34 #ifndef OPENVDB_MATH_MAT_HAS_BEEN_INCLUDED 35 #define OPENVDB_MATH_MAT_HAS_BEEN_INCLUDED 52 template<
unsigned SIZE,
typename T>
61 static unsigned numRows() {
return SIZE; }
71 for (
unsigned i(0); i < numElements(); ++i) {
78 for (
unsigned i = 0; i < numElements(); ++i) {
95 str(
unsigned indentation = 0)
const {
101 indent.append(indentation+1,
' ');
106 for (
unsigned i(0); i < SIZE; i++) {
111 for (
unsigned j(0); j < SIZE; j++) {
114 if (j) ret.append(
", ");
115 ret.append(std::to_string(mm[(i*SIZE)+j]));
142 void write(std::ostream& os)
const {
143 os.write(reinterpret_cast<const char*>(&mm),
sizeof(T)*SIZE*SIZE);
147 is.read(reinterpret_cast<char*>(&mm),
sizeof(T)*SIZE*SIZE);
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])));
161 for (
unsigned i = 0; i < numElements(); ++i) {
162 if (std::isnan(mm[i]))
return true;
169 for (
unsigned i = 0; i < numElements(); ++i) {
170 if (std::isinf(mm[i]))
return true;
177 for (
unsigned i = 0; i < numElements(); ++i) {
178 if (!std::isfinite(mm[i]))
return false;
185 for (
unsigned i = 0; i < numElements(); ++i) {
186 if (!
isZero(mm[i]))
return false;
196 template<
typename T>
class Quat;
197 template<
typename T>
class Vec3;
202 template<
class MatType>
205 typename MatType::value_type eps = static_cast<typename MatType::value_type>(1.0e-8))
207 using T =
typename MatType::value_type;
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);
234 if(MatType::numColumns() == 4)
padMat4(r);
243 template<
class MatType>
247 using T =
typename MatType::value_type;
248 T c =
static_cast<T
>(cos(angle));
249 T s =
static_cast<T
>(sin(angle));
252 result.setIdentity();
274 throw ValueError(
"Unrecognized rotation axis");
281 template<
class MatType>
285 using T =
typename MatType::value_type;
286 T txy, txz, tyz, sx, sy, sz;
291 T c(cos(
double(angle)));
292 T s(sin(
double(angle)));
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;
301 txy = axis[0]*axis[1] * t;
304 txz = axis[0]*axis[2] * t;
307 tyz = axis[1]*axis[2] * t;
312 result[0][1] = txy + sz;
313 result[1][0] = txy - sz;
315 result[0][2] = txz - sy;
316 result[2][0] = txz + sy;
318 result[1][2] = tyz + sx;
319 result[2][1] = tyz - sx;
321 if(MatType::numColumns() == 4)
padMat4(result);
322 return MatType(result);
363 template<
class MatType>
368 typename MatType::value_type eps = static_cast<typename MatType::value_type>(1.0e-8))
370 using ValueType =
typename MatType::value_type;
372 ValueType phi, theta, psi;
374 switch(rotationOrder)
378 theta = ValueType(M_PI_2);
379 phi = ValueType(0.5 * atan2(mat[1][2], mat[1][1]));
382 theta = ValueType(-M_PI_2);
383 phi = ValueType(0.5 * atan2(mat[1][2], mat[1][1]));
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])));
392 return V(phi, theta, psi);
395 theta = ValueType(M_PI_2);
396 phi = ValueType(0.5 * atan2(mat[0][1], mat[0][0]));
399 theta = ValueType(-M_PI/2);
400 phi = ValueType(0.5 * atan2(mat[0][1],mat[2][1]));
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])));
409 return V(theta, psi, phi);
413 theta = ValueType(M_PI_2);
414 phi = ValueType(0.5 * atan2(mat[2][0], mat[2][2]));
417 theta = ValueType(-M_PI/2);
418 phi = ValueType(0.5 * atan2(mat[2][0], mat[1][0]));
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])));
427 return V(psi, phi, theta);
432 theta = ValueType(0.0);
433 phi = ValueType(0.5 * atan2(mat[1][2], mat[1][1]));
436 theta = ValueType(M_PI);
437 psi = ValueType(0.5 * atan2(mat[2][1], -mat[1][1]));
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]),
446 return V(phi, psi, theta);
451 theta = ValueType(0.0);
452 phi = ValueType(0.5 * atan2(mat[0][1], mat[0][0]));
455 theta = ValueType(M_PI);
456 phi = ValueType(0.5 * atan2(mat[0][1], mat[0][0]));
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]),
465 return V(theta, psi, phi);
470 theta = ValueType(-M_PI_2);
471 phi = ValueType(0.5 * atan2(-mat[1][0], mat[0][0]));
474 theta = ValueType(M_PI_2);
475 phi = ValueType(0.5 * atan2(mat[1][0], mat[0][0]));
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])));
484 return V(theta, phi, psi);
489 theta = ValueType(-M_PI_2);
490 phi = ValueType(0.5 * atan2(-mat[1][0], mat[1][1]));
493 theta = ValueType(M_PI_2);
494 phi = ValueType(0.5 * atan2(mat[2][1], mat[2][0]));
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])));
503 return V(psi, theta, phi);
508 theta = ValueType(M_PI_2);
509 psi = ValueType(0.5 * atan2(mat[2][1], mat[2][2]));
512 theta = ValueType(-M_PI_2);
513 psi = ValueType(0.5 * atan2(- mat[2][1], mat[2][2]));
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])));
522 return V(phi, psi, theta);
531 template<
class MatType>
536 typename MatType::value_type eps=1.0e-8)
538 using T =
typename MatType::value_type;
567 Vec3<T> u, v, p(0.0, 0.0, 0.0);
569 double x =
Abs(v1[0]);
570 double y =
Abs(v1[1]);
571 double z =
Abs(v1[2]);
589 double udot = u.
dot(u);
590 double vdot = v.
dot(v);
592 double a = -2 / udot;
593 double b = -2 / vdot;
594 double c = 4 * u.
dot(v) / (udot * vdot);
597 result.setIdentity();
599 for (
int j = 0; j < 3; j++) {
600 for (
int i = 0; i < 3; i++)
602 a * u[i] * u[j] + b * v[i] * v[j] + c * v[j] * u[i];
608 if(MatType::numColumns() == 4)
padMat4(result);
612 double c = v1.
dot(v2);
613 double a = (1.0 - c) / cross.
dot(cross);
615 double a0 = a * cross[0];
616 double a1 = a * cross[1];
617 double a2 = a * cross[2];
619 double a01 = a0 * cross[1];
620 double a02 = a0 * cross[2];
621 double a12 = a1 * cross[2];
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];
635 if(MatType::numColumns() == 4)
padMat4(r);
643 template<
class MatType>
651 result.setIdentity();
661 template<
class MatType>
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());
676 template<
class MatType>
678 unit(
const MatType &mat,
typename MatType::value_type eps = 1.0e-8)
681 return unit(mat, eps, dud);
689 template<
class MatType>
693 typename MatType::value_type eps,
696 using T =
typename MatType::value_type;
699 for (
int i(0); i < 3; i++) {
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];
705 for (
int j=0; j<3; j++) result[i][j] = 0;
716 template <
class MatType>
720 int index0 =
static_cast<int>(axis0);
721 int index1 =
static_cast<int>(axis1);
724 result.setIdentity();
725 if (axis0 == axis1) {
726 result[index1][index0] = shear + 1;
728 result[index1][index0] =
shear;
736 template<
class MatType>
740 using T =
typename MatType::value_type;
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);
747 if(MatType::numColumns() == 4)
padMat4(r);
754 template<
class MatType>
759 using T =
typename MatType::value_type;
761 Vec3<T> horizontal(vertical.
unit().cross(forward).unit());
762 Vec3<T> up(forward.cross(horizontal).unit());
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();
770 if(MatType::numColumns() == 4)
padMat4(r);
779 template<
class MatType>
783 using T =
typename MatType::value_type;
786 Vec3<T> ourUnitAxis(source.row(axis).unit());
789 T parallel = unitDir.
dot(ourUnitAxis);
799 T angleBetween(
angle(unitDir, ourUnitAxis));
804 rotation.setToRotation(rotationAxis, angleBetween);
811 template<
class MatType>
815 dest[0][3] = dest[1][3] = dest[2][3] = 0;
816 dest[3][2] = dest[3][1] = dest[3][0] = 0;
825 template<
typename MatType>
827 sqrtSolve(
const MatType& aA, MatType& aB,
double aTol=0.01)
829 unsigned int iterations =
static_cast<unsigned int>(log(aTol)/log(0.5));
833 Z[0] = MatType::identity();
835 unsigned int current = 0;
836 for (
unsigned int iteration=0; iteration < iterations; iteration++) {
837 unsigned int last = current;
840 MatType invY = Y[last].inverse();
841 MatType invZ = Z[last].inverse();
843 Y[current] = 0.5 * (Y[last] + invZ);
844 Z[current] = 0.5 * (Z[last] + invY);
850 template<
typename MatType>
852 powSolve(
const MatType& aA, MatType& aB,
double aPower,
double aTol=0.01)
854 unsigned int iterations =
static_cast<unsigned int>(log(aTol)/log(0.5));
856 const bool inverted = (aPower < 0.0);
857 if (inverted) { aPower = -aPower; }
859 unsigned int whole =
static_cast<unsigned int>(aPower);
860 double fraction = aPower - whole;
862 MatType R = MatType::identity();
863 MatType partial = aA;
865 double contribution = 1.0;
866 for (
unsigned int iteration = 0; iteration < iterations; iteration++) {
869 if (fraction >= contribution) {
871 fraction -= contribution;
877 if (whole & 1) { R *= partial; }
879 if (whole) { partial *= partial; }
882 if (inverted) { aB = R.inverse(); }
888 template<
typename MatType>
892 return m.eq(MatType::identity());
897 template<
typename MatType>
901 using ValueType =
typename MatType::ValueType;
908 template<
typename MatType>
912 return m.eq(m.transpose());
917 template<
typename MatType>
921 using ValueType =
typename MatType::ValueType;
922 if (!
isApproxEqual(std::abs(m.det()), ValueType(1.0)))
return false;
924 MatType temp = m * m.transpose();
925 return temp.eq(MatType::identity());
930 template<
typename MatType>
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) {
939 temp += std::abs(mat(i,j));
943 return isApproxEqual(temp,
typename MatType::ValueType(0.0));
948 template<
typename MatType>
949 typename MatType::ValueType
952 int n = MatType::size;
953 typename MatType::ValueType norm = 0;
955 for(
int j = 0; j<n; ++j) {
956 typename MatType::ValueType column_sum = 0;
958 for (
int i = 0; i<n; ++i) {
959 column_sum += fabs(matrix(i,j));
969 template<
typename MatType>
970 typename MatType::ValueType
973 int n = MatType::size;
974 typename MatType::ValueType norm = 0;
976 for(
int i = 0; i<n; ++i) {
977 typename MatType::ValueType row_sum = 0;
979 for (
int j = 0; j<n; ++j) {
980 row_sum += fabs(matrix(i,j));
996 template<
typename MatType>
999 MatType& positive_hermitian,
unsigned int MAX_ITERATIONS=100)
1002 MatType new_unitary(input);
1003 MatType unitary_inv;
1007 unsigned int iteration(0);
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;
1017 unitary_inv = unitary.inverse();
1022 l1nm_of_u_inv =
lOneNorm(unitary_inv);
1024 gamma = sqrt( sqrt( (l1nm_of_u_inv * linf_of_u_inv ) / (l1nm_of_u * linf_of_u) ));
1026 new_unitary = 0.5*(gamma * unitary + (1./gamma) * unitary_inv.transpose() );
1029 unitary = new_unitary;
1032 if (iteration > MAX_ITERATIONS)
return false;
1036 positive_hermitian = unitary.transpose() * input;
1044 #endif // OPENVDB_MATH_MAT_HAS_BEEN_INCLUDED T & y()
Definition: Vec3.h:111
Definition: Exceptions.h:87
static MatType & padMat4(MatType &dest)
Write 0s along Mat4'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
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
bool isDiagonal(const MatType &mat)
Determine if a matrix is diagonal.
Definition: Mat.h:932
SIZE_
Definition: Mat.h:58
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
Vec3< T > unit(T eps=0) const
return normalized this, throws if null vector
Definition: Vec3.h:389
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
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: 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
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's upper 3×3's rows.
Definition: Mat.h:663
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
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
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
#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