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vtkMath Class Reference

performs common math operations. More...

#include <vtkMath.h>

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List of all members.

Public Methods

virtual const char * GetClassName ()
virtual int IsA (const char *type)

Static Public Methods

vtkMath * New ()
int IsTypeOf (const char *type)
vtkMath * SafeDownCast (vtkObject *o)
float Pi ()
float DegreesToRadians ()
double DoubleDegreesToRadians ()
float Dot (const float x[3], const float y[3])
double Dot (const double x[3], const double y[3])
void Cross (const float x[3], const float y[3], float z[3])
void Cross (const double x[3], const double y[3], double z[3])
float Norm (const float x[3])
double Norm (const double x[3])
float Normalize (float x[3])
double Normalize (double x[3])
void Perpendiculars (const double x[3], double y[3], double z[3], double theta)
void Perpendiculars (const float x[3], float y[3], float z[3], double theta)
float Distance2BetweenPoints (const float x[3], const float y[3])
double Distance2BetweenPoints (const double x[3], const double y[3])
float Dot2D (const float x[3], const float y[3])
double Dot2D (const double x[3], const double y[3])
float Norm2D (const float x[3])
double Norm2D (const double x[3])
float Normalize2D (float x[3])
double Normalize2D (double x[3])
float Determinant2x2 (const float c1[2], const float c2[2])
double Determinant2x2 (double a, double b, double c, double d)
void LUFactor3x3 (float A[3][3], int index[3])
void LUFactor3x3 (double A[3][3], int index[3])
void LUSolve3x3 (const float A[3][3], const int index[3], float x[3])
void LUSolve3x3 (const double A[3][3], const int index[3], double x[3])
void LinearSolve3x3 (const float A[3][3], const float x[3], float y[3])
void LinearSolve3x3 (const double A[3][3], const double x[3], double y[3])
void Multiply3x3 (const float A[3][3], const float x[3], float y[3])
void Multiply3x3 (const double A[3][3], const double x[3], double y[3])
void Multiply3x3 (const float A[3][3], const float B[3][3], float C[3][3])
void Multiply3x3 (const double A[3][3], const double B[3][3], double C[3][3])
void Transpose3x3 (const float A[3][3], float AT[3][3])
void Transpose3x3 (const double A[3][3], double AT[3][3])
void Invert3x3 (const float A[3][3], float AI[3][3])
void Invert3x3 (const double A[3][3], double AI[3][3])
void Identity3x3 (float A[3][3])
void Identity3x3 (double A[3][3])
double Determinant3x3 (float A[3][3])
double Determinant3x3 (double A[3][3])
float Determinant3x3 (const float c1[3], const float c2[3], const float c3[3])
double Determinant3x3 (double a1, double a2, double a3, double b1, double b2, double b3, double c1, double c2, double c3)
void QuaternionToMatrix3x3 (const float quat[4], float A[3][3])
void QuaternionToMatrix3x3 (const double quat[4], double A[3][3])
void Matrix3x3ToQuaternion (const float A[3][3], float quat[4])
void Matrix3x3ToQuaternion (const double A[3][3], double quat[4])
void Orthogonalize3x3 (const float A[3][3], float B[3][3])
void Orthogonalize3x3 (const double A[3][3], double B[3][3])
void Diagonalize3x3 (const float A[3][3], float w[3], float V[3][3])
void Diagonalize3x3 (const double A[3][3], double w[3], double V[3][3])
void SingularValueDecomposition3x3 (const float A[3][3], float U[3][3], float w[3], float VT[3][3])
void SingularValueDecomposition3x3 (const double A[3][3], double U[3][3], double w[3], double VT[3][3])
int SolveLinearSystem (double **A, double *x, int size)
int InvertMatrix (double **A, double **AI, int size)
int InvertMatrix (double **A, double **AI, int size, int *tmp1Size, double *tmp2Size)
int LUFactorLinearSystem (double **A, int *index, int size)
int LUFactorLinearSystem (double **A, int *index, int size, double *tmpSize)
void LUSolveLinearSystem (double **A, int *index, double *x, int size)
double EstimateMatrixCondition (double **A, int size)
void RandomSeed (long s)
float Random ()
float Random (float min, float max)
int Jacobi (float **a, float *w, float **v)
int Jacobi (double **a, double *w, double **v)
int JacobiN (float **a, int n, float *w, float **v)
int JacobiN (double **a, int n, double *w, double **v)
double * SolveCubic (double c0, double c1, double c2, double c3)
double * SolveQuadratic (double c0, double c1, double c2)
double * SolveLinear (double c0, double c1)
int SolveCubic (double c0, double c1, double c2, double c3, double *r1, double *r2, double *r3, int *num_roots)
int SolveQuadratic (double c0, double c1, double c2, double *r1, double *r2, int *num_roots)
int SolveLinear (double c0, double c1, double *r1, int *num_roots)
int SolveLeastSquares (int numberOfSamples, double **xt, int xOrder, double **yt, int yOrder, double **mt)

Protected Methods

 vtkMath ()
 ~vtkMath ()
 vtkMath (const vtkMath &)
void operator= (const vtkMath &)

Static Protected Attributes

long Seed

Detailed Description

performs common math operations.

Date:
2000/12/10 20:08:13
Revision:
1.66

vtkMath is provides methods to perform common math operations. These include providing constants such as Pi; conversion from degrees to radians; vector operations such as dot and cross products and vector norm; matrix determinant for 2x2 and 3x3 matrices; and random number generation.

Examples:
vtkMath (examples)

Definition at line 61 of file vtkMath.h.


Constructor & Destructor Documentation

vtkMath::vtkMath   [inline, protected]
 

Definition at line 384 of file vtkMath.h.

vtkMath::~vtkMath   [inline, protected]
 

Definition at line 385 of file vtkMath.h.

vtkMath::vtkMath const vtkMath &    [inline, protected]
 

Definition at line 386 of file vtkMath.h.


Member Function Documentation

vtkMath* vtkMath::New   [static]
 

Create an object with Debug turned off, modified time initialized to zero, and reference counting on.

Reimplemented from vtkObject.

virtual const char* vtkMath::GetClassName   [virtual]
 

Return the class name as a string. This method is defined in all subclasses of vtkObject with the vtkTypeMacro found in vtkSetGet.h.

Reimplemented from vtkObject.

int vtkMath::IsTypeOf const char *    type [static]
 

Return 1 if this class type is the same type of (or a subclass of) the named class. Returns 0 otherwise. This method works in combination with vtkTypeMacro found in vtkSetGet.h.

Reimplemented from vtkObject.

virtual int vtkMath::IsA const char *    type [virtual]
 

Return 1 if this class is the same type of (or a subclass of) the named class. Returns 0 otherwise. This method works in combination with vtkTypeMacro found in vtkSetGet.h.

Reimplemented from vtkObject.

vtkMath* vtkMath::SafeDownCast vtkObject   o [static]
 

Will cast the supplied object to vtkObject* is this is a safe operation (i.e., a safe downcast); otherwise NULL is returned. This method is defined in all subclasses of vtkObject with the vtkTypeMacro found in vtkSetGet.h.

Reimplemented from vtkObject.

float vtkMath::Pi   [inline, static]
 

Useful constants.

Definition at line 68 of file vtkMath.h.

float vtkMath::DegreesToRadians   [inline, static]
 

Definition at line 69 of file vtkMath.h.

double vtkMath::DoubleDegreesToRadians   [inline, static]
 

Useful constants. (double-precision version)

Definition at line 72 of file vtkMath.h.

float vtkMath::Dot const float    x[3],
const float    y[3]
[inline, static]
 

Dot product of two 3-vectors (float version).

Definition at line 75 of file vtkMath.h.

double vtkMath::Dot const double    x[3],
const double    y[3]
[inline, static]
 

Dot product of two 3-vectors (double-precision version).

Definition at line 79 of file vtkMath.h.

void vtkMath::Cross const float    x[3],
const float    y[3],
float    z[3]
[inline, static]
 

Cross product of two 3-vectors. Result vector in z[3].

Definition at line 479 of file vtkMath.h.

void vtkMath::Cross const double    x[3],
const double    y[3],
double    z[3]
[inline, static]
 

Cross product of two 3-vectors. Result vector in z[3]. (double-precision version)

Definition at line 488 of file vtkMath.h.

float vtkMath::Norm const float    x[3] [inline, static]
 

Compute the norm of 3-vector.

Definition at line 90 of file vtkMath.h.

Referenced by Normalize().

double vtkMath::Norm const double    x[3] [inline, static]
 

Compute the norm of 3-vector (double-precision version).

Definition at line 94 of file vtkMath.h.

float vtkMath::Normalize float    x[3] [inline, static]
 

Normalize (in place) a 3-vector. Returns norm of vector.

Definition at line 392 of file vtkMath.h.

double vtkMath::Normalize double    x[3] [inline, static]
 

Normalize (in place) a 3-vector. Returns norm of vector (double-precision version).

Definition at line 404 of file vtkMath.h.

void vtkMath::Perpendiculars const double    x[3],
double    y[3],
double    z[3],
double    theta
[static]
 

Given a unit vector x, find two unit vectors y and z such that x cross y = z (i.e. the vectors are perpendicular to each other). There is an infinite number of such vectors, specify an angle theta to choose one set. If you want only one perpendicular vector, specify NULL for z.

void vtkMath::Perpendiculars const float    x[3],
float    y[3],
float    z[3],
double    theta
[static]
 

float vtkMath::Distance2BetweenPoints const float    x[3],
const float    y[3]
[inline, static]
 

Compute distance squared between two points.

Definition at line 460 of file vtkMath.h.

Referenced by vtkTriangle::TriangleArea().

double vtkMath::Distance2BetweenPoints const double    x[3],
const double    y[3]
[inline, static]
 

Compute distance squared between two points (double precision version).

Definition at line 466 of file vtkMath.h.

float vtkMath::Dot2D const float    x[3],
const float    y[3]
[inline, static]
 

Dot product of two 2-vectors. The third (z) component is ignored.

Definition at line 122 of file vtkMath.h.

double vtkMath::Dot2D const double    x[3],
const double    y[3]
[inline, static]
 

Dot product of two 2-vectors. The third (z) component is ignored (double-precision version).

Definition at line 127 of file vtkMath.h.

float vtkMath::Norm2D const float    x[3] [inline, static]
 

Compute the norm of a 2-vector. Ignores z-component.

Definition at line 131 of file vtkMath.h.

Referenced by Normalize2D().

double vtkMath::Norm2D const double    x[3] [inline, static]
 

Compute the norm of a 2-vector. Ignores z-component (double-precision version).

Definition at line 136 of file vtkMath.h.

float vtkMath::Normalize2D float    x[3] [inline, static]
 

Normalize (in place) a 2-vector. Returns norm of vector. Ignores z-component.

Definition at line 417 of file vtkMath.h.

double vtkMath::Normalize2D double    x[3] [inline, static]
 

Normalize (in place) a 2-vector. Returns norm of vector. Ignores z-component (double-precision version).

Definition at line 430 of file vtkMath.h.

float vtkMath::Determinant2x2 const float    c1[2],
const float    c2[2]
[inline, static]
 

Compute determinant of 2x2 matrix. Two columns of matrix are input.

Definition at line 148 of file vtkMath.h.

Referenced by Determinant3x3().

double vtkMath::Determinant2x2 double    a,
double    b,
double    c,
double    d
[inline, static]
 

Calculate the determinant of a 2x2 matrix: | a b | | c d |

Definition at line 152 of file vtkMath.h.

void vtkMath::LUFactor3x3 float    A[3][3],
int    index[3]
[static]
 

LU Factorization of a 3x3 matrix. The diagonal elements are the multiplicative inverse of those in the standard LU factorization.

void vtkMath::LUFactor3x3 double    A[3][3],
int    index[3]
[static]
 

void vtkMath::LUSolve3x3 const float    A[3][3],
const int    index[3],
float    x[3]
[static]
 

LU back substitution for a 3x3 matrix. The diagonal elements are the multiplicative inverse of those in the standard LU factorization.

void vtkMath::LUSolve3x3 const double    A[3][3],
const int    index[3],
double    x[3]
[static]
 

void vtkMath::LinearSolve3x3 const float    A[3][3],
const float    x[3],
float    y[3]
[static]
 

Solve Ay = x for y and place the result in x. The matrix A is destroyed in the process.

void vtkMath::LinearSolve3x3 const double    A[3][3],
const double    x[3],
double    y[3]
[static]
 

void vtkMath::Multiply3x3 const float    A[3][3],
const float    x[3],
float    y[3]
[static]
 

Multiply a vector by a 3x3 matrix. The result is placed in x.

void vtkMath::Multiply3x3 const double    A[3][3],
const double    x[3],
double    y[3]
[static]
 

void vtkMath::Multiply3x3 const float    A[3][3],
const float    B[3][3],
float    C[3][3]
[static]
 

Mutltiply one 3x3 matrix by another according to C = BA (if the order seems odd, it is simply consistent with vtkMatrix4x4::Multiply4x4()).

void vtkMath::Multiply3x3 const double    A[3][3],
const double    B[3][3],
double    C[3][3]
[static]
 

void vtkMath::Transpose3x3 const float    A[3][3],
float    AT[3][3]
[static]
 

Transpose a 3x3 matrix.

void vtkMath::Transpose3x3 const double    A[3][3],
double    AT[3][3]
[static]
 

void vtkMath::Invert3x3 const float    A[3][3],
float    AI[3][3]
[static]
 

Invert a 3x3 matrix.

void vtkMath::Invert3x3 const double    A[3][3],
double    AI[3][3]
[static]
 

void vtkMath::Identity3x3 float    A[3][3] [static]
 

Set A to the identity matrix.

void vtkMath::Identity3x3 double    A[3][3] [static]
 

double vtkMath::Determinant3x3 float    A[3][3] [inline, static]
 

Return the determinant of a 3x3 matrix.

Definition at line 507 of file vtkMath.h.

double vtkMath::Determinant3x3 double    A[3][3] [inline, static]
 

Definition at line 512 of file vtkMath.h.

float vtkMath::Determinant3x3 const float    c1[3],
const float    c2[3],
const float    c3[3]
[inline, static]
 

Compute determinant of 3x3 matrix. Three columns of matrix are input.

Definition at line 443 of file vtkMath.h.

double vtkMath::Determinant3x3 double    a1,
double    a2,
double    a3,
double    b1,
double    b2,
double    b3,
double    c1,
double    c2,
double    c3
[inline, static]
 

Calculate the determinant of a 3x3 matrix in the form: | a1, b1, c1 | | a2, b2, c2 | | a3, b3, c3 |

Definition at line 451 of file vtkMath.h.

void vtkMath::QuaternionToMatrix3x3 const float    quat[4],
float    A[3][3]
[static]
 

Convert a quaternion to a 3x3 rotation matrix. The quaternion does not have to be normalized beforehand.

void vtkMath::QuaternionToMatrix3x3 const double    quat[4],
double    A[3][3]
[static]
 

void vtkMath::Matrix3x3ToQuaternion const float    A[3][3],
float    quat[4]
[static]
 

Convert a 3x3 matrix into a quaternion. This will provide the best possible answer even if the matrix is not a pure rotation matrix. The method used is that of B.K.P. Horn.

void vtkMath::Matrix3x3ToQuaternion const double    A[3][3],
double    quat[4]
[static]
 

void vtkMath::Orthogonalize3x3 const float    A[3][3],
float    B[3][3]
[static]
 

Orthogonalize a 3x3 matrix and put the result in B. The result is a 3x3 rotation matrix with a positive determinant. Adapted from a method by B.K.P. Horn that uses diagonalization of an appropriately constructed 4x4 matrix.

void vtkMath::Orthogonalize3x3 const double    A[3][3],
double    B[3][3]
[static]
 

void vtkMath::Diagonalize3x3 const float    A[3][3],
float    w[3],
float    V[3][3]
[static]
 

Diagonalize a symmetric 3x3 matrix and return the eigenvalues in w and the eigenvectors in the columns of V. The matrix V will have a positive determinant, and the three eigenvectors will be aligned as closely as possible with the x, y, and z axes.

void vtkMath::Diagonalize3x3 const double    A[3][3],
double    w[3],
double    V[3][3]
[static]
 

void vtkMath::SingularValueDecomposition3x3 const float    A[3][3],
float    U[3][3],
float    w[3],
float    VT[3][3]
[static]
 

Perform singular value decomposition on a 3x3 matrix. This is not done using a conventional SVD algorithm, instead it is done using Orthogonalize3x3 and Diagonalize3x3. The determinants of both matrices are guaranteed to be positive.

void vtkMath::SingularValueDecomposition3x3 const double    A[3][3],
double    U[3][3],
double    w[3],
double    VT[3][3]
[static]
 

int vtkMath::SolveLinearSystem double **    A,
double *    x,
int    size
[static]
 

Solve linear equations Ax = b using Crout's method. Input is square matrix A and load vector x. Solution x is written over load vector. The dimension of the matrix is specified in size. If error is found, method returns a 0.

int vtkMath::InvertMatrix double **    A,
double **    AI,
int    size
[static]
 

Invert input square matrix A into matrix AI. Note that A is modified during the inversion. The size variable is the dimension of the matrix. Returns 0 if inverse not computed.

int vtkMath::InvertMatrix double **    A,
double **    AI,
int    size,
int *    tmp1Size,
double *    tmp2Size
[static]
 

Thread safe version of InvertMatrix method. Working memory arrays tmp1SIze and tmp2Size of length size must be passed in.

int vtkMath::LUFactorLinearSystem double **    A,
int *    index,
int    size
[static]
 

Factor linear equations Ax = b using LU decomposition A = LU where L is lower triangular matrix and U is upper triangular matrix. Input is square matrix A, integer array of pivot indices index[0->n-1], and size of square matrix n. Output factorization LU is in matrix A. If error is found, method returns 0.

int vtkMath::LUFactorLinearSystem double **    A,
int *    index,
int    size,
double *    tmpSize
[static]
 

Thread safe version of LUFactorLinearSystem method. Working memory array tmpSize of length size must be passed in.

void vtkMath::LUSolveLinearSystem double **    A,
int *    index,
double *    x,
int    size
[static]
 

Solve linear equations Ax = b using LU decomposition A = LU where L is lower triangular matrix and U is upper triangular matrix. Input is factored matrix A=LU, integer array of pivot indices index[0->n-1], load vector x[0->n-1], and size of square matrix n. Note that A=LU and index[] are generated from method LUFactorLinearSystem). Also, solution vector is written directly over input load vector.

double vtkMath::EstimateMatrixCondition double **    A,
int    size
[static]
 

Estimate the condition number of a LU factored matrix. Used to judge the accuracy of the solution. The matrix A must have been previously factored using the method LUFactorLinearSystem. The condition number is the ratio of the infinity matrix norm (i.e., maximum value of matrix component) divided by the minimum diagonal value. (This works for triangular matrices only: see Conte and de Boor, Elementary Numerical Analysis.)

void vtkMath::RandomSeed long    s [static]
 

Initialize seed value. NOTE: Random() has the bad property that the first random number returned after RandomSeed() is called is proportional to the seed value! To help solve this, call RandomSeed() a few times inside seed. This doesn't ruin the repeatability of Random().

float vtkMath::Random   [static]
 

Generate random numbers between 0.0 and 1.0. This is used to provide portability across different systems.

Referenced by Random().

float vtkMath::Random float    min,
float    max
[inline, static]
 

Generate random number between (min,max).

Definition at line 473 of file vtkMath.h.

int vtkMath::Jacobi float **    a,
float *    w,
float **    v
[static]
 

Jacobi iteration for the solution of eigenvectors/eigenvalues of a 3x3 real symmetric matrix. Square 3x3 matrix a; output eigenvalues in w; and output eigenvectors in v. Resulting eigenvalues/vectors are sorted in decreasing order; eigenvectors are normalized.

int vtkMath::Jacobi double **    a,
double *    w,
double **    v
[static]
 

int vtkMath::JacobiN float **    a,
int    n,
float *    w,
float **    v
[static]
 

JacobiN iteration for the solution of eigenvectors/eigenvalues of a nxn real symmetric matrix. Square nxn matrix a; size of matrix in n; output eigenvalues in w; and output eigenvectors in v. Resulting eigenvalues/vectors are sorted in decreasing order; eigenvectors are normalized. w and v need to be allocated previously

int vtkMath::JacobiN double **    a,
int    n,
double *    w,
double **    v
[static]
 

double* vtkMath::SolveCubic double    c0,
double    c1,
double    c2,
double    c3
[static]
 

Solves a cubic equation c0*t^3 + c1*t^2 + c2*t + c3 = 0 when c0, c1, c2, and c3 are REAL. Solution is motivated by Numerical Recipes In C 2nd Ed. Return array contains number of (real) roots (counting multiple roots as one) followed by roots themselves. The value in roots[4] is a integer giving further information about the roots (see return codes for int SolveCubic()).

double* vtkMath::SolveQuadratic double    c0,
double    c1,
double    c2
[static]
 

Solves a quadratic equation c1*t^2 + c2*t + c3 = 0 when c1, c2, and c3 are REAL. Solution is motivated by Numerical Recipes In C 2nd Ed. Return array contains number of (real) roots (counting multiple roots as one) followed by roots themselves. Note that roots[3] contains a return code further describing solution - see documentation for SolveCubic() for meaning of return codes.

double* vtkMath::SolveLinear double    c0,
double    c1
[static]
 

Solves a linear equation c2*t + c3 = 0 when c2 and c3 are REAL. Solution is motivated by Numerical Recipes In C 2nd Ed. Return array contains number of roots followed by roots themselves.

int vtkMath::SolveCubic double    c0,
double    c1,
double    c2,
double    c3,
double *    r1,
double *    r2,
double *    r3,
int *    num_roots
[static]
 

Solves a cubic equation when c0, c1, c2, And c3 Are REAL. Solution is motivated by Numerical Recipes In C 2nd Ed. Roots and number of real roots are stored in user provided variables r1, r2, r3, and num_roots. Note that the function can return the following integer values describing the roots: (0)-no solution; (-1)-infinite number of solutions; (1)-one distinct real root of multiplicity 3 (stored in r1); (2)-two distinct real roots, one of multiplicity 2 (stored in r1 & r2); (3)-three distinct real roots; (-2)-quadratic equation with complex conjugate solution (real part of root returned in r1, imaginary in r2); (-3)-one real root and a complex conjugate pair (real root in r1 and real part of pair in r2 and imaginary in r3).

int vtkMath::SolveQuadratic double    c0,
double    c1,
double    c2,
double *    r1,
double *    r2,
int *    num_roots
[static]
 

Solves A Quadratic Equation c1*t^2 + c2*t + c3 = 0 when c1, c2, and c3 are REAL. Solution is motivated by Numerical Recipes In C 2nd Ed. Roots and number of roots are stored in user provided variables r1, r2, num_roots

int vtkMath::SolveLinear double    c0,
double    c1,
double *    r1,
int *    num_roots
[static]
 

Solves a linear equation c2*t + c3 = 0 when c2 and c3 are REAL. Solution is motivated by Numerical Recipes In C 2nd Ed. Root and number of (real) roots are stored in user provided variables r2 and num_roots.

int vtkMath::SolveLeastSquares int    numberOfSamples,
double **    xt,
int    xOrder,
double **    yt,
int    yOrder,
double **    mt
[static]
 

Solves for the least squares best fit matrix for the equation X'M' = Y'. Uses pseudoinverse to get the ordinary least squares. The inputs and output are transposed matrices. Dimensions: X' is numberOfSamples by xOrder, Y' is numberOfSamples by yOrder, M' dimension is xOrder by yOrder. M' should be pre-allocated. All matrices are row major. The resultant matrix M' should be pre-multiplied to X' to get Y', or transposed and then post multiplied to X to get Y

void vtkMath::operator= const vtkMath &    [inline, protected]
 

Definition at line 387 of file vtkMath.h.


Member Data Documentation

long vtkMath::Seed [static, protected]
 

Definition at line 389 of file vtkMath.h.


The documentation for this class was generated from the following file:
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