VTK-5.0.0/Common/vtkMath.h

/*=========================================================================

  Program:   Visualization Toolkit
  Module:    $RCSfile: vtkMath.h,v $

  Copyright (c) Ken Martin, Will Schroeder, Bill Lorensen
  All rights reserved.
  See Copyright.txt or http://www.kitware.com/Copyright.htm for details.

     This software is distributed WITHOUT ANY WARRANTY; without even
     the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
     PURPOSE.  See the above copyright notice for more information.

=========================================================================*/
// .NAME vtkMath - performs common math operations
// .SECTION Description
// 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.

#ifndef __vtkMath_h
#define __vtkMath_h

#include "vtkObject.h"

class vtkDataArray;

class VTK_COMMON_EXPORT vtkMath : public vtkObject
{
public:
  static vtkMath *New();
  vtkTypeRevisionMacro(vtkMath,vtkObject);
  void PrintSelf(ostream& os, vtkIndent indent);

  // Description:
  // Useful constants.
  static float Pi() {return 3.14159265358979f;};
  static float DegreesToRadians() {return 0.017453292f;};
  static float RadiansToDegrees() {return 57.2957795131f;};

  // Description:
  // Useful constants. (double-precision version)
  static double DoubleDegreesToRadians() {return 0.017453292519943295;};
  static double DoublePi() {return 3.1415926535897932384626;};
  static double DoubleRadiansToDegrees() {return 57.29577951308232;};

  // Description:
  // Rounds a float to the nearest integer.
  static int Round(float f) {
    return static_cast<int>(f + (f >= 0 ? 0.5 : -0.5)); }
  static int Round(double f) {
    return static_cast<int>(f + (f >= 0 ? 0.5 : -0.5)); }

  static int Floor(double x);
  
  // Description:
  // Dot product of two 3-vectors (float version).
  static float Dot(const float x[3], const float y[3]) {
    return (x[0]*y[0] + x[1]*y[1] + x[2]*y[2]);};

  // Description:
  // Dot product of two 3-vectors (double-precision version).
  static double Dot(const double x[3], const double y[3]) {
    return (x[0]*y[0] + x[1]*y[1] + x[2]*y[2]);};
  
  // Description:
  // Cross product of two 3-vectors. Result vector in z[3].
  static void Cross(const float x[3], const float y[3], float z[3]);

  // Description:
  // Cross product of two 3-vectors. Result vector in z[3]. (double-precision
  // version)
  static void Cross(const double x[3], const double y[3], double z[3]);

  // Description:
  // Compute the norm of n-vector.
  static float Norm(const float* x, int n); 
  static double Norm(const double* x, int n); 

  // Description:
  // Compute the norm of 3-vector.
  static float Norm(const float x[3]) {
    return static_cast<float> (sqrt(x[0]*x[0] + x[1]*x[1] + x[2]*x[2]));};
  
  // Description:
  // Compute the norm of 3-vector (double-precision version).
  static double Norm(const double x[3]) {
    return sqrt(x[0]*x[0] + x[1]*x[1] + x[2]*x[2]);};
  
  // Description:
  // Normalize (in place) a 3-vector. Returns norm of vector.
  static float Normalize(float x[3]);

  // Description:
  // Normalize (in place) a 3-vector. Returns norm of vector
  // (double-precision version).
  static double Normalize(double x[3]);

  // Description:
  // 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.
  static void Perpendiculars(const double x[3], double y[3], double z[3], 
                             double theta);
  static void Perpendiculars(const float x[3], float y[3], float z[3],
                             double theta);

  // Description:
  // Compute distance squared between two points.
  static float Distance2BetweenPoints(const float x[3], const float y[3]);

  // Description:
  // Compute distance squared between two points (double precision version).
  static double Distance2BetweenPoints(const double x[3], const double y[3]);

  // Description:
  // Dot product of two 2-vectors. The third (z) component is ignored.
  static float Dot2D(const float x[3], const float y[3]) {
    return (x[0]*y[0] + x[1]*y[1]);};
  
  // Description:
  // Dot product of two 2-vectors. The third (z) component is
  // ignored (double-precision version).
  static double Dot2D(const double x[3], const double y[3]) {
    return (x[0]*y[0] + x[1]*y[1]);};

  // Description:
  // Compute the norm of a 2-vector. Ignores z-component.
  static float Norm2D(const float x[3]) {
    return static_cast<float> (sqrt(x[0]*x[0] + x[1]*x[1]));};

  // Description:
  // Compute the norm of a 2-vector. Ignores z-component
  // (double-precision version).
  static double Norm2D(const double x[3]) {
    return sqrt(x[0]*x[0] + x[1]*x[1]);};

  // Description:
  // Normalize (in place) a 2-vector. Returns norm of vector. Ignores
  // z-component.
  static float Normalize2D(float x[3]);

  // Description:
  // Normalize (in place) a 2-vector. Returns norm of vector. Ignores
  // z-component (double-precision version).
  static double Normalize2D(double x[3]);

  // Description:
  // Compute determinant of 2x2 matrix. Two columns of matrix are input.
  static float Determinant2x2(const float c1[2], const float c2[2]) {
    return (c1[0]*c2[1] - c2[0]*c1[1]);};

  // Description:
  // Calculate the determinant of a 2x2 matrix: | a b | | c d |
  static double Determinant2x2(double a, double b, double c, double d) {
    return (a * d - b * c);};
  static double Determinant2x2(const double c1[2], const double c2[2]) {
    return (c1[0]*c2[1] - c2[0]*c1[1]);};

  // Description:
  // LU Factorization of a 3x3 matrix.  The diagonal elements are the
  // multiplicative inverse of those in the standard LU factorization.
  static void LUFactor3x3(float A[3][3], int index[3]);
  static void LUFactor3x3(double A[3][3], int index[3]);

  // Description:
  // LU back substitution for a 3x3 matrix.  The diagonal elements are the
  // multiplicative inverse of those in the standard LU factorization.
  static void LUSolve3x3(const float A[3][3], const int index[3], 
                         float x[3]);
  static void LUSolve3x3(const double A[3][3], const int index[3], 
                         double x[3]);

  // Description:
  // Solve Ay = x for y and place the result in y.  The matrix A is
  // destroyed in the process.
  static void LinearSolve3x3(const float A[3][3], const float x[3], 
                             float y[3]);
  static void LinearSolve3x3(const double A[3][3], const double x[3], 
                             double y[3]);

  // Description:
  // Multiply a vector by a 3x3 matrix.  The result is placed in out.
  static void Multiply3x3(const float A[3][3], const float in[3], 
                          float out[3]);
  static void Multiply3x3(const double A[3][3], const double in[3], 
                          double out[3]);
  
  // Description:
  // Multiply one 3x3 matrix by another according to C = AB.
  static void Multiply3x3(const float A[3][3], const float B[3][3], 
                          float C[3][3]);
  static void Multiply3x3(const double A[3][3], const double B[3][3], 
                          double C[3][3]);

  // Description:
  // Transpose a 3x3 matrix.
  static void Transpose3x3(const float A[3][3], float AT[3][3]);
  static void Transpose3x3(const double A[3][3], double AT[3][3]);

  // Description:
  // Invert a 3x3 matrix.
  static void Invert3x3(const float A[3][3], float AI[3][3]);
  static void Invert3x3(const double A[3][3], double AI[3][3]);

  // Description:
  // Set A to the identity matrix.
  static void Identity3x3(float A[3][3]);
  static void Identity3x3(double A[3][3]);

  // Description:
  // Return the determinant of a 3x3 matrix.
  static double Determinant3x3(float A[3][3]);
  static double Determinant3x3(double A[3][3]);

  // Description:
  // Compute determinant of 3x3 matrix. Three columns of matrix are input.
  static float Determinant3x3(const float c1[3], 
                              const float c2[3], 
                              const float c3[3]);

  // Description:
  // Compute determinant of 3x3 matrix. Three columns of matrix are input.
  static double Determinant3x3(const double c1[3], 
                               const double c2[3], 
                               const double c3[3]);

  // Description:
  // Calculate the determinant of a 3x3 matrix in the form:
  //     | a1,  b1,  c1 |
  //     | a2,  b2,  c2 |
  //     | a3,  b3,  c3 |
  static double Determinant3x3(double a1, double a2, double a3, 
                               double b1, double b2, double b3, 
                               double c1, double c2, double c3);

  // Description:
  // Convert a quaternion to a 3x3 rotation matrix.  The quaternion
  // does not have to be normalized beforehand.
  static void QuaternionToMatrix3x3(const float quat[4], float A[3][3]); 
  static void QuaternionToMatrix3x3(const double quat[4], double A[3][3]); 

  // Description:
  // 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.
  static void Matrix3x3ToQuaternion(const float A[3][3], float quat[4]);
  static void Matrix3x3ToQuaternion(const double A[3][3], double quat[4]);
  
  // Description:
  // Orthogonalize a 3x3 matrix and put the result in B.  If matrix A
  // has a negative determinant, then B will be a rotation plus a flip
  // i.e. it will have a determinant of -1.
  static void Orthogonalize3x3(const float A[3][3], float B[3][3]);
  static void Orthogonalize3x3(const double A[3][3], double B[3][3]);

  // Description:
  // 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.
  static void Diagonalize3x3(const float A[3][3], float w[3], float V[3][3]);
  static void Diagonalize3x3(const double A[3][3],double w[3],double V[3][3]);

  // Description:
  // 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.  Both output matrices U and VT
  // will have positive determinants, and the w values will be arranged
  // such that the three rows of VT are aligned as closely as possible
  // with the x, y, and z axes respectively.  If the determinant of A is
  // negative, then the three w values will be negative.
  static void SingularValueDecomposition3x3(const float A[3][3],
                                            float U[3][3], float w[3],
                                            float VT[3][3]);
  static void SingularValueDecomposition3x3(const double A[3][3],
                                            double U[3][3], double w[3],
                                            double VT[3][3]);

  // Description:
  // 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.
  static int SolveLinearSystem(double **A, double *x, int size);

  // Description:
  // 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.
  static int InvertMatrix(double **A, double **AI, int size);

  // Description:
  // Thread safe version of InvertMatrix method.
  // Working memory arrays tmp1SIze and tmp2Size
  // of length size must be passed in.
  static int InvertMatrix(double **A, double **AI, int size,
                          int *tmp1Size, double *tmp2Size);

  // Description:
  // 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. 
  static int LUFactorLinearSystem(double **A, int *index, int size);

  // Description:
  // Thread safe version of LUFactorLinearSystem method.
  // Working memory array tmpSize of length size
  // must be passed in.
  static int LUFactorLinearSystem(double **A, int *index, int size,
                                  double *tmpSize);

  // Description:
  // 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.
  static void LUSolveLinearSystem(double **A, int *index, 
                                  double *x, int size);

  // Description:
  // 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.)
  static double EstimateMatrixCondition(double **A, int size);

  // Description:
  // 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().
  static void RandomSeed(long s);  

  // Description:
  // Generate random numbers between 0.0 and 1.0.
  // This is used to provide portability across different systems.
  static double Random();  

  // Description:
  // Generate random number between (min,max).
  static double Random(double min, double max);

  // Description:
  // 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.
  static int Jacobi(float **a, float *w, float **v);
  static int Jacobi(double **a, double *w, double **v);

  // Description:
  // 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
  static int JacobiN(float **a, int n, float *w, float **v);
  static int JacobiN(double **a, int n, double *w, double **v);

  // Description:
  // 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()).
  static double* SolveCubic(double c0, double c1, double c2, double c3);

  // Description:
  // 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.
  static double* SolveQuadratic(double c0, double c1, double c2);

  // Description:
  // 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.
  static double* SolveLinear(double c0, double c1);

  // Description:
  // 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).
  static int SolveCubic(double c0, double c1, double c2, double c3, 
                        double *r1, double *r2, double *r3, int *num_roots);

  // Description:
  // 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
  static int SolveQuadratic(double c0, double c1, double c2, 
                            double *r1, double *r2, int *num_roots);
  
  // Description:
  // 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.
  static int SolveLinear(double c0, double c1, double *r1, int *num_roots);

  // Description:
  // Solves for the least squares best fit matrix for the homogeneous equation X'M' = 0'.
  // Uses the method described on pages 40-41 of Computer Vision by 
  // Forsyth and Ponce, which is that the solution is the eigenvector 
  // associated with the minimum eigenvalue of T(X)X, where T(X) is the
  // transpose of X.
  // The inputs and output are transposed matrices.
  //    Dimensions: X' is numberOfSamples by xOrder,
  //                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 0', or transposed and
  // then post multiplied to X to get 0
  static int SolveHomogeneousLeastSquares(int numberOfSamples, double **xt, int xOrder,
                                double **mt);


  // Description:
  // 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
  // By default, this method checks for the homogeneous condition where Y==0, and
  // if so, invokes SolveHomogeneousLeastSquares. For better performance when
  // the system is known not to be homogeneous, invoke with checkHomogeneous=0.
  static int SolveLeastSquares(int numberOfSamples, double **xt, int xOrder,
                               double **yt, int yOrder, double **mt, int checkHomogeneous=1);

  // Description:
  // Convert color in RGB format (Red, Green, Blue) to HSV format
  // (Hue, Saturation, Value). The input color is not modified.
  static void RGBToHSV(float rgb[3], float hsv[3])
    { RGBToHSV(rgb[0], rgb[1], rgb[2], hsv, hsv+1, hsv+2); }
  static void RGBToHSV(float r, float g, float b, float *h, float *s, float *v);
  static double* RGBToHSV(double rgb[3]);
  static double* RGBToHSV(double r, double g, double b);
  static void RGBToHSV(double rgb[3], double hsv[3])
    { RGBToHSV(rgb[0], rgb[1], rgb[2], hsv, hsv+1, hsv+2); }
  static void RGBToHSV(double r, double g, double b, double *h, double *s, double *v);

  // Description:
  // Convert color in HSV format (Hue, Saturation, Value) to RGB
  // format (Red, Green, Blue). The input color is not modified.
  static void HSVToRGB(float hsv[3], float rgb[3])
    { HSVToRGB(hsv[0], hsv[1], hsv[2], rgb, rgb+1, rgb+2); }
  static void HSVToRGB(float h, float s, float v, float *r, float *g, float *b);
  static double* HSVToRGB(double hsv[3]);
  static double* HSVToRGB(double h, double s, double v);
  static void HSVToRGB(double hsv[3], double rgb[3])
    { HSVToRGB(hsv[0], hsv[1], hsv[2], rgb, rgb+1, rgb+2); }
  static void HSVToRGB(double h, double s, double v, double *r, double *g, double *b);

  // Description:
  // Convert color from Lab to XYZ system, and vice-versa
  static void LabToXYZ(double lab[3], double xyz[3]);
  static void XYZToRGB(double xyz[3], double rgb[3]);

  // Description:
  // Set the bounds to an uninitialized state
  static void UninitializeBounds(double bounds[6]){
    bounds[0] = 1.0;
    bounds[1] = -1.0;
    bounds[2] = 1.0;
    bounds[3] = -1.0;
    bounds[4] = 1.0;
    bounds[5] = -1.0;
  }
  
  // Description:
  // Are the bounds initialized?
  static int AreBoundsInitialized(double bounds[6]){
    if (bounds[1]-bounds[0]<0.0)
      {
      return 0;
      }
    return 1;
  }

  // Description:
  // Clamp some values against a range
  // The method without 'clamped_values' will perform in-place clamping.
  static void ClampValue(double *value, const double range[2]);
  static void ClampValue(double value, const double range[2], double *clamped_value);
  static void ClampValues(
    double *values, int nb_values, const double range[2]);
  static void ClampValues(
    const double *values, int nb_values, const double range[2], double *clamped_values);

  // Description:
  // Return the scalar type that is most likely to have enough precision 
  // to store a given range of data once it has been scaled and shifted 
  // (i.e. [range_min * scale + shift, range_max * scale + shift]. 
  // If any one of the parameters is not an integer number (decimal part != 0),
  // the search will default to float types only (float or double)
  // Return -1 on error or no scalar type found.
  static int GetScalarTypeFittingRange(
    double range_min, double range_max, 
    double scale = 1.0, double shift = 0.0);

  // Description:
  // Get a vtkDataArray's scalar range for a given component. 
  // If the vtkDataArray's data type is unsigned char (VTK_UNSIGNED_CHAR)
  // the range is adjusted to the whole data type range [0, 255.0]. 
  // Same goes for unsigned short (VTK_UNSIGNED_SHORT) but the upper bound 
  // is also adjusted down to 4095.0 if was between ]255, 4095.0].
  // Return 1 on success, 0 otherwise.
  static int GetAdjustedScalarRange(
    vtkDataArray *array, int comp, double range[2]);
 
  // Description:
  // Return true if first 3D extent is within second 3D extent
  // Extent is x-min, x-max, y-min, y-max, z-min, z-max
  static int ExtentIsWithinOtherExtent(int extent1[6], int extent2[6]);
  
  // Description:
  // Return true if first 3D bounds is within the second 3D bounds
  // Bounds is x-min, x-max, y-min, y-max, z-min, z-max
  // Delta is the error margin along each axis (usually a small number)
  static int BoundsIsWithinOtherBounds(double bounds1[6], double bounds2[6], double delta[3]);
  
  // Description:
  // Return true if point is within the given 3D bounds
  // Bounds is x-min, x-max, y-min, y-max, z-min, z-max
  // Delta is the error margin along each axis (usually a small number)
  static int PointIsWithinBounds(double point[3], double bounds[6], double delta[3]);
  

protected:
  vtkMath() {};
  ~vtkMath() {};
  
  static long Seed;
private:
  vtkMath(const vtkMath&);  // Not implemented.
  void operator=(const vtkMath&);  // Not implemented.
};

//----------------------------------------------------------------------------
inline int vtkMath::Floor(double x)
{
#if defined i386 || defined _M_IX86
  union { int i[2]; double d; } u;
  // use 52-bit precision of IEEE double to round (x - 0.25) to 
  // the nearest multiple of 0.5, according to prevailing rounding
  // mode which is IEEE round-to-nearest,even
  u.d = (x - 0.25) + 3377699720527872.0; // (2**51)*1.5
  // extract mantissa, use shift to divide by 2 and hence get rid
  // of the bit that gets messed up because the FPU uses
  // round-to-nearest,even mode instead of round-to-nearest,+infinity
  return u.i[0] >> 1;
#else
  return (int)floor(x);
#endif
}

//----------------------------------------------------------------------------
inline float vtkMath::Normalize(float x[3])
{
  float den; 
  if ( (den = vtkMath::Norm(x)) != 0.0 )
    {
    for (int i=0; i < 3; i++)
      {
      x[i] /= den;
      }
    }
  return den;
}

//----------------------------------------------------------------------------
inline double vtkMath::Normalize(double x[3])
{
  double den; 
  if ( (den = vtkMath::Norm(x)) != 0.0 )
    {
    for (int i=0; i < 3; i++)
      {
      x[i] /= den;
      }
    }
  return den;
}

//----------------------------------------------------------------------------
inline float vtkMath::Normalize2D(float x[3])
{
  float den; 
  if ( (den = vtkMath::Norm2D(x)) != 0.0 )
    {
    for (int i=0; i < 2; i++)
      {
      x[i] /= den;
      }
    }
  return den;
}

//----------------------------------------------------------------------------
inline double vtkMath::Normalize2D(double x[3])
{
  double den; 
  if ( (den = vtkMath::Norm2D(x)) != 0.0 )
    {
    for (int i=0; i < 2; i++)
      {
      x[i] /= den;
      }
    }
  return den;
}

//----------------------------------------------------------------------------
inline float vtkMath::Determinant3x3(const float c1[3], 
                                     const float c2[3], 
                                     const float c3[3])
{
  return c1[0]*c2[1]*c3[2] + c2[0]*c3[1]*c1[2] + c3[0]*c1[1]*c2[2] -
         c1[0]*c3[1]*c2[2] - c2[0]*c1[1]*c3[2] - c3[0]*c2[1]*c1[2];
}

//----------------------------------------------------------------------------
inline double vtkMath::Determinant3x3(const double c1[3], 
                                      const double c2[3], 
                                      const double c3[3])
{
  return c1[0]*c2[1]*c3[2] + c2[0]*c3[1]*c1[2] + c3[0]*c1[1]*c2[2] -
         c1[0]*c3[1]*c2[2] - c2[0]*c1[1]*c3[2] - c3[0]*c2[1]*c1[2];
}

//----------------------------------------------------------------------------
inline double vtkMath::Determinant3x3(double a1, double a2, double a3, 
                                      double b1, double b2, double b3, 
                                      double c1, double c2, double c3)
{
    return ( a1 * vtkMath::Determinant2x2( b2, b3, c2, c3 )
           - b1 * vtkMath::Determinant2x2( a2, a3, c2, c3 )
           + c1 * vtkMath::Determinant2x2( a2, a3, b2, b3 ) );
}

//----------------------------------------------------------------------------
inline float vtkMath::Distance2BetweenPoints(const float x[3], 
                                             const float y[3])
{
  return ((x[0]-y[0])*(x[0]-y[0]) + (x[1]-y[1])*(x[1]-y[1]) +
          (x[2]-y[2])*(x[2]-y[2]));
}

//----------------------------------------------------------------------------
inline double vtkMath::Distance2BetweenPoints(const double x[3], 
                                              const double y[3])
{
  return ((x[0]-y[0])*(x[0]-y[0]) + (x[1]-y[1])*(x[1]-y[1]) +
          (x[2]-y[2])*(x[2]-y[2]));
}

//----------------------------------------------------------------------------
inline double vtkMath::Random(double min, double max)
{
  return (min + vtkMath::Random()*(max-min));
}

//----------------------------------------------------------------------------
// Cross product of two 3-vectors. Result vector in z[3].
inline void vtkMath::Cross(const float x[3], const float y[3], float z[3])
{
  float Zx = x[1]*y[2] - x[2]*y[1]; 
  float Zy = x[2]*y[0] - x[0]*y[2];
  float Zz = x[0]*y[1] - x[1]*y[0];
  z[0] = Zx; z[1] = Zy; z[2] = Zz; 
}

//----------------------------------------------------------------------------
// Cross product of two 3-vectors. Result vector in z[3].
inline void vtkMath::Cross(const double x[3], const double y[3], double z[3])
{
  double Zx = x[1]*y[2] - x[2]*y[1]; 
  double Zy = x[2]*y[0] - x[0]*y[2];
  double Zz = x[0]*y[1] - x[1]*y[0];
  z[0] = Zx; z[1] = Zy; z[2] = Zz; 
}

//BTX
//----------------------------------------------------------------------------
template<class T>
inline double vtkDeterminant3x3(T A[3][3])
{
  return A[0][0]*A[1][1]*A[2][2] + A[1][0]*A[2][1]*A[0][2] + 
         A[2][0]*A[0][1]*A[1][2] - A[0][0]*A[2][1]*A[1][2] - 
         A[1][0]*A[0][1]*A[2][2] - A[2][0]*A[1][1]*A[0][2];
}
//ETX

//----------------------------------------------------------------------------
inline double vtkMath::Determinant3x3(float A[3][3])
{
  return vtkDeterminant3x3(A);
}

//----------------------------------------------------------------------------
inline double vtkMath::Determinant3x3(double A[3][3])
{
  return vtkDeterminant3x3(A);
}

//----------------------------------------------------------------------------
inline void vtkMath::ClampValue(double *value, const double range[2])
{
  if (value && range)
    {
    if (*value < range[0])
      {
      *value = range[0];
      }
    else if (*value > range[1])
      {
      *value = range[1];
      }
    }
}

//----------------------------------------------------------------------------
inline void vtkMath::ClampValue(
  double value, const double range[2], double *clamped_value)
{
  if (range && clamped_value)
    {
    if (value < range[0])
      {
      *clamped_value = range[0];
      }
    else if (value > range[1])
      {
      *clamped_value = range[1];
      }
    else
      {
      *clamped_value = value;
      }
    }
}

#endif