OpenVDB: Mat.h Source File

OpenVDB 5.0.0

Mat.h
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33
34 #ifndef OPENVDB_MATH_MAT_HAS_BEEN_INCLUDED
35 #define OPENVDB_MATH_MAT_HAS_BEEN_INCLUDED
36
37 #include "Math.h"
38 #include <openvdb/Exceptions.h>
39 #include <algorithm> // for std::max()
40 #include <cmath>
41 #include <iostream>
42 #include <string>
43
44
45 namespace openvdb {
46 OPENVDB_USE_VERSION_NAMESPACE
47 namespace OPENVDB_VERSION_NAME {
48 namespace math {
49
52 template<unsigned SIZE, typename T>
53 class Mat
54 {
55 public:
56 using value_type = T;
57 using ValueType = T;
58 enum SIZE_ { size = SIZE };
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
67 Mat() { }
68
70 Mat(Mat const &src) {
71 for (unsigned i(0); i < numElements(); ++i) {
72 mm[i] = src.mm[i];
73 }
74 }
75
76 Mat& operator=(Mat const& src) {
77 if (&src != this) {
78 for (unsigned i = 0; i < numElements(); ++i) {
79 mm[i] = src.mm[i];
80 }
81 }
82 return *this;
83 }
84
94 std::string
95 str(unsigned indentation = 0) const {
96
97 std::string ret;
98 std::string indent;
99
100 // We add +1 since we're indenting one for the first '['
101 indent.append(indentation+1, ' ');
102
103 ret.append("[");
104
105 // For each row,
106 for (unsigned i(0); i < SIZE; i++) {
107
108 ret.append("[");
109
110 // For each column
111 for (unsigned j(0); j < SIZE; j++) {
112
113 // Put a comma after everything except the last
114 if (j) ret.append(", ");
115 ret.append(std::to_string(mm[(i*SIZE)+j]));
116 }
117
118 ret.append("]");
119
120 // At the end of every row (except the last)...
121 if (i < SIZE - 1) {
122 // ...suffix the row bracket with a comma, newline, and advance indentation.
123 ret.append(",\n");
124 ret.append(indent);
125 }
126 }
127
128 ret.append("]");
129
130 return ret;
131 }
132
134 friend std::ostream& operator<<(
135 std::ostream& ostr,
136 const Mat<SIZE, T>& m)
137 {
138 ostr << m.str();
139 return ostr;
140 }
141
142 void write(std::ostream& os) const {
143 os.write(reinterpret_cast<const char*>(&mm), sizeof(T)*SIZE*SIZE);
144 }
145
146 void read(std::istream& is) {
147 is.read(reinterpret_cast<char*>(&mm), sizeof(T)*SIZE*SIZE);
148 }
149
151 T absMax() const {
152 T x = static_cast<T>(std::fabs(mm[0]));
153 for (unsigned i = 1; i < numElements(); ++i) {
154 x = std::max(x, static_cast<T>(std::fabs(mm[i])));
155 }
156 return x;
157 }
158
160 bool isNan() const {
161 for (unsigned i = 0; i < numElements(); ++i) {
162 if (std::isnan(mm[i])) return true;
163 }
164 return false;
165 }
166
168 bool isInfinite() const {
169 for (unsigned i = 0; i < numElements(); ++i) {
170 if (std::isinf(mm[i])) return true;
171 }
172 return false;
173 }
174
176 bool isFinite() const {
177 for (unsigned i = 0; i < numElements(); ++i) {
178 if (!std::isfinite(mm[i])) return false;
179 }
180 return true;
181 }
182
184 bool isZero() const {
185 for (unsigned i = 0; i < numElements(); ++i) {
186 if (!isZero(mm[i])) return false;
187 }
188 return true;
189 }
190
191 protected:
192 T mm[SIZE*SIZE];
193 };
194
195
196 template<typename T> class Quat;
197 template<typename T> class Vec3;
198
202 template<class MatType>
203 MatType
204 rotation(const Quat<typename MatType::value_type> &q,
205 typename MatType::value_type eps = static_cast<typename MatType::value_type>(1.0e-8))
206 {
207 using T = typename MatType::value_type;
208
209 T qdot(q.dot(q));
210 T s(0);
211
212 if (!isApproxEqual(qdot, T(0.0),eps)) {
213 s = T(2.0 / qdot);
214 }
215
216 T x = s*q.x();
217 T y = s*q.y();
218 T z = s*q.z();
219 T wx = x*q.w();
220 T wy = y*q.w();
221 T wz = z*q.w();
222 T xx = x*q.x();
223 T xy = y*q.x();
224 T xz = z*q.x();
225 T yy = y*q.y();
226 T yz = z*q.y();
227 T zz = z*q.z();
228
229 MatType r;
230 r[0][0]=T(1) - (yy+zz); r[0][1]=xy + wz; r[0][2]=xz - wy;
231 r[1][0]=xy - wz; r[1][1]=T(1) - (xx+zz); r[1][2]=yz + wx;
232 r[2][0]=xz + wy; r[2][1]=yz - wx; r[2][2]=T(1) - (xx+yy);
233
234 if(MatType::numColumns() == 4) padMat4(r);
235 return r;
236 }
237
238
239
243 template<class MatType>
244 MatType
245 rotation(Axis axis, typename MatType::value_type angle)
246 {
247 using T = typename MatType::value_type;
248 T c = static_cast<T>(cos(angle));
249 T s = static_cast<T>(sin(angle));
250
251 MatType result;
252 result.setIdentity();
253
254 switch (axis) {
255 case X_AXIS:
256 result[1][1] = c;
257 result[1][2] = s;
258 result[2][1] = -s;
259 result[2][2] = c;
260 return result;
261 case Y_AXIS:
262 result[0][0] = c;
263 result[0][2] = -s;
264 result[2][0] = s;
265 result[2][2] = c;
266 return result;
267 case Z_AXIS:
268 result[0][0] = c;
269 result[0][1] = s;
270 result[1][0] = -s;
271 result[1][1] = c;
272 return result;
273 default:
274 throw ValueError("Unrecognized rotation axis");
275 }
276 }
277
278
281 template<class MatType>
282 MatType
283 rotation(const Vec3<typename MatType::value_type> &_axis, typename MatType::value_type angle)
284 {
285 using T = typename MatType::value_type;
286 T txy, txz, tyz, sx, sy, sz;
287
288 Vec3<T> axis(_axis.unit());
289
290 // compute trig properties of angle:
291 T c(cos(double(angle)));
292 T s(sin(double(angle)));
293 T t(1 - c);
294
295 MatType result;
296 // handle diagonal elements
297 result[0][0] = axis[0]*axis[0] * t + c;
298 result[1][1] = axis[1]*axis[1] * t + c;
299 result[2][2] = axis[2]*axis[2] * t + c;
300
301 txy = axis[0]*axis[1] * t;
302 sz = axis[2] * s;
303
304 txz = axis[0]*axis[2] * t;
305 sy = axis[1] * s;
306
307 tyz = axis[1]*axis[2] * t;
308 sx = axis[0] * s;
309
310 // right handed space
311 // Contribution from rotation about 'z'
312 result[0][1] = txy + sz;
313 result[1][0] = txy - sz;
314 // Contribution from rotation about 'y'
315 result[0][2] = txz - sy;
316 result[2][0] = txz + sy;
317 // Contribution from rotation about 'x'
318 result[1][2] = tyz + sx;
319 result[2][1] = tyz - sx;
320
321 if(MatType::numColumns() == 4) padMat4(result);
322 return MatType(result);
323 }
324
325
363 template<class MatType>
364 Vec3<typename MatType::value_type>
365 eulerAngles(
366 const MatType& mat,
367 RotationOrder rotationOrder,
368 typename MatType::value_type eps = static_cast<typename MatType::value_type>(1.0e-8))
369 {
370 using ValueType = typename MatType::value_type;
371 using V = Vec3<ValueType>;
372 ValueType phi, theta, psi;
373
374 switch(rotationOrder)
375 {
376 case XYZ_ROTATION:
377 if (isApproxEqual(mat[2][0], ValueType(1.0), eps)) {
378 theta = ValueType(M_PI_2);
379 phi = ValueType(0.5 * atan2(mat[1][2], mat[1][1]));
380 psi = phi;
381 } else if (isApproxEqual(mat[2][0], ValueType(-1.0), eps)) {
382 theta = ValueType(-M_PI_2);
383 phi = ValueType(0.5 * atan2(mat[1][2], mat[1][1]));
384 psi = -phi;
385 } else {
386 psi = ValueType(atan2(-mat[1][0],mat[0][0]));
387 phi = ValueType(atan2(-mat[2][1],mat[2][2]));
388 theta = ValueType(atan2(mat[2][0],
389 sqrt( mat[2][1]*mat[2][1] +
390 mat[2][2]*mat[2][2])));
391 }
392 return V(phi, theta, psi);
393 case ZXY_ROTATION:
394 if (isApproxEqual(mat[1][2], ValueType(1.0), eps)) {
395 theta = ValueType(M_PI_2);
396 phi = ValueType(0.5 * atan2(mat[0][1], mat[0][0]));
397 psi = phi;
398 } else if (isApproxEqual(mat[1][2], ValueType(-1.0), eps)) {
399 theta = ValueType(-M_PI/2);
400 phi = ValueType(0.5 * atan2(mat[0][1],mat[2][1]));
401 psi = -phi;
402 } else {
403 psi = ValueType(atan2(-mat[0][2], mat[2][2]));
404 phi = ValueType(atan2(-mat[1][0], mat[1][1]));
405 theta = ValueType(atan2(mat[1][2],
406 sqrt(mat[0][2] * mat[0][2] +
407 mat[2][2] * mat[2][2])));
408 }
409 return V(theta, psi, phi);
410
411 case YZX_ROTATION:
412 if (isApproxEqual(mat[0][1], ValueType(1.0), eps)) {
413 theta = ValueType(M_PI_2);
414 phi = ValueType(0.5 * atan2(mat[2][0], mat[2][2]));
415 psi = phi;
416 } else if (isApproxEqual(mat[0][1], ValueType(-1.0), eps)) {
417 theta = ValueType(-M_PI/2);
418 phi = ValueType(0.5 * atan2(mat[2][0], mat[1][0]));
419 psi = -phi;
420 } else {
421 psi = ValueType(atan2(-mat[2][1], mat[1][1]));
422 phi = ValueType(atan2(-mat[0][2], mat[0][0]));
423 theta = ValueType(atan2(mat[0][1],
424 sqrt(mat[0][0] * mat[0][0] +
425 mat[0][2] * mat[0][2])));
426 }
427 return V(psi, phi, theta);
428
429 case XZX_ROTATION:
430
431 if (isApproxEqual(mat[0][0], ValueType(1.0), eps)) {
432 theta = ValueType(0.0);
433 phi = ValueType(0.5 * atan2(mat[1][2], mat[1][1]));
434 psi = phi;
435 } else if (isApproxEqual(mat[0][0], ValueType(-1.0), eps)) {
436 theta = ValueType(M_PI);
437 psi = ValueType(0.5 * atan2(mat[2][1], -mat[1][1]));
438 phi = - psi;
439 } else {
440 psi = ValueType(atan2(mat[2][0], -mat[1][0]));
441 phi = ValueType(atan2(mat[0][2], mat[0][1]));
442 theta = ValueType(atan2(sqrt(mat[0][1] * mat[0][1] +
443 mat[0][2] * mat[0][2]),
444 mat[0][0]));
445 }
446 return V(phi, psi, theta);
447
448 case ZXZ_ROTATION:
449
450 if (isApproxEqual(mat[2][2], ValueType(1.0), eps)) {
451 theta = ValueType(0.0);
452 phi = ValueType(0.5 * atan2(mat[0][1], mat[0][0]));
453 psi = phi;
454 } else if (isApproxEqual(mat[2][2], ValueType(-1.0), eps)) {
455 theta = ValueType(M_PI);
456 phi = ValueType(0.5 * atan2(mat[0][1], mat[0][0]));
457 psi = -phi;
458 } else {
459 psi = ValueType(atan2(mat[0][2], mat[1][2]));
460 phi = ValueType(atan2(mat[2][0], -mat[2][1]));
461 theta = ValueType(atan2(sqrt(mat[0][2] * mat[0][2] +
462 mat[1][2] * mat[1][2]),
463 mat[2][2]));
464 }
465 return V(theta, psi, phi);
466
467 case YXZ_ROTATION:
468
469 if (isApproxEqual(mat[2][1], ValueType(1.0), eps)) {
470 theta = ValueType(-M_PI_2);
471 phi = ValueType(0.5 * atan2(-mat[1][0], mat[0][0]));
472 psi = phi;
473 } else if (isApproxEqual(mat[2][1], ValueType(-1.0), eps)) {
474 theta = ValueType(M_PI_2);
475 phi = ValueType(0.5 * atan2(mat[1][0], mat[0][0]));
476 psi = -phi;
477 } else {
478 psi = ValueType(atan2(mat[0][1], mat[1][1]));
479 phi = ValueType(atan2(mat[2][0], mat[2][2]));
480 theta = ValueType(atan2(-mat[2][1],
481 sqrt(mat[0][1] * mat[0][1] +
482 mat[1][1] * mat[1][1])));
483 }
484 return V(theta, phi, psi);
485
486 case ZYX_ROTATION:
487
488 if (isApproxEqual(mat[0][2], ValueType(1.0), eps)) {
489 theta = ValueType(-M_PI_2);
490 phi = ValueType(0.5 * atan2(-mat[1][0], mat[1][1]));
491 psi = phi;
492 } else if (isApproxEqual(mat[0][2], ValueType(-1.0), eps)) {
493 theta = ValueType(M_PI_2);
494 phi = ValueType(0.5 * atan2(mat[2][1], mat[2][0]));
495 psi = -phi;
496 } else {
497 psi = ValueType(atan2(mat[1][2], mat[2][2]));
498 phi = ValueType(atan2(mat[0][1], mat[0][0]));
499 theta = ValueType(atan2(-mat[0][2],
500 sqrt(mat[0][1] * mat[0][1] +
501 mat[0][0] * mat[0][0])));
502 }
503 return V(psi, theta, phi);
504
505 case XZY_ROTATION:
506
507 if (isApproxEqual(mat[1][0], ValueType(-1.0), eps)) {
508 theta = ValueType(M_PI_2);
509 psi = ValueType(0.5 * atan2(mat[2][1], mat[2][2]));
510 phi = -psi;
511 } else if (isApproxEqual(mat[1][0], ValueType(1.0), eps)) {
512 theta = ValueType(-M_PI_2);
513 psi = ValueType(0.5 * atan2(- mat[2][1], mat[2][2]));
514 phi = psi;
515 } else {
516 psi = ValueType(atan2(mat[2][0], mat[0][0]));
517 phi = ValueType(atan2(mat[1][2], mat[1][1]));
518 theta = ValueType(atan2(- mat[1][0],
519 sqrt(mat[1][1] * mat[1][1] +
520 mat[1][2] * mat[1][2])));
521 }
522 return V(phi, psi, theta);
523 }
524
525 OPENVDB_THROW(NotImplementedError, "Euler extraction sequence not implemented");
526 }
527
528
531 template<class MatType>
532 MatType
533 rotation(
534 const Vec3<typename MatType::value_type>& _v1,
535 const Vec3<typename MatType::value_type>& _v2,
536 typename MatType::value_type eps=1.0e-8)
537 {
538 using T = typename MatType::value_type;
539 Vec3<T> v1(_v1);
540 Vec3<T> v2(_v2);
541
542 // Check if v1 and v2 are unit length
543 if (!isApproxEqual(1.0, v1.dot(v1), eps)) {
544 v1.normalize();
545 }
546 if (!isApproxEqual(1.0, v2.dot(v2), eps)) {
547 v2.normalize();
548 }
549
550 Vec3<T> cross;
551 cross.cross(v1, v2);
552
553 if (isApproxEqual(cross[0], 0.0, eps) &&
554 isApproxEqual(cross[1], 0.0, eps) &&
555 isApproxEqual(cross[2], 0.0, eps)) {
556
557
558 // Given two unit vectors v1 and v2 that are nearly parallel, build a
559 // rotation matrix that maps v1 onto v2. First find which principal axis
560 // p is closest to perpendicular to v1. Find a reflection that exchanges
561 // v1 and p, and find a reflection that exchanges p2 and v2. The desired
562 // rotation matrix is the composition of these two reflections. See the
563 // paper "Efficiently Building a Matrix to Rotate One Vector to
564 // Another" by Tomas Moller and John Hughes in Journal of Graphics
565 // Tools Vol 4, No 4 for details.
566
567 Vec3<T> u, v, p(0.0, 0.0, 0.0);
568
569 double x = Abs(v1[0]);
570 double y = Abs(v1[1]);
571 double z = Abs(v1[2]);
572
573 if (x < y) {
574 if (z < x) {
575 p[2] = 1;
576 } else {
577 p[0] = 1;
578 }
579 } else {
580 if (z < y) {
581 p[2] = 1;
582 } else {
583 p[1] = 1;
584 }
585 }
586 u = p - v1;
587 v = p - v2;
588
589 double udot = u.dot(u);
590 double vdot = v.dot(v);
591
592 double a = -2 / udot;
593 double b = -2 / vdot;
594 double c = 4 * u.dot(v) / (udot * vdot);
595
596 MatType result;
597 result.setIdentity();
598
599 for (int j = 0; j < 3; j++) {
600 for (int i = 0; i < 3; i++)
601 result[i][j] =
602 a * u[i] * u[j] + b * v[i] * v[j] + c * v[j] * u[i];
603 }
604 result[0][0] += 1.0;
605 result[1][1] += 1.0;
606 result[2][2] += 1.0;
607
608 if(MatType::numColumns() == 4) padMat4(result);
609 return result;
610
611 } else {
612 double c = v1.dot(v2);
613 double a = (1.0 - c) / cross.dot(cross);
614
615 double a0 = a * cross[0];
616 double a1 = a * cross[1];
617 double a2 = a * cross[2];
618
619 double a01 = a0 * cross[1];
620 double a02 = a0 * cross[2];
621 double a12 = a1 * cross[2];
622
623 MatType r;
624
625 r[0][0] = c + a0 * cross[0];
626 r[0][1] = a01 + cross[2];
627 r[0][2] = a02 - cross[1],
628 r[1][0] = a01 - cross[2];
629 r[1][1] = c + a1 * cross[1];
630 r[1][2] = a12 + cross[0];
631 r[2][0] = a02 + cross[1];
632 r[2][1] = a12 - cross[0];
633 r[2][2] = c + a2 * cross[2];
634
635 if(MatType::numColumns() == 4) padMat4(r);
636 return r;
637
638 }
639 }
640
641
643 template<class MatType>
644 MatType
645 scale(const Vec3<typename MatType::value_type>& s)
646 {
647 // Gets identity, then sets top 3 diagonal
648 // Inefficient by 3 sets.
649
650 MatType result;
651 result.setIdentity();
652 result[0][0] = s[0];
653 result[1][1] = s[1];
654 result[2][2] = s[2];
655
656 return result;
657 }
658
659
661 template<class MatType>
662 Vec3<typename MatType::value_type>
663 getScale(const MatType &mat)
664 {
665 using V = Vec3<typename MatType::value_type>;
666 return V(
667 V(mat[0][0], mat[0][1], mat[0][2]).length(),
668 V(mat[1][0], mat[1][1], mat[1][2]).length(),
669 V(mat[2][0], mat[2][1], mat[2][2]).length());
670 }
671
672
676 template<class MatType>
677 MatType
678 unit(const MatType &mat, typename MatType::value_type eps = 1.0e-8)
679 {
680 Vec3<typename MatType::value_type> dud;
681 return unit(mat, eps, dud);
682 }
683
684
689 template<class MatType>
690 MatType
691 unit(
692 const MatType &in,
693 typename MatType::value_type eps,
694 Vec3<typename MatType::value_type>& scaling)
695 {
696 using T = typename MatType::value_type;
697 MatType result(in);
698
699 for (int i(0); i < 3; i++) {
700 try {
701 const Vec3<T> u(
702 Vec3<T>(in[i][0], in[i][1], in[i][2]).unit(eps, scaling[i]));
703 for (int j=0; j<3; j++) result[i][j] = u[j];
704 } catch (ArithmeticError&) {
705 for (int j=0; j<3; j++) result[i][j] = 0;
706 }
707 }
708 return result;
709 }
710
711
716 template <class MatType>
717 MatType
718 shear(Axis axis0, Axis axis1, typename MatType::value_type shear)
719 {
720 int index0 = static_cast<int>(axis0);
721 int index1 = static_cast<int>(axis1);
722
723 MatType result;
724 result.setIdentity();
725 if (axis0 == axis1) {
726 result[index1][index0] = shear + 1;
727 } else {
728 result[index1][index0] = shear;
729 }
730
731 return result;
732 }
733
734
736 template<class MatType>
737 MatType
738 skew(const Vec3<typename MatType::value_type> &skew)
739 {
740 using T = typename MatType::value_type;
741
742 MatType r;
743 r[0][0] = T(0); r[0][1] = skew.z(); r[0][2] = -skew.y();
744 r[1][0] = -skew.z(); r[1][1] = T(0); r[2][1] = skew.x();
745 r[2][0] = skew.y(); r[2][1] = -skew.x(); r[2][2] = T(0);
746
747 if(MatType::numColumns() == 4) padMat4(r);
748 return r;
749 }
750
751
754 template<class MatType>
755 MatType
756 aim(const Vec3<typename MatType::value_type>& direction,
757 const Vec3<typename MatType::value_type>& vertical)
758 {
759 using T = typename MatType::value_type;
760 Vec3<T> forward(direction.unit());
761 Vec3<T> horizontal(vertical.unit().cross(forward).unit());
762 Vec3<T> up(forward.cross(horizontal).unit());
763
764 MatType r;
765
766 r[0][0]=horizontal.x(); r[0][1]=horizontal.y(); r[0][2]=horizontal.z();
767 r[1][0]=up.x(); r[1][1]=up.y(); r[1][2]=up.z();
768 r[2][0]=forward.x(); r[2][1]=forward.y(); r[2][2]=forward.z();
769
770 if(MatType::numColumns() == 4) padMat4(r);
771 return r;
772 }
773
779 template<class MatType>
780 inline MatType
781 snapMatBasis(const MatType& source, Axis axis, const Vec3<typename MatType::value_type>& direction)
782 {
783 using T = typename MatType::value_type;
784
785 Vec3<T> unitDir(direction.unit());
786 Vec3<T> ourUnitAxis(source.row(axis).unit());
787
788 // Are the two parallel?
789 T parallel = unitDir.dot(ourUnitAxis);
790
791 // Already snapped!
792 if (isApproxEqual(parallel, T(1.0))) return source;
793
794 if (isApproxEqual(parallel, T(-1.0))) {
795 OPENVDB_THROW(ValueError, "Cannot snap to inverse axis");
796 }
797
798 // Find angle between our basis and the one specified
799 T angleBetween(angle(unitDir, ourUnitAxis));
800 // Caclulate axis to rotate along
801 Vec3<T> rotationAxis = unitDir.cross(ourUnitAxis);
802
803 MatType rotation;
804 rotation.setToRotation(rotationAxis, angleBetween);
805
806 return source * rotation;
807 }
808
811 template<class MatType>
812 static MatType&
813 padMat4(MatType& dest)
814 {
815 dest[0][3] = dest[1][3] = dest[2][3] = 0;
816 dest[3][2] = dest[3][1] = dest[3][0] = 0;
817 dest[3][3] = 1;
818
819 return dest;
820 }
821
822
825 template<typename MatType>
826 inline void
827 sqrtSolve(const MatType& aA, MatType& aB, double aTol=0.01)
828 {
829 unsigned int iterations = static_cast<unsigned int>(log(aTol)/log(0.5));
830
831 MatType Y[2], Z[2];
832 Y[0] = aA;
833 Z[0] = MatType::identity();
834
835 unsigned int current = 0;
836 for (unsigned int iteration=0; iteration < iterations; iteration++) {
837 unsigned int last = current;
838 current = !current;
839
840 MatType invY = Y[last].inverse();
841 MatType invZ = Z[last].inverse();
842
843 Y[current] = 0.5 * (Y[last] + invZ);
844 Z[current] = 0.5 * (Z[last] + invY);
845 }
846 aB = Y[current];
847 }
848
849
850 template<typename MatType>
851 inline void
852 powSolve(const MatType& aA, MatType& aB, double aPower, double aTol=0.01)
853 {
854 unsigned int iterations = static_cast<unsigned int>(log(aTol)/log(0.5));
855
856 const bool inverted = (aPower < 0.0);
857 if (inverted) { aPower = -aPower; }
858
859 unsigned int whole = static_cast<unsigned int>(aPower);
860 double fraction = aPower - whole;
861
862 MatType R = MatType::identity();
863 MatType partial = aA;
864
865 double contribution = 1.0;
866 for (unsigned int iteration = 0; iteration < iterations; iteration++) {
867 sqrtSolve(partial, partial, aTol);
868 contribution *= 0.5;
869 if (fraction >= contribution) {
870 R *= partial;
871 fraction -= contribution;
872 }
873 }
874
875 partial = aA;
876 while (whole) {
877 if (whole & 1) { R *= partial; }
878 whole >>= 1;
879 if (whole) { partial *= partial; }
880 }
881
882 if (inverted) { aB = R.inverse(); }
883 else { aB = R; }
884 }
885
886
888 template<typename MatType>
889 inline bool
890 isIdentity(const MatType& m)
891 {
892 return m.eq(MatType::identity());
893 }
894
895
897 template<typename MatType>
898 inline bool
899 isInvertible(const MatType& m)
900 {
901 using ValueType = typename MatType::ValueType;
902 return !isApproxEqual(m.det(), ValueType(0));
903 }
904
905
908 template<typename MatType>
909 inline bool
910 isSymmetric(const MatType& m)
911 {
912 return m.eq(m.transpose());
913 }
914
915
917 template<typename MatType>
918 inline bool
919 isUnitary(const MatType& m)
920 {
921 using ValueType = typename MatType::ValueType;
922 if (!isApproxEqual(std::abs(m.det()), ValueType(1.0))) return false;
923 // check that the matrix transpose is the inverse
924 MatType temp = m * m.transpose();
925 return temp.eq(MatType::identity());
926 }
927
928
930 template<typename MatType>
931 inline bool
932 isDiagonal(const MatType& mat)
933 {
934 int n = MatType::size;
935 typename MatType::ValueType temp(0);
936 for (int i = 0; i < n; ++i) {
937 for (int j = 0; j < n; ++j) {
938 if (i != j) {
939 temp += std::abs(mat(i,j));
940 }
941 }
942 }
943 return isApproxEqual(temp, typename MatType::ValueType(0.0));
944 }
945
946
948 template<typename MatType>
949 typename MatType::ValueType
950 lInfinityNorm(const MatType& matrix)
951 {
952 int n = MatType::size;
953 typename MatType::ValueType norm = 0;
954
955 for( int j = 0; j<n; ++j) {
956 typename MatType::ValueType column_sum = 0;
957
958 for (int i = 0; i<n; ++i) {
959 column_sum += fabs(matrix(i,j));
960 }
961 norm = std::max(norm, column_sum);
962 }
963
964 return norm;
965 }
966
967
969 template<typename MatType>
970 typename MatType::ValueType
971 lOneNorm(const MatType& matrix)
972 {
973 int n = MatType::size;
974 typename MatType::ValueType norm = 0;
975
976 for( int i = 0; i<n; ++i) {
977 typename MatType::ValueType row_sum = 0;
978
979 for (int j = 0; j<n; ++j) {
980 row_sum += fabs(matrix(i,j));
981 }
982 norm = std::max(norm, row_sum);
983 }
984
985 return norm;
986 }
987
988
996 template<typename MatType>
997 bool
998 polarDecomposition(const MatType& input, MatType& unitary,
999 MatType& positive_hermitian, unsigned int MAX_ITERATIONS=100)
1000 {
1001 unitary = input;
1002 MatType new_unitary(input);
1003 MatType unitary_inv;
1004
1005 if (fabs(unitary.det()) < math::Tolerance<typename MatType::ValueType>::value()) return false;
1006
1007 unsigned int iteration(0);
1008
1009 typename MatType::ValueType linf_of_u;
1010 typename MatType::ValueType l1nm_of_u;
1011 typename MatType::ValueType linf_of_u_inv;
1012 typename MatType::ValueType l1nm_of_u_inv;
1013 typename MatType::ValueType l1_error = 100;
1014 double gamma;
1015
1016 do {
1017 unitary_inv = unitary.inverse();
1018 linf_of_u = lInfinityNorm(unitary);
1019 l1nm_of_u = lOneNorm(unitary);
1020
1021 linf_of_u_inv = lInfinityNorm(unitary_inv);
1022 l1nm_of_u_inv = lOneNorm(unitary_inv);
1023
1024 gamma = sqrt( sqrt( (l1nm_of_u_inv * linf_of_u_inv ) / (l1nm_of_u * linf_of_u) ));
1025
1026 new_unitary = 0.5*(gamma * unitary + (1./gamma) * unitary_inv.transpose() );
1027
1028 l1_error = lInfinityNorm(unitary - new_unitary);
1029 unitary = new_unitary;
1030
1032 if (iteration > MAX_ITERATIONS) return false;
1033 iteration++;
1034 } while (l1_error > math::Tolerance<typename MatType::ValueType>::value());
1035
1036 positive_hermitian = unitary.transpose() * input;
1037 return true;
1038 }
1039
1040 } // namespace math
1041 } // namespace OPENVDB_VERSION_NAME
1042 } // namespace openvdb
1043
1044 #endif // OPENVDB_MATH_MAT_HAS_BEEN_INCLUDED
1045
1046 // Copyright (c) 2012-2017 DreamWorks Animation LLC
1047 // All rights reserved. This software is distributed under the
1048 // Mozilla Public License 2.0 ( http://www.mozilla.org/MPL/2.0/ )
openvdb::v5_0::math::Vec3::y
T & y()
Definition: Vec3.h:111
openvdb::v5_0::NotImplementedError
Definition: Exceptions.h:87
openvdb::v5_0::math::ZXZ_ROTATION
Definition: Math.h:867
openvdb::v5_0::math::Quat
Definition: Mat.h:196
openvdb::v5_0::math::padMat4
static MatType & padMat4(MatType &dest)
Write 0s along Mat4&#39;s last row and column, and a 1 on its diagonal.
Definition: Mat.h:813
openvdb::v5_0::math::Mat::numColumns
static unsigned numColumns()
Definition: Mat.h:62
openvdb::v5_0::math::Mat::Mat
Mat(Mat const &src)
Copy constructor. Used when the class signature matches exactly.
Definition: Mat.h:70
openvdb::v5_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:718
openvdb::v5_0::math::rotation
MatType rotation(const Vec3< typename MatType::value_type > &_v1, const Vec3< typename MatType::value_type > &_v2, typename MatType::value_type eps=1.0e-8)
Return a rotation matrix that maps v1 onto v2 about the cross product of v1 and v2.
Definition: Mat.h:533
openvdb::v5_0::tools::composite::max
const std::enable_if<!VecTraits< T >::IsVec, T >::type & max(const T &a, const T &b)
Definition: Composite.h:133
Math.h
General-purpose arithmetic and comparison routines, most of which accept arbitrary value types (or at...
openvdb::v5_0::math::RotationOrder
RotationOrder
Definition: Math.h:859
openvdb::v5_0::math::Mat::isNan
bool isNan() const
True if a Nan is present in this matrix.
Definition: Mat.h:160
openvdb::v5_0::math::Mat::absMax
T absMax() const
Return the maximum of the absolute of all elements in this matrix.
Definition: Mat.h:151
OPENVDB_THROW
#define OPENVDB_THROW(exception, message)
Definition: Exceptions.h:108
openvdb::v5_0::math::YZX_ROTATION
Definition: Math.h:863
openvdb::v5_0::math::isDiagonal
bool isDiagonal(const MatType &mat)
Determine if a matrix is diagonal.
Definition: Mat.h:932
openvdb::v5_0::math::Mat< 3, T >::SIZE_
SIZE_
Definition: Mat.h:58
openvdb::v5_0::math::YXZ_ROTATION
Definition: Math.h:862
openvdb::v5_0::math::Quat::w
T & w()
Definition: Quat.h:227
openvdb::v5_0::math::sqrtSolve
void sqrtSolve(const MatType &aA, MatType &aB, double aTol=0.01)
Solve for A=B*B, given A.
Definition: Mat.h:827
openvdb::v5_0::math::XZY_ROTATION
Definition: Math.h:861
openvdb::v5_0::math::Vec3::unit
Vec3< T > unit(T eps=0) const
return normalized this, throws if null vector
Definition: Vec3.h:389
openvdb::v5_0::math::XYZ_ROTATION
Definition: Math.h:860
openvdb::v5_0::math::Mat::isInfinite
bool isInfinite() const
True if an Inf is present in this matrix.
Definition: Mat.h:168
openvdb::v5_0::math::Quat::x
T & x()
Reference to the component, e.g. q.x() = 4.5f;.
Definition: Quat.h:224
openvdb::v5_0::math::Quat::dot
T dot(const Quat &q) const
Dot product.
Definition: Quat.h:493
openvdb::v5_0::math::isApproxEqual
bool isApproxEqual(const Type &a, const Type &b)
Return true if a is equal to b to within the default floating-point comparison tolerance.
Definition: Math.h:354
openvdb::v5_0::math::Abs
Coord Abs(const Coord &xyz)
Definition: Coord.h:513
openvdb::v5_0::math::Mat< 3, T >::value_type
T value_type
Definition: Mat.h:56
openvdb::v5_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 ...
Definition: Mat.h:691
openvdb::v5_0::math::XZX_ROTATION
Definition: Math.h:866
openvdb::v5_0::math::angle
T angle(const Vec2< T > &v1, const Vec2< T > &v2)
Definition: Vec2.h:472
openvdb::v5_0::math::lOneNorm
MatType::ValueType lOneNorm(const MatType &matrix)
Return the L1 norm of an N×N matrix.
Definition: Mat.h:971
openvdb::v5_0::math::Tolerance
Tolerance for floating-point comparison.
Definition: Math.h:117
Exceptions.h
openvdb::v5_0::math::Mat::mm
T mm[SIZE *SIZE]
Definition: Mat.h:192
OPENVDB_VERSION_NAME
#define OPENVDB_VERSION_NAME
The version namespace name for this library version.
Definition: version.h:136
openvdb::v5_0::math::Axis
Axis
Definition: Math.h:852
openvdb::v5_0::math::Vec3::dot
T dot(const Vec3< T > &v) const
Dot product.
Definition: Vec3.h:216
openvdb::v5_0::math::ZYX_ROTATION
Definition: Math.h:865
openvdb::v5_0::ValueError
Definition: Exceptions.h:91
openvdb::v5_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:365
openvdb::v5_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 se...
Definition: Mat.h:998
openvdb::v5_0::math::Mat::numElements
static unsigned numElements()
Definition: Mat.h:63
openvdb::v5_0::math::isUnitary
bool isUnitary(const MatType &m)
Determine if a matrix is unitary (i.e., rotation or reflection).
Definition: Mat.h:919
openvdb::v5_0::math::Vec3::x
T & x()
Reference to the component, e.g. v.x() = 4.5f;.
Definition: Vec3.h:110
openvdb::v5_0::math::isInvertible
bool isInvertible(const MatType &m)
Determine if a matrix is invertible.
Definition: Mat.h:899
openvdb::v5_0::math::Mat::Mat
Mat()
Definition: Mat.h:67
openvdb
Definition: Exceptions.h:39
openvdb::v5_0::math::isIdentity
bool isIdentity(const MatType &m)
Determine if a matrix is an identity matrix.
Definition: Mat.h:890
openvdb::v5_0::math::Vec3::normalize
bool normalize(T eps=T(1.0e-7))
this = normalized this
Definition: Vec3.h:377
openvdb::v5_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:134
openvdb::v5_0::math::Mat
Definition: Mat.h:53
openvdb::v5_0::math::Z_AXIS
Definition: Math.h:855
openvdb::v5_0::math::Mat::operator=
Mat & operator=(Mat const &src)
Definition: Mat.h:76
openvdb::v5_0::ArithmeticError
Definition: Exceptions.h:82
openvdb::v5_0::math::getScale
Vec3< typename MatType::value_type > getScale(const MatType &mat)
Return a Vec3 representing the lengths of the passed matrix&#39;s upper 3×3&#39;s rows.
Definition: Mat.h:663
openvdb::v5_0::math::Y_AXIS
Definition: Math.h:854
openvdb::v5_0::math::isZero
bool isZero(const Type &x)
Return true if x is exactly equal to zero.
Definition: Math.h:308
openvdb::v5_0::math::Vec3::cross
Vec3< T > cross(const Vec3< T > &v) const
Return the cross product of "this" vector and v;.
Definition: Vec3.h:245
openvdb::v5_0::math::powSolve
void powSolve(const MatType &aA, MatType &aB, double aPower, double aTol=0.01)
Definition: Mat.h:852
openvdb::v5_0::math::ZXY_ROTATION
Definition: Math.h:864
openvdb::v5_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:738
openvdb::v5_0::math::Mat::read
void read(std::istream &is)
Definition: Mat.h:146
openvdb::v5_0::math::Mat::str
std::string str(unsigned indentation=0) const
Definition: Mat.h:95
openvdb::v5_0::math::scale
MatType scale(const Vec3< typename MatType::value_type > &s)
Return a matrix that scales by s.
Definition: Mat.h:645
openvdb::v5_0::math::Mat::numRows
static unsigned numRows()
Definition: Mat.h:61
openvdb::v5_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:756
openvdb::v5_0::math::X_AXIS
Definition: Math.h:853
openvdb::v5_0::math::Mat::isFinite
bool isFinite() const
True if no Nan or Inf values are present.
Definition: Mat.h:176
openvdb::v5_0::math::Mat::isZero
bool isZero() const
True if all elements are exactly zero.
Definition: Mat.h:184
openvdb::v5_0::math::Vec3
Definition: Mat.h:197
OPENVDB_USE_VERSION_NAMESPACE
#define OPENVDB_USE_VERSION_NAMESPACE
Definition: version.h:188
openvdb::v5_0::math::lInfinityNorm
MatType::ValueType lInfinityNorm(const MatType &matrix)
Return the L∞ norm of an N×N matrix.
Definition: Mat.h:950
openvdb::v5_0::math::isSymmetric
bool isSymmetric(const MatType &m)
Determine if a matrix is symmetric.
Definition: Mat.h:910
openvdb::v5_0::math::Quat::z
T & z()
Definition: Quat.h:226
openvdb::v5_0::math::Quat::y
T & y()
Definition: Quat.h:225
openvdb::v5_0::math::Mat::write
void write(std::ostream &os) const
Definition: Mat.h:142
openvdb::v5_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:781
openvdb::v5_0::math::Mat< 3, T >::ValueType
T ValueType
Definition: Mat.h:57
openvdb::v5_0::math::Vec3::z
T & z()
Definition: Vec3.h:112