][1]));
508 theta = ValueType(atan2(- mat[1][0],
509 sqrt(mat[1][1] * mat[1][1] +
510 mat[1][2] * mat[1][2])));
511 }
512 return V(phi, psi, theta);
513 }
514
515 OPENVDB_THROW(NotImplementedError, "Euler extraction sequence not implemented");
516}
517
518
519/// @brief Return a rotation matrix that maps @a v1 onto @a v2
520/// about the cross product of @a v1 and @a v2.
521/// <a name="rotation_v1_v2"></a>
522template<typename MatType, typename ValueType1, typename ValueType2>
523inline MatType
524rotation(
525 const Vec3<ValueType1>& _v1,
526 const Vec3<ValueType2>& _v2,
527 typename MatType::value_type eps = static_cast<typename MatType::value_type>(1.0e-8))
528{
529 using T = typename MatType::value_type;
530
531 Vec3<T> v1(_v1);
532 Vec3<T> v2(_v2);
533
534 // Check if v1 and v2 are unit length
535 if (!isApproxEqual(T(1), v1.dot(v1), eps)) {
536 v1.normalize();
537 }
538 if (!isApproxEqual(T(1), v2.dot(v2), eps)) {
539 v2.normalize();
540 }
541
542 Vec3<T> cross;
543 cross.cross(v1, v2);
544
545 if (isApproxEqual(cross[0], zeroVal<T>(), eps) &&
546 isApproxEqual(cross[1], zeroVal<T>(), eps) &&
547 isApproxEqual(cross[2], zeroVal<T>(), eps)) {
548
549
550 // Given two unit vectors v1 and v2 that are nearly parallel, build a
551 // rotation matrix that maps v1 onto v2. First find which principal axis
552 // p is closest to perpendicular to v1. Find a reflection that exchanges
553 // v1 and p, and find a reflection that exchanges p2 and v2. The desired
554 // rotation matrix is the composition of these two reflections. See the
555 // paper "Efficiently Building a Matrix to Rotate One Vector to
556 // Another" by Tomas Moller and John Hughes in Journal of Graphics
557 // Tools Vol 4, No 4 for details.
558
559 Vec3<T> u, v, p(0.0, 0.0, 0.0);
560
561 double x = Abs(v1[0]);
562 double y = Abs(v1[1]);
563 double z = Abs(v1[2]);
564
565 if (x < y) {
566 if (z < x) {
567 p[2] = 1;
568 } else {
569 p[0] = 1;
570 }
571 } else {
572 if (z < y) {
573 p[2] = 1;
574 } else {
575 p[1] = 1;
576 }
577 }
578 u = p - v1;
579 v = p - v2;
580
581 double udot = u.dot(u);
582 double vdot = v.dot(v);
583
584 double a = -2 / udot;
585 double b = -2 / vdot;
586 double c = 4 * u.dot(v) / (udot * vdot);
587
588 MatType result;
589 result.setIdentity();
590
591 for (int j = 0; j < 3; j++) {
592 for (int i = 0; i < 3; i++)
593 result[i][j] = static_cast<T>(
594 a * u[i] * u[j] + b * v[i] * v[j] + c * v[j] * u[i]);
595 }
596 result[0][0] += 1.0;
597 result[1][1] += 1.0;
598 result[2][2] += 1.0;
599
600 if(MatType::numColumns() == 4) padMat4(result);
601 return result;
602
603 } else {
604 double c = v1.dot(v2);
605 double a = (1.0 - c) / cross.dot(cross);
606
607 double a0 = a * cross[0];
608 double a1 = a * cross[1];
609 double a2 = a * cross[2];
610
611 double a01 = a0 * cross[1];
612 double a02 = a0 * cross[2];
613 double a12 = a1 * cross[2];
614
615 MatType r;
616
617 r[0][0] = static_cast<T>(c + a0 * cross[0]);
618 r[0][1] = static_cast<T>(a01 + cross[2]);
619 r[0][2] = static_cast<T>(a02 - cross[1]);
620 r[1][0] = static_cast<T>(a01 - cross[2]);
621 r[1][1] = static_cast<T>(c + a1 * cross[1]);
622 r[1][2] = static_cast<T>(a12 + cross[0]);
623 r[2][0] = static_cast<T>(a02 + cross[1]);
624 r[2][1] = static_cast<T>(a12 - cross[0]);
625 r[2][2] = static_cast<T>(c + a2 * cross[2]);
626
627 if(MatType::numColumns() == 4) padMat4(r);
628 return r;
629
630 }
631}
632
633
634/// Return a matrix that scales by @a s.
635template<class MatType>
636MatType
637scale(const Vec3<typename MatType::value_type>& s)
638{
639 // Gets identity, then sets top 3 diagonal
640 // Inefficient by 3 sets.
641
642 MatType result;
643 result.setIdentity();
644 result[0][0] = s[0];
645 result[1][1] = s[1];
646 result[2][2] = s[2];
647
648 return result;
649}
650
651
652/// Return a Vec3 representing the lengths of the passed matrix's upper 3&times;3's rows.
653template<class MatType>
654 Vec3<typename MatType::value_type>
655getScale(const MatType &mat)
656{
657 using V = Vec3<typename MatType::value_type>;
658 return V(
659 V(mat[0][0], mat[0][1], mat[0][2]).length(),
660 V(mat[1][0], mat[1][1], mat[1][2]).length(),
661 V(mat[2][0], mat[2][1], mat[2][2]).length());
662}
663
664
665/// @brief Return a copy of the given matrix with its upper 3&times;3 rows normalized.
666/// @details This can be geometrically interpreted as a matrix with no scaling
667/// along its major axes.
668template<class MatType>
669MatType
670 unit(const MatType &mat, typename MatType::value_type eps = 1.0e-8)
671{
672 Vec3<typename MatType::value_type> dud;
673 return unit(mat, eps, dud);
674}
675
676
677/// @brief Return a copy of the given matrix with its upper 3&times;3 rows normalized,
678/// and return the length of each of these rows in @a scaling.
679/// @details This can be geometrically interpretted as a matrix with no scaling
680/// along its major axes, and the scaling in the input vector
681template<class MatType>
682MatType
683unit(
684 const MatType &in,
685 typename MatType::value_type eps,
686 Vec3<typename MatType::value_type>& scaling)
687{
688 using T = typename MatType::value_type;
689 MatType result(in);
690
691 for (int i(0); i < 3; i++) {
692 try {
693 const Vec3<T> u(
694 Vec3<T>(in[i][0], in[i][1], in[i][2]).unit(eps, scaling[i]));
695 for (int j=0; j<3; j++) result[i][j] = u[j];
696 } catch (ArithmeticError&) {
697 for (int j=0; j<3; j++) result[i][j] = 0;
698 }
699 }
700 return result;
701}
702
703
704/// @brief Set the matrix to a shear along @a axis0 by a fraction of @a axis1.
705/// @param axis0 The fixed axis of the shear.
706/// @param axis1 The shear axis.
707/// @param shear The shear factor.
708template <class MatType>
709MatType
710shear(Axis axis0, Axis axis1, typename MatType::value_type shear)
711{
712 int index0 = static_cast<int>(axis0);
713 int index1 = static_cast<int>(axis1);
714
715 MatType result;
716 result.setIdentity();
717 if (axis0 == axis1) {
718 result[index1][index0] = shear + 1;
719 } else {
720 result[index1][index0] = shear;
721 }
722
723 return result;
724}
725
726
727/// Return a matrix as the cross product of the given vector.
728template<class MatType>
729MatType
730skew(const Vec3<typename MatType::value_type> &skew)
731{
732 using T = typename MatType::value_type;
733
734 MatType r;
735 r[0][0] = T(0); r[0][1] = skew.z(); r[0][2] = -skew.y();
736 r[1][0] = -skew.z(); r[1][1] = T(0); r[2][1] = skew.x();
737 r[2][0] = skew.y(); r[2][1] = -skew.x(); r[2][2] = T(0);
738
739 if(MatType::numColumns() == 4) padMat4(r);
740 return r;
741}
742
743
744/// @brief Return an orientation matrix such that z points along @a direction,
745/// and y is along the @a direction / @a vertical plane.
746template<class MatType>
747MatType
748aim(const Vec3<typename MatType::value_type>& direction,
749 const Vec3<typename MatType::value_type>& vertical)
750{
751 using T = typename MatType::value_type;
752 Vec3<T> forward(direction.unit());
753 Vec3<T> horizontal(vertical.unit().cross(forward).unit());
754 Vec3<T> up(forward.cross(horizontal).unit());
755
756 MatType r;
757
758 r[0][0]=horizontal.x(); r[0][1]=horizontal.y(); r[0][2]=horizontal.z();
759 r[1][0]=up.x(); r[1][1]=up.y(); r[1][2]=up.z();
760 r[2][0]=forward.x(); r[2][1]=forward.y(); r[2][2]=forward.z();
761
762 if(MatType::numColumns() == 4) padMat4(r);
763 return r;
764}
765
766/// @brief This function snaps a specific axis to a specific direction,
767/// preserving scaling.
768/// @details It does this using minimum energy, thus posing a unique solution if
769/// basis & direction aren't parallel.
770/// @note @a direction need not be unit.
771template<class MatType>
772inline MatType
773snapMatBasis(const MatType& source, Axis axis, const Vec3<typename MatType::value_type>& direction)
774{
775 using T = typename MatType::value_type;
776
777 Vec3<T> unitDir(direction.unit());
778 Vec3<T> ourUnitAxis(source.row(axis).unit());
779
780 // Are the two parallel?
781 T parallel = unitDir.dot(ourUnitAxis);
782
783 // Already snapped!
784 if (isApproxEqual(parallel, T(1.0))) return source;
785
786 if (isApproxEqual(parallel, T(-1.0))) {
787 OPENVDB_THROW(ValueError, "Cannot snap to inverse axis");
788 }
789
790 // Find angle between our basis and the one specified
791 T angleBetween(angle(unitDir, ourUnitAxis));
792 // Caclulate axis to rotate along
793 Vec3<T> rotationAxis = unitDir.cross(ourUnitAxis);
794
795 MatType rotation;
796 rotation.setToRotation(rotationAxis, angleBetween);
797
798 return source * rotation;
799}
800
801/// @brief Write 0s along Mat4's last row and column, and a 1 on its diagonal.
802/// @details Useful initialization when we're initializing just the 3&times;3 block.
803template<class MatType>
804inline MatType&
805padMat4(MatType& dest)
806{
807 dest[0][3] = dest[1][3] = dest[2][3] = 0;
808 dest[3][2] = dest[3][1] = dest[3][0] = 0;
809 dest[3][3] = 1;
810
811 return dest;
812}
813
814
815/// @brief Solve for A=B*B, given A.
816/// @details Denman-Beavers square root iteration
817template<typename MatType>
818inline void
819sqrtSolve(const MatType& aA, MatType& aB, double aTol=0.01)
820{
821 unsigned int iterations = static_cast<unsigned int>(log(aTol)/log(0.5));
822
823 MatType Y[2], Z[2];
824 Y[0] = aA;
825 Z[0] = MatType::identity();
826
827 unsigned int current = 0;
828 for (unsigned int iteration=0; iteration < iterations; iteration++) {
829 unsigned int last = current;
830 current = !current;
831
832 MatType invY = Y[last].inverse();
833 MatType invZ = Z[last].inverse();
834
835 Y[current] = 0.5 * (Y[last] + invZ);
836 Z[current] = 0.5 * (Z[last] + invY);
837 }
838 aB = Y[current];
839}
840
841
842template<typename MatType>
843inline void
844powSolve(const MatType& aA, MatType& aB, double aPower, double aTol=0.01)
845{
846 unsigned int iterations = static_cast<unsigned int>(log(aTol)/log(0.5));
847
848 const bool inverted = (aPower < 0.0);
849 if (inverted) { aPower = -aPower; }
850
851 unsigned int whole = static_cast<unsigned int>(aPower);
852 double fraction = aPower - whole;
853
854 MatType R = MatType::identity();
855 MatType partial = aA;
856
857 double contribution = 1.0;
858 for (unsigned int iteration = 0; iteration < iterations; iteration++) {
859 sqrtSolve(partial, partial, aTol);
860 contribution *= 0.5;
861 if (fraction >= contribution) {
862 R *= partial;
863 fraction -= contribution;
864 }
865 }
866
867 partial = aA;
868 while (whole) {
869 if (whole & 1) { R *= partial; }
870 whole >>= 1;
871 if (whole) { partial *= partial; }
872 }
873
874 if (inverted) { aB = R.inverse(); }
875 else { aB = R; }
876}
877
878
879/// @brief Determine if a matrix is an identity matrix.
880template<typename MatType>
881inline bool
882isIdentity(const MatType& m)
883{
884 return m.eq(MatType::identity());
885}
886
887
888/// @brief Determine if a matrix is invertible.
889template<typename MatType>
890inline bool
891isInvertible(const MatType& m)
892{
893 using ValueType = typename MatType::ValueType;
894 return !isApproxEqual(m.det(), ValueType(0));
895}
896
897
898/// @brief Determine if a matrix is symmetric.
899/// @details This implicitly uses math::isApproxEqual() to determine equality.
900template<typename MatType>
901inline bool
902isSymmetric(const MatType& m)
903{
904 return m.eq(m.transpose());
905}
906
907
908/// Determine if a matrix is unitary (i.e., rotation or reflection).
909template<typename MatType>
910inline bool
911isUnitary(const MatType& m)
912{
913 using ValueType = typename MatType::ValueType;
914 if (!isApproxEqual(std::abs(m.det()), ValueType(1.0))) return false;
915 // check that the matrix transpose is the inverse
916 MatType temp = m * m.transpose();
917 return temp.eq(MatType::identity());
918}
919
920
921/// Determine if a matrix is diagonal.
922template<typename MatType>
923inline bool
924isDiagonal(const MatType& mat)
925{
926 int n = MatType::size;
927 typename MatType::ValueType temp(0);
928 for (int i = 0; i < n; ++i) {
929 for (int j = 0; j < n; ++j) {
930 if (i != j) {
931 temp += std::abs(mat(i,j));
932 }
933 }
934 }
935 return isApproxEqual(temp, typename MatType::ValueType(0.0));
936}
937
938
939/// Return the <i>L</i><sub>&infin;</sub> norm of an <i>N</i>&times;<i>N</i> matrix.
940template<typename MatType>
941typename MatType::ValueType
942lInfinityNorm(const MatType& matrix)
943{
944 int n = MatType::size;
945 typename MatType::ValueType norm = 0;
946
947 for( int j = 0; j<n; ++j) {
948 typename MatType::ValueType column_sum = 0;
949
950 for (int i = 0; i<n; ++i) {
951 column_sum += std::fabs(matrix(i,j));
952 }
953 norm = std::max(norm, column_sum);
954 }
955
956 return norm;
957}
958
959
960/// Return the <i>L</i><sub>1</sub> norm of an <i>N</i>&times;<i>N</i> matrix.
961template<typename MatType>
962typename MatType::ValueType
963lOneNorm(const MatType& matrix)
964{
965 int n = MatType::size;
966 typename MatType::ValueType norm = 0;
967
968 for( int i = 0; i<n; ++i) {
969 typename MatType::ValueType row_sum = 0;
970
971 for (int j = 0; j<n; ++j) {
972 row_sum += std::fabs(matrix(i,j));
973 }
974 norm = std::max(norm, row_sum);
975 }
976
977 return norm;
978}
979
980
981/// @brief Decompose an invertible 3&times;3 matrix into a unitary matrix
982/// followed by a symmetric matrix (positive semi-definite Hermitian),
983/// i.e., M = U * S.
984/// @details If det(U) = 1 it is a rotation, otherwise det(U) = -1,
985/// meaning there is some part reflection.
986/// See "Computing the polar decomposition with applications"
987/// Higham, N.J. - SIAM J. Sc. Stat Comput 7(4):1160-1174
988template<typename MatType>
989bool
990polarDecomposition(const MatType& input, MatType& unitary,
991 MatType& positive_hermitian, unsigned int MAX_ITERATIONS=100)
992{
993 unitary = input;
994 MatType new_unitary(input);
995 MatType unitary_inv;
996
997 if (fabs(unitary.det()) < math::Tolerance<typename MatType::ValueType>::value()) return false;
998
999 unsigned int iteration(0);
1000
1001 typename MatType::ValueType linf_of_u;
1002 typename MatType::ValueType l1nm_of_u;
1003 typename MatType::ValueType linf_of_u_inv;
1004 typename MatType::ValueType l1nm_of_u_inv;
1005 typename MatType::ValueType l1_error = 100;
1006 double gamma;
1007
1008 do {
1009 unitary_inv = unitary.inverse();
1010 linf_of_u = lInfinityNorm(unitary);
1011 l1nm_of_u = lOneNorm(unitary);
1012
1013 linf_of_u_inv = lInfinityNorm(unitary_inv);
1014 l1nm_of_u_inv = lOneNorm(unitary_inv);
1015
1016 gamma = sqrt( sqrt( (l1nm_of_u_inv * linf_of_u_inv ) / (l1nm_of_u * linf_of_u) ));
1017
1018 new_unitary = 0.5*(gamma * unitary + (1./gamma) * unitary_inv.transpose() );
1019
1020 l1_error = lInfinityNorm(unitary - new_unitary);
1021 unitary = new_unitary;
1022
1023 /// this generally converges in less than ten iterations
1024 if (iteration > MAX_ITERATIONS) return false;
1025 iteration++;
1026 } while (l1_error > math::Tolerance<typename MatType::ValueType>::value());
1027
1028 positive_hermitian = unitary.transpose() * input;
1029 return true;
1030}
1031
1032////////////////////////////////////////
1033
1034/// @return true if m0 < m1, comparing components in order of significance.
1035template<unsigned SIZE, typename T>
1036inline bool
1037cwiseLessThan(const Mat<SIZE, T>& m0, const Mat<SIZE, T>& m1)
1038{
1039 const T* m0p = m0.asPointer();
1040 const T* m1p = m1.asPointer();
1041 constexpr unsigned size = SIZE*SIZE;
1042 for (unsigned i = 0; i < size-1; ++i, ++m0p, ++m1p) {
1043 if (!math::isExactlyEqual(*m0p, *m1p)) return *m0p < *m1p;
1044 }
1045 return *m0p < *m1p;
1046}
1047
1048/// @return true if m0 > m1, comparing components in order of significance.
1049template<unsigned SIZE, typename T>
1050inline bool
1051cwiseGreaterThan(const Mat<SIZE, T>& m0, const Mat<SIZE, T>& m1)
1052{
1053 const T* m0p = m0.asPointer();
1054 const T* m1p = m1.asPointer();
1055 constexpr unsigned size = SIZE*SIZE;
1056 for (unsigned i = 0; i < size-1; ++i, ++m0p, ++m1p) {
1057 if (!math::isExactlyEqual(*m0p, *m1p)) return *m0p > *m1p;
1058 }
1059 return *m0p > *m1p;
1060}
1061
1062} // namespace math
1063} // namespace OPENVDB_VERSION_NAME
1064} // namespace openvdb
1065
1066#endif // OPENVDB_MATH_MAT_HAS_BEEN_INCLUDED
Math.h
General-purpose arithmetic and comparison routines, most of which accept arbitrary value types (or at...
openvdb::v9_0::ArithmeticError
Definition: Exceptions.h:56
openvdb::v9_0::NotImplementedError
Definition: Exceptions.h:61
openvdb::v9_0::ValueError
Definition: Exceptions.h:65
openvdb::v9_0::math::Mat
Definition: Mat.h:27
openvdb::v9_0::math::Mat::mm
T mm[SIZE *SIZE]
Definition: Mat.h:182
openvdb::v9_0::math::Mat::isZero
bool isZero() const
True if all elements are exactly zero.
Definition: Mat.h:174
openvdb::v9_0::math::Mat::operator[]
T * operator[](int i)
Array style reference to ith row.
Definition: Mat.h:128
openvdb::v9_0::math::Mat::operator[]
const T * operator[](int i) const
Definition: Mat.h:129
openvdb::v9_0::math::Mat::Mat
Mat()=default
openvdb::v9_0::math::Mat::write
void write(std::ostream &os) const
Definition: Mat.h:132
openvdb::v9_0::math::Mat::numColumns
static unsigned numColumns()
Definition: Mat.h:62
openvdb::v9_0::math::Mat::asPointer
T * asPointer()
Direct access to the internal data.
Definition: Mat.h:123
openvdb::v9_0::math::Mat::numRows
static unsigned numRows()
Definition: Mat.h:61
openvdb::v9_0::math::Mat::str
std::string str(unsigned indentation=0) const
Definition: Mat.h:75
openvdb::v9_0::math::Mat::absMax
T absMax() const
Return the maximum of the absolute of all elements in this matrix.
Definition: Mat.h:141
openvdb::v9_0::math::Mat::isInfinite
bool isInfinite() const
True if an Inf is present in this matrix.
Definition: Mat.h:158
openvdb::v9_0::math::Mat::asPointer
const T * asPointer() const
Definition: Mat.h:124
openvdb::v9_0::math::Mat::operator<<
friend std::ostream & operator<<(std::ostream &ostr, const Mat< SIZE, T > &m)
Write a Mat to an output stream.
Definition: Mat.h:114
openvdb::v9_0::math::Mat::SIZE_
SIZE_
Definition: Mat.h:31
openvdb::v9_0::math::Mat::numElements
static unsigned numElements()
Definition: Mat.h:63
openvdb::v9_0::math::Mat::ValueType
T ValueType
Definition: Mat.h:30
openvdb::v9_0::math::Mat::isNan
bool isNan() const
True if a Nan is present in this matrix.
Definition: Mat.h:150
openvdb::v9_0::math::Mat::value_type
T value_type
Definition: Mat.h:29
openvdb::v9_0::math::Mat::read
void read(std::istream &is)
Definition: Mat.h:136
openvdb::v9_0::math::Mat::isFinite
bool isFinite() const
True if no Nan or Inf values are present.
Definition: Mat.h:166
openvdb::v9_0::math::Quat
Definition: Quat.h:79
openvdb::v9_0::math::Quat::x
T & x()
Reference to the component, e.g. q.x() = 4.5f;.
Definition: Quat.h:218
openvdb::v9_0::math::Quat::y
T & y()
Definition: Quat.h:219
openvdb::v9_0::math::Quat::z
T & z()
Definition: Quat.h:220
openvdb::v9_0::math::Quat::w
T & w()
Definition: Quat.h:221
openvdb::v9_0::math::Quat::dot
T dot(const Quat &q) const
Dot product.
Definition: Quat.h:476
openvdb::v9_0::math::Vec3
Definition: Vec3.h:24
openvdb::v9_0::math::Vec3::x
T & x()
Reference to the component, e.g. v.x() = 4.5f;.
Definition: Vec3.h:89
openvdb::v9_0::math::Vec3::dot
T dot(const Vec3< T > &v) const
Dot product.
Definition: Vec3.h:195
openvdb::v9_0::math::Vec3::y
T & y()
Definition: Vec3.h:90
openvdb::v9_0::math::Vec3::z
T & z()
Definition: Vec3.h:91
openvdb::v9_0::math::Vec3::cross
Vec3< T > cross(const Vec3< T > &v) const
Return the cross product of "this" vector and v;.
Definition: Vec3.h:224
openvdb::v9_0::math::Vec3::unit
Vec3< T > unit(T eps=0) const
return normalized this, throws if null vector
Definition: Vec3.h:378
openvdb::v9_0::math::Vec3::normalize
bool normalize(T eps=T(1.0e-7))
this = normalized this
Definition: Vec3.h:366
nanovdb::Abs
T Abs(T x)
Definition: NanoVDB.h:747
openvdb::v9_0::math::cwiseLessThan
bool cwiseLessThan(const Mat< SIZE, T > &m0, const Mat< SIZE, T > &m1)
Definition: Mat.h:1037
openvdb::v9_0::math::isFinite
bool isFinite(const float x)
Return true if x is finite.
Definition: Math.h:376
openvdb::v9_0::math::powSolve
void powSolve(const MatType &aA, MatType &aB, double aPower, double aTol=0.01)
Definition: Mat.h:844
openvdb::v9_0::math::aim
MatType aim(const Vec3< typename MatType::value_type > &direction, const Vec3< typename MatType::value_type > &vertical)
Return an orientation matrix such that z points along direction, and y is along the direction / verti...
Definition: Mat.h:748
openvdb::v9_0::math::eulerAngles
Vec3< typename MatType::value_type > eulerAngles(const MatType &mat, RotationOrder rotationOrder, typename MatType::value_type eps=static_cast< typename MatType::value_type >(1.0e-8))
Return the Euler angles composing the given rotation matrix.
Definition: Mat.h:355
openvdb::v9_0::math::skew
MatType skew(const Vec3< typename MatType::value_type > &skew)
Return a matrix as the cross product of the given vector.
Definition: Mat.h:730
openvdb::v9_0::math::isExactlyEqual
bool isExactlyEqual(const T0 &a, const T1 &b)
Return true if a is exactly equal to b.
Definition: Math.h:444
openvdb::v9_0::math::padMat4
MatType & padMat4(MatType &dest)
Write 0s along Mat4's last row and column, and a 1 on its diagonal.
Definition: Mat.h:805
openvdb::v9_0::math::isInvertible
bool isInvertible(const MatType &m)
Determine if a matrix is invertible.
Definition: Mat.h:891
openvdb::v9_0::math::rotation
MatType rotation(const Vec3< ValueType1 > &_v1, const Vec3< ValueType2 > &_v2, typename MatType::value_type eps=static_cast< typename MatType::value_type >(1.0e-8))
Return a rotation matrix that maps v1 onto v2 about the cross product of v1 and v2.
Definition: Mat.h:524
openvdb::v9_0::math::unit
MatType unit(const MatType &in, typename MatType::value_type eps, Vec3< typename MatType::value_type > &scaling)
Return a copy of the given matrix with its upper 3×3 rows normalized, and return the length of each o...
Definition: Mat.h:683
openvdb::v9_0::math::isUnitary
bool isUnitary(const MatType &m)
Determine if a matrix is unitary (i.e., rotation or reflection).
Definition: Mat.h:911
openvdb::v9_0::math::isSymmetric
bool isSymmetric(const MatType &m)
Determine if a matrix is symmetric.
Definition: Mat.h:902
openvdb::v9_0::math::isInfinite
bool isInfinite(const float x)
Return true if x is an infinity value (either positive infinity or negative infinity).
Definition: Math.h:386
openvdb::v9_0::math::isZero
bool isZero(const Type &x)
Return true if x is exactly equal to zero.
Definition: Math.h:338
openvdb::v9_0::math::angle
T angle(const Vec2< T > &v1, const Vec2< T > &v2)
Definition: Vec2.h:450
openvdb::v9_0::math::getScale
Vec3< typename MatType::value_type > getScale(const MatType &mat)
Return a Vec3 representing the lengths of the passed matrix's upper 3×3's rows.
Definition: Mat.h:655
openvdb::v9_0::math::shear
MatType shear(Axis axis0, Axis axis1, typename MatType::value_type shear)
Set the matrix to a shear along axis0 by a fraction of axis1.
Definition: Mat.h:710
openvdb::v9_0::math::isApproxEqual
bool isApproxEqual(const Type &a, const Type &b, const Type &tolerance)
Return true if a is equal to b to within the given tolerance.
Definition: Math.h:407
openvdb::v9_0::math::lOneNorm
MatType::ValueType lOneNorm(const MatType &matrix)
Return the L1 norm of an N×N matrix.
Definition: Mat.h:963
openvdb::v9_0::math::cwiseGreaterThan
bool cwiseGreaterThan(const Mat< SIZE, T > &m0, const Mat< SIZE, T > &m1)
Definition: Mat.h:1051
openvdb::v9_0::math::scale
MatType scale(const Vec3< typename MatType::value_type > &s)
Return a matrix that scales by s.
Definition: Mat.h:637
openvdb::v9_0::math::sqrtSolve
void sqrtSolve(const MatType &aA, MatType &aB, double aTol=0.01)
Solve for A=B*B, given A.
Definition: Mat.h:819
openvdb::v9_0::math::polarDecomposition
bool polarDecomposition(const MatType &input, MatType &unitary, MatType &positive_hermitian, unsigned int MAX_ITERATIONS=100)
Decompose an invertible 3×3 matrix into a unitary matrix followed by a symmetric matrix (positive sem...
Definition: Mat.h:990
openvdb::v9_0::math::snapMatBasis
MatType snapMatBasis(const MatType &source, Axis axis, const Vec3< typename MatType::value_type > &direction)
This function snaps a specific axis to a specific direction, preserving scaling.
Definition: Mat.h:773
openvdb::v9_0::math::isNan
bool isNan(const float x)
Return true if x is a NaN (Not-A-Number) value.
Definition: Math.h:396
openvdb::v9_0::math::Axis
Axis
Definition: Math.h:904
openvdb::v9_0::math::Z_AXIS
@ Z_AXIS
Definition: Math.h:907
openvdb::v9_0::math::X_AXIS
@ X_AXIS
Definition: Math.h:905
openvdb::v9_0::math::Y_AXIS
@ Y_AXIS
Definition: Math.h:906
openvdb::v9_0::math::isDiagonal
bool isDiagonal(const MatType &mat)
Determine if a matrix is diagonal.
Definition: Mat.h:924
openvdb::v9_0::math::lInfinityNorm
MatType::ValueType lInfinityNorm(const MatType &matrix)
Return the L∞ norm of an N×N matrix.
Definition: Mat.h:942
openvdb::v9_0::math::RotationOrder
RotationOrder
Definition: Math.h:911
openvdb::v9_0::math::YXZ_ROTATION
@ YXZ_ROTATION
Definition: Math.h:914
openvdb::v9_0::math::ZXY_ROTATION
@ ZXY_ROTATION
Definition: Math.h:916
openvdb::v9_0::math::YZX_ROTATION
@ YZX_ROTATION
Definition: Math.h:915
openvdb::v9_0::math::ZXZ_ROTATION
@ ZXZ_ROTATION
Definition: Math.h:919
openvdb::v9_0::math::XZX_ROTATION
@ XZX_ROTATION
Definition: Math.h:918
openvdb::v9_0::math::XYZ_ROTATION
@ XYZ_ROTATION
Definition: Math.h:912
openvdb::v9_0::math::ZYX_ROTATION
@ ZYX_ROTATION
Definition: Math.h:917
openvdb::v9_0::math::XZY_ROTATION
@ XZY_ROTATION
Definition: Math.h:913
openvdb::v9_0::math::isIdentity
bool isIdentity(const MatType &m)
Determine if a matrix is an identity matrix.
Definition: Mat.h:882
openvdb::v9_0::tools::composite::max
const std::enable_if<!VecTraits< T >::IsVec, T >::type & max(const T &a, const T &b)
Definition: Composite.h:107
openvdb
Definition: Exceptions.h:13
Exceptions.h
OPENVDB_THROW
#define OPENVDB_THROW(exception, message)
Definition: Exceptions.h:74
openvdb::v9_0::math::Tolerance
Tolerance for floating-point comparison.
Definition: Math.h:147
OPENVDB_VERSION_NAME
#define OPENVDB_VERSION_NAME
The version namespace name for this library version.
Definition: version.h.in:116
OPENVDB_USE_VERSION_NAMESPACE
#define OPENVDB_USE_VERSION_NAMESPACE
Definition: version.h.in:202