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 {
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
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>
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
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
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>
663 getScale(const MatType &mat)
664 {
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 {
681  return unit(mat, eps, dud);
682 }
683 
684 
689 template<class MatType>
690 MatType
692  const MatType &in,
693  typename MatType::value_type eps,
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
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
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++;
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/ )
T & y()
Definition: Vec3.h:111
Definition: Exceptions.h:87
Definition: Math.h:867
Definition: Mat.h:196
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
static unsigned numColumns()
Definition: Mat.h:62
Mat(Mat const &src)
Copy constructor. Used when the class signature matches exactly.
Definition: Mat.h:70
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
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
const std::enable_if<!VecTraits< T >::IsVec, T >::type & max(const T &a, const T &b)
Definition: Composite.h:133
General-purpose arithmetic and comparison routines, most of which accept arbitrary value types (or at...
RotationOrder
Definition: Math.h:859
bool isNan() const
True if a Nan is present in this matrix.
Definition: Mat.h:160
T absMax() const
Return the maximum of the absolute of all elements in this matrix.
Definition: Mat.h:151
#define OPENVDB_THROW(exception, message)
Definition: Exceptions.h:108
Definition: Math.h:863
bool isDiagonal(const MatType &mat)
Determine if a matrix is diagonal.
Definition: Mat.h:932
Definition: Math.h:862
T & w()
Definition: Quat.h:227
void sqrtSolve(const MatType &aA, MatType &aB, double aTol=0.01)
Solve for A=B*B, given A.
Definition: Mat.h:827
Definition: Math.h:861
Vec3< T > unit(T eps=0) const
return normalized this, throws if null vector
Definition: Vec3.h:389
Definition: Math.h:860
bool isInfinite() const
True if an Inf is present in this matrix.
Definition: Mat.h:168
T & x()
Reference to the component, e.g. q.x() = 4.5f;.
Definition: Quat.h:224
T dot(const Quat &q) const
Dot product.
Definition: Quat.h:493
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
Coord Abs(const Coord &xyz)
Definition: Coord.h:513
T value_type
Definition: Mat.h:56
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:691
Definition: Math.h:866
T angle(const Vec2< T > &v1, const Vec2< T > &v2)
Definition: Vec2.h:472
MatType::ValueType lOneNorm(const MatType &matrix)
Return the L1 norm of an N×N matrix.
Definition: Mat.h:971
Tolerance for floating-point comparison.
Definition: Math.h:117
T mm[SIZE *SIZE]
Definition: Mat.h:192
#define OPENVDB_VERSION_NAME
The version namespace name for this library version.
Definition: version.h:136
Axis
Definition: Math.h:852
T dot(const Vec3< T > &v) const
Dot product.
Definition: Vec3.h:216
Definition: Math.h:865
Definition: Exceptions.h:91
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
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:998
static unsigned numElements()
Definition: Mat.h:63
bool isUnitary(const MatType &m)
Determine if a matrix is unitary (i.e., rotation or reflection).
Definition: Mat.h:919
T & x()
Reference to the component, e.g. v.x() = 4.5f;.
Definition: Vec3.h:110
bool isInvertible(const MatType &m)
Determine if a matrix is invertible.
Definition: Mat.h:899
Mat()
Definition: Mat.h:67
Definition: Exceptions.h:39
bool isIdentity(const MatType &m)
Determine if a matrix is an identity matrix.
Definition: Mat.h:890
bool normalize(T eps=T(1.0e-7))
this = normalized this
Definition: Vec3.h:377
friend std::ostream & operator<<(std::ostream &ostr, const Mat< SIZE, T > &m)
Write a Mat to an output stream.
Definition: Mat.h:134
Definition: Mat.h:53
Definition: Math.h:855
Mat & operator=(Mat const &src)
Definition: Mat.h:76
Definition: Exceptions.h:82
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
Definition: Math.h:854
bool isZero(const Type &x)
Return true if x is exactly equal to zero.
Definition: Math.h:308
Vec3< T > cross(const Vec3< T > &v) const
Return the cross product of "this" vector and v;.
Definition: Vec3.h:245
void powSolve(const MatType &aA, MatType &aB, double aPower, double aTol=0.01)
Definition: Mat.h:852
Definition: Math.h:864
MatType skew(const Vec3< typename MatType::value_type > &skew)
Return a matrix as the cross product of the given vector.
Definition: Mat.h:738
void read(std::istream &is)
Definition: Mat.h:146
std::string str(unsigned indentation=0) const
Definition: Mat.h:95
MatType scale(const Vec3< typename MatType::value_type > &s)
Return a matrix that scales by s.
Definition: Mat.h:645
static unsigned numRows()
Definition: Mat.h:61
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
Definition: Math.h:853
bool isFinite() const
True if no Nan or Inf values are present.
Definition: Mat.h:176
bool isZero() const
True if all elements are exactly zero.
Definition: Mat.h:184
Definition: Mat.h:197
#define OPENVDB_USE_VERSION_NAMESPACE
Definition: version.h:188
MatType::ValueType lInfinityNorm(const MatType &matrix)
Return the L∞ norm of an N×N matrix.
Definition: Mat.h:950
bool isSymmetric(const MatType &m)
Determine if a matrix is symmetric.
Definition: Mat.h:910
T & z()
Definition: Quat.h:226
T & y()
Definition: Quat.h:225
void write(std::ostream &os) const
Definition: Mat.h:142
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
T ValueType
Definition: Mat.h:57
T & z()
Definition: Vec3.h:112