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  1. /*
  2.  * Mesa 3-D graphics library
  3.  *
  4.  * Copyright (C) 1999-2005  Brian Paul   All Rights Reserved.
  5.  *
  6.  * Permission is hereby granted, free of charge, to any person obtaining a
  7.  * copy of this software and associated documentation files (the "Software"),
  8.  * to deal in the Software without restriction, including without limitation
  9.  * the rights to use, copy, modify, merge, publish, distribute, sublicense,
  10.  * and/or sell copies of the Software, and to permit persons to whom the
  11.  * Software is furnished to do so, subject to the following conditions:
  12.  *
  13.  * The above copyright notice and this permission notice shall be included
  14.  * in all copies or substantial portions of the Software.
  15.  *
  16.  * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
  17.  * OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
  18.  * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL
  19.  * THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
  20.  * OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
  21.  * ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
  22.  * OTHER DEALINGS IN THE SOFTWARE.
  23.  */
  24.  
  25.  
  26. /**
  27.  * \file m_matrix.c
  28.  * Matrix operations.
  29.  *
  30.  * \note
  31.  * -# 4x4 transformation matrices are stored in memory in column major order.
  32.  * -# Points/vertices are to be thought of as column vectors.
  33.  * -# Transformation of a point p by a matrix M is: p' = M * p
  34.  */
  35.  
  36.  
  37. #include "c99_math.h"
  38. #include "main/glheader.h"
  39. #include "main/imports.h"
  40. #include "main/macros.h"
  41.  
  42. #include "m_matrix.h"
  43.  
  44.  
  45. /**
  46.  * \defgroup MatFlags MAT_FLAG_XXX-flags
  47.  *
  48.  * Bitmasks to indicate different kinds of 4x4 matrices in GLmatrix::flags
  49.  */
  50. /*@{*/
  51. #define MAT_FLAG_IDENTITY       0     /**< is an identity matrix flag.
  52.                                        *   (Not actually used - the identity
  53.                                        *   matrix is identified by the absence
  54.                                        *   of all other flags.)
  55.                                        */
  56. #define MAT_FLAG_GENERAL        0x1   /**< is a general matrix flag */
  57. #define MAT_FLAG_ROTATION       0x2   /**< is a rotation matrix flag */
  58. #define MAT_FLAG_TRANSLATION    0x4   /**< is a translation matrix flag */
  59. #define MAT_FLAG_UNIFORM_SCALE  0x8   /**< is an uniform scaling matrix flag */
  60. #define MAT_FLAG_GENERAL_SCALE  0x10  /**< is a general scaling matrix flag */
  61. #define MAT_FLAG_GENERAL_3D     0x20  /**< general 3D matrix flag */
  62. #define MAT_FLAG_PERSPECTIVE    0x40  /**< is a perspective proj matrix flag */
  63. #define MAT_FLAG_SINGULAR       0x80  /**< is a singular matrix flag */
  64. #define MAT_DIRTY_TYPE          0x100  /**< matrix type is dirty */
  65. #define MAT_DIRTY_FLAGS         0x200  /**< matrix flags are dirty */
  66. #define MAT_DIRTY_INVERSE       0x400  /**< matrix inverse is dirty */
  67.  
  68. /** angle preserving matrix flags mask */
  69. #define MAT_FLAGS_ANGLE_PRESERVING (MAT_FLAG_ROTATION | \
  70.                                     MAT_FLAG_TRANSLATION | \
  71.                                     MAT_FLAG_UNIFORM_SCALE)
  72.  
  73. /** geometry related matrix flags mask */
  74. #define MAT_FLAGS_GEOMETRY (MAT_FLAG_GENERAL | \
  75.                             MAT_FLAG_ROTATION | \
  76.                             MAT_FLAG_TRANSLATION | \
  77.                             MAT_FLAG_UNIFORM_SCALE | \
  78.                             MAT_FLAG_GENERAL_SCALE | \
  79.                             MAT_FLAG_GENERAL_3D | \
  80.                             MAT_FLAG_PERSPECTIVE | \
  81.                             MAT_FLAG_SINGULAR)
  82.  
  83. /** length preserving matrix flags mask */
  84. #define MAT_FLAGS_LENGTH_PRESERVING (MAT_FLAG_ROTATION | \
  85.                                      MAT_FLAG_TRANSLATION)
  86.  
  87.  
  88. /** 3D (non-perspective) matrix flags mask */
  89. #define MAT_FLAGS_3D (MAT_FLAG_ROTATION | \
  90.                       MAT_FLAG_TRANSLATION | \
  91.                       MAT_FLAG_UNIFORM_SCALE | \
  92.                       MAT_FLAG_GENERAL_SCALE | \
  93.                       MAT_FLAG_GENERAL_3D)
  94.  
  95. /** dirty matrix flags mask */
  96. #define MAT_DIRTY          (MAT_DIRTY_TYPE | \
  97.                             MAT_DIRTY_FLAGS | \
  98.                             MAT_DIRTY_INVERSE)
  99.  
  100. /*@}*/
  101.  
  102.  
  103. /**
  104.  * Test geometry related matrix flags.
  105.  *
  106.  * \param mat a pointer to a GLmatrix structure.
  107.  * \param a flags mask.
  108.  *
  109.  * \returns non-zero if all geometry related matrix flags are contained within
  110.  * the mask, or zero otherwise.
  111.  */
  112. #define TEST_MAT_FLAGS(mat, a)  \
  113.     ((MAT_FLAGS_GEOMETRY & (~(a)) & ((mat)->flags) ) == 0)
  114.  
  115.  
  116.  
  117. /**
  118.  * Names of the corresponding GLmatrixtype values.
  119.  */
  120. static const char *types[] = {
  121.    "MATRIX_GENERAL",
  122.    "MATRIX_IDENTITY",
  123.    "MATRIX_3D_NO_ROT",
  124.    "MATRIX_PERSPECTIVE",
  125.    "MATRIX_2D",
  126.    "MATRIX_2D_NO_ROT",
  127.    "MATRIX_3D"
  128. };
  129.  
  130.  
  131. /**
  132.  * Identity matrix.
  133.  */
  134. static GLfloat Identity[16] = {
  135.    1.0, 0.0, 0.0, 0.0,
  136.    0.0, 1.0, 0.0, 0.0,
  137.    0.0, 0.0, 1.0, 0.0,
  138.    0.0, 0.0, 0.0, 1.0
  139. };
  140.  
  141.  
  142.  
  143. /**********************************************************************/
  144. /** \name Matrix multiplication */
  145. /*@{*/
  146.  
  147. #define A(row,col)  a[(col<<2)+row]
  148. #define B(row,col)  b[(col<<2)+row]
  149. #define P(row,col)  product[(col<<2)+row]
  150.  
  151. /**
  152.  * Perform a full 4x4 matrix multiplication.
  153.  *
  154.  * \param a matrix.
  155.  * \param b matrix.
  156.  * \param product will receive the product of \p a and \p b.
  157.  *
  158.  * \warning Is assumed that \p product != \p b. \p product == \p a is allowed.
  159.  *
  160.  * \note KW: 4*16 = 64 multiplications
  161.  *
  162.  * \author This \c matmul was contributed by Thomas Malik
  163.  */
  164. static void matmul4( GLfloat *product, const GLfloat *a, const GLfloat *b )
  165. {
  166.    GLint i;
  167.    for (i = 0; i < 4; i++) {
  168.       const GLfloat ai0=A(i,0),  ai1=A(i,1),  ai2=A(i,2),  ai3=A(i,3);
  169.       P(i,0) = ai0 * B(0,0) + ai1 * B(1,0) + ai2 * B(2,0) + ai3 * B(3,0);
  170.       P(i,1) = ai0 * B(0,1) + ai1 * B(1,1) + ai2 * B(2,1) + ai3 * B(3,1);
  171.       P(i,2) = ai0 * B(0,2) + ai1 * B(1,2) + ai2 * B(2,2) + ai3 * B(3,2);
  172.       P(i,3) = ai0 * B(0,3) + ai1 * B(1,3) + ai2 * B(2,3) + ai3 * B(3,3);
  173.    }
  174. }
  175.  
  176. /**
  177.  * Multiply two matrices known to occupy only the top three rows, such
  178.  * as typical model matrices, and orthogonal matrices.
  179.  *
  180.  * \param a matrix.
  181.  * \param b matrix.
  182.  * \param product will receive the product of \p a and \p b.
  183.  */
  184. static void matmul34( GLfloat *product, const GLfloat *a, const GLfloat *b )
  185. {
  186.    GLint i;
  187.    for (i = 0; i < 3; i++) {
  188.       const GLfloat ai0=A(i,0),  ai1=A(i,1),  ai2=A(i,2),  ai3=A(i,3);
  189.       P(i,0) = ai0 * B(0,0) + ai1 * B(1,0) + ai2 * B(2,0);
  190.       P(i,1) = ai0 * B(0,1) + ai1 * B(1,1) + ai2 * B(2,1);
  191.       P(i,2) = ai0 * B(0,2) + ai1 * B(1,2) + ai2 * B(2,2);
  192.       P(i,3) = ai0 * B(0,3) + ai1 * B(1,3) + ai2 * B(2,3) + ai3;
  193.    }
  194.    P(3,0) = 0;
  195.    P(3,1) = 0;
  196.    P(3,2) = 0;
  197.    P(3,3) = 1;
  198. }
  199.  
  200. #undef A
  201. #undef B
  202. #undef P
  203.  
  204. /**
  205.  * Multiply a matrix by an array of floats with known properties.
  206.  *
  207.  * \param mat pointer to a GLmatrix structure containing the left multiplication
  208.  * matrix, and that will receive the product result.
  209.  * \param m right multiplication matrix array.
  210.  * \param flags flags of the matrix \p m.
  211.  *
  212.  * Joins both flags and marks the type and inverse as dirty.  Calls matmul34()
  213.  * if both matrices are 3D, or matmul4() otherwise.
  214.  */
  215. static void matrix_multf( GLmatrix *mat, const GLfloat *m, GLuint flags )
  216. {
  217.    mat->flags |= (flags | MAT_DIRTY_TYPE | MAT_DIRTY_INVERSE);
  218.  
  219.    if (TEST_MAT_FLAGS(mat, MAT_FLAGS_3D))
  220.       matmul34( mat->m, mat->m, m );
  221.    else
  222.       matmul4( mat->m, mat->m, m );
  223. }
  224.  
  225. /**
  226.  * Matrix multiplication.
  227.  *
  228.  * \param dest destination matrix.
  229.  * \param a left matrix.
  230.  * \param b right matrix.
  231.  *
  232.  * Joins both flags and marks the type and inverse as dirty.  Calls matmul34()
  233.  * if both matrices are 3D, or matmul4() otherwise.
  234.  */
  235. void
  236. _math_matrix_mul_matrix( GLmatrix *dest, const GLmatrix *a, const GLmatrix *b )
  237. {
  238.    dest->flags = (a->flags |
  239.                   b->flags |
  240.                   MAT_DIRTY_TYPE |
  241.                   MAT_DIRTY_INVERSE);
  242.  
  243.    if (TEST_MAT_FLAGS(dest, MAT_FLAGS_3D))
  244.       matmul34( dest->m, a->m, b->m );
  245.    else
  246.       matmul4( dest->m, a->m, b->m );
  247. }
  248.  
  249. /**
  250.  * Matrix multiplication.
  251.  *
  252.  * \param dest left and destination matrix.
  253.  * \param m right matrix array.
  254.  *
  255.  * Marks the matrix flags with general flag, and type and inverse dirty flags.
  256.  * Calls matmul4() for the multiplication.
  257.  */
  258. void
  259. _math_matrix_mul_floats( GLmatrix *dest, const GLfloat *m )
  260. {
  261.    dest->flags |= (MAT_FLAG_GENERAL |
  262.                    MAT_DIRTY_TYPE |
  263.                    MAT_DIRTY_INVERSE |
  264.                    MAT_DIRTY_FLAGS);
  265.  
  266.    matmul4( dest->m, dest->m, m );
  267. }
  268.  
  269. /*@}*/
  270.  
  271.  
  272. /**********************************************************************/
  273. /** \name Matrix output */
  274. /*@{*/
  275.  
  276. /**
  277.  * Print a matrix array.
  278.  *
  279.  * \param m matrix array.
  280.  *
  281.  * Called by _math_matrix_print() to print a matrix or its inverse.
  282.  */
  283. static void print_matrix_floats( const GLfloat m[16] )
  284. {
  285.    int i;
  286.    for (i=0;i<4;i++) {
  287.       _mesa_debug(NULL,"\t%f %f %f %f\n", m[i], m[4+i], m[8+i], m[12+i] );
  288.    }
  289. }
  290.  
  291. /**
  292.  * Dumps the contents of a GLmatrix structure.
  293.  *
  294.  * \param m pointer to the GLmatrix structure.
  295.  */
  296. void
  297. _math_matrix_print( const GLmatrix *m )
  298. {
  299.    GLfloat prod[16];
  300.  
  301.    _mesa_debug(NULL, "Matrix type: %s, flags: %x\n", types[m->type], m->flags);
  302.    print_matrix_floats(m->m);
  303.    _mesa_debug(NULL, "Inverse: \n");
  304.    print_matrix_floats(m->inv);
  305.    matmul4(prod, m->m, m->inv);
  306.    _mesa_debug(NULL, "Mat * Inverse:\n");
  307.    print_matrix_floats(prod);
  308. }
  309.  
  310. /*@}*/
  311.  
  312.  
  313. /**
  314.  * References an element of 4x4 matrix.
  315.  *
  316.  * \param m matrix array.
  317.  * \param c column of the desired element.
  318.  * \param r row of the desired element.
  319.  *
  320.  * \return value of the desired element.
  321.  *
  322.  * Calculate the linear storage index of the element and references it.
  323.  */
  324. #define MAT(m,r,c) (m)[(c)*4+(r)]
  325.  
  326.  
  327. /**********************************************************************/
  328. /** \name Matrix inversion */
  329. /*@{*/
  330.  
  331. /**
  332.  * Swaps the values of two floating point variables.
  333.  *
  334.  * Used by invert_matrix_general() to swap the row pointers.
  335.  */
  336. #define SWAP_ROWS(a, b) { GLfloat *_tmp = a; (a)=(b); (b)=_tmp; }
  337.  
  338. /**
  339.  * Compute inverse of 4x4 transformation matrix.
  340.  *
  341.  * \param mat pointer to a GLmatrix structure. The matrix inverse will be
  342.  * stored in the GLmatrix::inv attribute.
  343.  *
  344.  * \return GL_TRUE for success, GL_FALSE for failure (\p singular matrix).
  345.  *
  346.  * \author
  347.  * Code contributed by Jacques Leroy jle@star.be
  348.  *
  349.  * Calculates the inverse matrix by performing the gaussian matrix reduction
  350.  * with partial pivoting followed by back/substitution with the loops manually
  351.  * unrolled.
  352.  */
  353. static GLboolean invert_matrix_general( GLmatrix *mat )
  354. {
  355.    const GLfloat *m = mat->m;
  356.    GLfloat *out = mat->inv;
  357.    GLfloat wtmp[4][8];
  358.    GLfloat m0, m1, m2, m3, s;
  359.    GLfloat *r0, *r1, *r2, *r3;
  360.  
  361.    r0 = wtmp[0], r1 = wtmp[1], r2 = wtmp[2], r3 = wtmp[3];
  362.  
  363.    r0[0] = MAT(m,0,0), r0[1] = MAT(m,0,1),
  364.    r0[2] = MAT(m,0,2), r0[3] = MAT(m,0,3),
  365.    r0[4] = 1.0, r0[5] = r0[6] = r0[7] = 0.0,
  366.  
  367.    r1[0] = MAT(m,1,0), r1[1] = MAT(m,1,1),
  368.    r1[2] = MAT(m,1,2), r1[3] = MAT(m,1,3),
  369.    r1[5] = 1.0, r1[4] = r1[6] = r1[7] = 0.0,
  370.  
  371.    r2[0] = MAT(m,2,0), r2[1] = MAT(m,2,1),
  372.    r2[2] = MAT(m,2,2), r2[3] = MAT(m,2,3),
  373.    r2[6] = 1.0, r2[4] = r2[5] = r2[7] = 0.0,
  374.  
  375.    r3[0] = MAT(m,3,0), r3[1] = MAT(m,3,1),
  376.    r3[2] = MAT(m,3,2), r3[3] = MAT(m,3,3),
  377.    r3[7] = 1.0, r3[4] = r3[5] = r3[6] = 0.0;
  378.  
  379.    /* choose pivot - or die */
  380.    if (fabsf(r3[0])>fabsf(r2[0])) SWAP_ROWS(r3, r2);
  381.    if (fabsf(r2[0])>fabsf(r1[0])) SWAP_ROWS(r2, r1);
  382.    if (fabsf(r1[0])>fabsf(r0[0])) SWAP_ROWS(r1, r0);
  383.    if (0.0 == r0[0])  return GL_FALSE;
  384.  
  385.    /* eliminate first variable     */
  386.    m1 = r1[0]/r0[0]; m2 = r2[0]/r0[0]; m3 = r3[0]/r0[0];
  387.    s = r0[1]; r1[1] -= m1 * s; r2[1] -= m2 * s; r3[1] -= m3 * s;
  388.    s = r0[2]; r1[2] -= m1 * s; r2[2] -= m2 * s; r3[2] -= m3 * s;
  389.    s = r0[3]; r1[3] -= m1 * s; r2[3] -= m2 * s; r3[3] -= m3 * s;
  390.    s = r0[4];
  391.    if (s != 0.0) { r1[4] -= m1 * s; r2[4] -= m2 * s; r3[4] -= m3 * s; }
  392.    s = r0[5];
  393.    if (s != 0.0) { r1[5] -= m1 * s; r2[5] -= m2 * s; r3[5] -= m3 * s; }
  394.    s = r0[6];
  395.    if (s != 0.0) { r1[6] -= m1 * s; r2[6] -= m2 * s; r3[6] -= m3 * s; }
  396.    s = r0[7];
  397.    if (s != 0.0) { r1[7] -= m1 * s; r2[7] -= m2 * s; r3[7] -= m3 * s; }
  398.  
  399.    /* choose pivot - or die */
  400.    if (fabsf(r3[1])>fabsf(r2[1])) SWAP_ROWS(r3, r2);
  401.    if (fabsf(r2[1])>fabsf(r1[1])) SWAP_ROWS(r2, r1);
  402.    if (0.0 == r1[1])  return GL_FALSE;
  403.  
  404.    /* eliminate second variable */
  405.    m2 = r2[1]/r1[1]; m3 = r3[1]/r1[1];
  406.    r2[2] -= m2 * r1[2]; r3[2] -= m3 * r1[2];
  407.    r2[3] -= m2 * r1[3]; r3[3] -= m3 * r1[3];
  408.    s = r1[4]; if (0.0 != s) { r2[4] -= m2 * s; r3[4] -= m3 * s; }
  409.    s = r1[5]; if (0.0 != s) { r2[5] -= m2 * s; r3[5] -= m3 * s; }
  410.    s = r1[6]; if (0.0 != s) { r2[6] -= m2 * s; r3[6] -= m3 * s; }
  411.    s = r1[7]; if (0.0 != s) { r2[7] -= m2 * s; r3[7] -= m3 * s; }
  412.  
  413.    /* choose pivot - or die */
  414.    if (fabsf(r3[2])>fabsf(r2[2])) SWAP_ROWS(r3, r2);
  415.    if (0.0 == r2[2])  return GL_FALSE;
  416.  
  417.    /* eliminate third variable */
  418.    m3 = r3[2]/r2[2];
  419.    r3[3] -= m3 * r2[3], r3[4] -= m3 * r2[4],
  420.    r3[5] -= m3 * r2[5], r3[6] -= m3 * r2[6],
  421.    r3[7] -= m3 * r2[7];
  422.  
  423.    /* last check */
  424.    if (0.0 == r3[3]) return GL_FALSE;
  425.  
  426.    s = 1.0F/r3[3];             /* now back substitute row 3 */
  427.    r3[4] *= s; r3[5] *= s; r3[6] *= s; r3[7] *= s;
  428.  
  429.    m2 = r2[3];                 /* now back substitute row 2 */
  430.    s  = 1.0F/r2[2];
  431.    r2[4] = s * (r2[4] - r3[4] * m2), r2[5] = s * (r2[5] - r3[5] * m2),
  432.    r2[6] = s * (r2[6] - r3[6] * m2), r2[7] = s * (r2[7] - r3[7] * m2);
  433.    m1 = r1[3];
  434.    r1[4] -= r3[4] * m1, r1[5] -= r3[5] * m1,
  435.    r1[6] -= r3[6] * m1, r1[7] -= r3[7] * m1;
  436.    m0 = r0[3];
  437.    r0[4] -= r3[4] * m0, r0[5] -= r3[5] * m0,
  438.    r0[6] -= r3[6] * m0, r0[7] -= r3[7] * m0;
  439.  
  440.    m1 = r1[2];                 /* now back substitute row 1 */
  441.    s  = 1.0F/r1[1];
  442.    r1[4] = s * (r1[4] - r2[4] * m1), r1[5] = s * (r1[5] - r2[5] * m1),
  443.    r1[6] = s * (r1[6] - r2[6] * m1), r1[7] = s * (r1[7] - r2[7] * m1);
  444.    m0 = r0[2];
  445.    r0[4] -= r2[4] * m0, r0[5] -= r2[5] * m0,
  446.    r0[6] -= r2[6] * m0, r0[7] -= r2[7] * m0;
  447.  
  448.    m0 = r0[1];                 /* now back substitute row 0 */
  449.    s  = 1.0F/r0[0];
  450.    r0[4] = s * (r0[4] - r1[4] * m0), r0[5] = s * (r0[5] - r1[5] * m0),
  451.    r0[6] = s * (r0[6] - r1[6] * m0), r0[7] = s * (r0[7] - r1[7] * m0);
  452.  
  453.    MAT(out,0,0) = r0[4]; MAT(out,0,1) = r0[5],
  454.    MAT(out,0,2) = r0[6]; MAT(out,0,3) = r0[7],
  455.    MAT(out,1,0) = r1[4]; MAT(out,1,1) = r1[5],
  456.    MAT(out,1,2) = r1[6]; MAT(out,1,3) = r1[7],
  457.    MAT(out,2,0) = r2[4]; MAT(out,2,1) = r2[5],
  458.    MAT(out,2,2) = r2[6]; MAT(out,2,3) = r2[7],
  459.    MAT(out,3,0) = r3[4]; MAT(out,3,1) = r3[5],
  460.    MAT(out,3,2) = r3[6]; MAT(out,3,3) = r3[7];
  461.  
  462.    return GL_TRUE;
  463. }
  464. #undef SWAP_ROWS
  465.  
  466. /**
  467.  * Compute inverse of a general 3d transformation matrix.
  468.  *
  469.  * \param mat pointer to a GLmatrix structure. The matrix inverse will be
  470.  * stored in the GLmatrix::inv attribute.
  471.  *
  472.  * \return GL_TRUE for success, GL_FALSE for failure (\p singular matrix).
  473.  *
  474.  * \author Adapted from graphics gems II.
  475.  *
  476.  * Calculates the inverse of the upper left by first calculating its
  477.  * determinant and multiplying it to the symmetric adjust matrix of each
  478.  * element. Finally deals with the translation part by transforming the
  479.  * original translation vector using by the calculated submatrix inverse.
  480.  */
  481. static GLboolean invert_matrix_3d_general( GLmatrix *mat )
  482. {
  483.    const GLfloat *in = mat->m;
  484.    GLfloat *out = mat->inv;
  485.    GLfloat pos, neg, t;
  486.    GLfloat det;
  487.  
  488.    /* Calculate the determinant of upper left 3x3 submatrix and
  489.     * determine if the matrix is singular.
  490.     */
  491.    pos = neg = 0.0;
  492.    t =  MAT(in,0,0) * MAT(in,1,1) * MAT(in,2,2);
  493.    if (t >= 0.0) pos += t; else neg += t;
  494.  
  495.    t =  MAT(in,1,0) * MAT(in,2,1) * MAT(in,0,2);
  496.    if (t >= 0.0) pos += t; else neg += t;
  497.  
  498.    t =  MAT(in,2,0) * MAT(in,0,1) * MAT(in,1,2);
  499.    if (t >= 0.0) pos += t; else neg += t;
  500.  
  501.    t = -MAT(in,2,0) * MAT(in,1,1) * MAT(in,0,2);
  502.    if (t >= 0.0) pos += t; else neg += t;
  503.  
  504.    t = -MAT(in,1,0) * MAT(in,0,1) * MAT(in,2,2);
  505.    if (t >= 0.0) pos += t; else neg += t;
  506.  
  507.    t = -MAT(in,0,0) * MAT(in,2,1) * MAT(in,1,2);
  508.    if (t >= 0.0) pos += t; else neg += t;
  509.  
  510.    det = pos + neg;
  511.  
  512.    if (fabsf(det) < 1e-25)
  513.       return GL_FALSE;
  514.  
  515.    det = 1.0F / det;
  516.    MAT(out,0,0) = (  (MAT(in,1,1)*MAT(in,2,2) - MAT(in,2,1)*MAT(in,1,2) )*det);
  517.    MAT(out,0,1) = (- (MAT(in,0,1)*MAT(in,2,2) - MAT(in,2,1)*MAT(in,0,2) )*det);
  518.    MAT(out,0,2) = (  (MAT(in,0,1)*MAT(in,1,2) - MAT(in,1,1)*MAT(in,0,2) )*det);
  519.    MAT(out,1,0) = (- (MAT(in,1,0)*MAT(in,2,2) - MAT(in,2,0)*MAT(in,1,2) )*det);
  520.    MAT(out,1,1) = (  (MAT(in,0,0)*MAT(in,2,2) - MAT(in,2,0)*MAT(in,0,2) )*det);
  521.    MAT(out,1,2) = (- (MAT(in,0,0)*MAT(in,1,2) - MAT(in,1,0)*MAT(in,0,2) )*det);
  522.    MAT(out,2,0) = (  (MAT(in,1,0)*MAT(in,2,1) - MAT(in,2,0)*MAT(in,1,1) )*det);
  523.    MAT(out,2,1) = (- (MAT(in,0,0)*MAT(in,2,1) - MAT(in,2,0)*MAT(in,0,1) )*det);
  524.    MAT(out,2,2) = (  (MAT(in,0,0)*MAT(in,1,1) - MAT(in,1,0)*MAT(in,0,1) )*det);
  525.  
  526.    /* Do the translation part */
  527.    MAT(out,0,3) = - (MAT(in,0,3) * MAT(out,0,0) +
  528.                      MAT(in,1,3) * MAT(out,0,1) +
  529.                      MAT(in,2,3) * MAT(out,0,2) );
  530.    MAT(out,1,3) = - (MAT(in,0,3) * MAT(out,1,0) +
  531.                      MAT(in,1,3) * MAT(out,1,1) +
  532.                      MAT(in,2,3) * MAT(out,1,2) );
  533.    MAT(out,2,3) = - (MAT(in,0,3) * MAT(out,2,0) +
  534.                      MAT(in,1,3) * MAT(out,2,1) +
  535.                      MAT(in,2,3) * MAT(out,2,2) );
  536.  
  537.    return GL_TRUE;
  538. }
  539.  
  540. /**
  541.  * Compute inverse of a 3d transformation matrix.
  542.  *
  543.  * \param mat pointer to a GLmatrix structure. The matrix inverse will be
  544.  * stored in the GLmatrix::inv attribute.
  545.  *
  546.  * \return GL_TRUE for success, GL_FALSE for failure (\p singular matrix).
  547.  *
  548.  * If the matrix is not an angle preserving matrix then calls
  549.  * invert_matrix_3d_general for the actual calculation. Otherwise calculates
  550.  * the inverse matrix analyzing and inverting each of the scaling, rotation and
  551.  * translation parts.
  552.  */
  553. static GLboolean invert_matrix_3d( GLmatrix *mat )
  554. {
  555.    const GLfloat *in = mat->m;
  556.    GLfloat *out = mat->inv;
  557.  
  558.    if (!TEST_MAT_FLAGS(mat, MAT_FLAGS_ANGLE_PRESERVING)) {
  559.       return invert_matrix_3d_general( mat );
  560.    }
  561.  
  562.    if (mat->flags & MAT_FLAG_UNIFORM_SCALE) {
  563.       GLfloat scale = (MAT(in,0,0) * MAT(in,0,0) +
  564.                        MAT(in,0,1) * MAT(in,0,1) +
  565.                        MAT(in,0,2) * MAT(in,0,2));
  566.  
  567.       if (scale == 0.0)
  568.          return GL_FALSE;
  569.  
  570.       scale = 1.0F / scale;
  571.  
  572.       /* Transpose and scale the 3 by 3 upper-left submatrix. */
  573.       MAT(out,0,0) = scale * MAT(in,0,0);
  574.       MAT(out,1,0) = scale * MAT(in,0,1);
  575.       MAT(out,2,0) = scale * MAT(in,0,2);
  576.       MAT(out,0,1) = scale * MAT(in,1,0);
  577.       MAT(out,1,1) = scale * MAT(in,1,1);
  578.       MAT(out,2,1) = scale * MAT(in,1,2);
  579.       MAT(out,0,2) = scale * MAT(in,2,0);
  580.       MAT(out,1,2) = scale * MAT(in,2,1);
  581.       MAT(out,2,2) = scale * MAT(in,2,2);
  582.    }
  583.    else if (mat->flags & MAT_FLAG_ROTATION) {
  584.       /* Transpose the 3 by 3 upper-left submatrix. */
  585.       MAT(out,0,0) = MAT(in,0,0);
  586.       MAT(out,1,0) = MAT(in,0,1);
  587.       MAT(out,2,0) = MAT(in,0,2);
  588.       MAT(out,0,1) = MAT(in,1,0);
  589.       MAT(out,1,1) = MAT(in,1,1);
  590.       MAT(out,2,1) = MAT(in,1,2);
  591.       MAT(out,0,2) = MAT(in,2,0);
  592.       MAT(out,1,2) = MAT(in,2,1);
  593.       MAT(out,2,2) = MAT(in,2,2);
  594.    }
  595.    else {
  596.       /* pure translation */
  597.       memcpy( out, Identity, sizeof(Identity) );
  598.       MAT(out,0,3) = - MAT(in,0,3);
  599.       MAT(out,1,3) = - MAT(in,1,3);
  600.       MAT(out,2,3) = - MAT(in,2,3);
  601.       return GL_TRUE;
  602.    }
  603.  
  604.    if (mat->flags & MAT_FLAG_TRANSLATION) {
  605.       /* Do the translation part */
  606.       MAT(out,0,3) = - (MAT(in,0,3) * MAT(out,0,0) +
  607.                         MAT(in,1,3) * MAT(out,0,1) +
  608.                         MAT(in,2,3) * MAT(out,0,2) );
  609.       MAT(out,1,3) = - (MAT(in,0,3) * MAT(out,1,0) +
  610.                         MAT(in,1,3) * MAT(out,1,1) +
  611.                         MAT(in,2,3) * MAT(out,1,2) );
  612.       MAT(out,2,3) = - (MAT(in,0,3) * MAT(out,2,0) +
  613.                         MAT(in,1,3) * MAT(out,2,1) +
  614.                         MAT(in,2,3) * MAT(out,2,2) );
  615.    }
  616.    else {
  617.       MAT(out,0,3) = MAT(out,1,3) = MAT(out,2,3) = 0.0;
  618.    }
  619.  
  620.    return GL_TRUE;
  621. }
  622.  
  623. /**
  624.  * Compute inverse of an identity transformation matrix.
  625.  *
  626.  * \param mat pointer to a GLmatrix structure. The matrix inverse will be
  627.  * stored in the GLmatrix::inv attribute.
  628.  *
  629.  * \return always GL_TRUE.
  630.  *
  631.  * Simply copies Identity into GLmatrix::inv.
  632.  */
  633. static GLboolean invert_matrix_identity( GLmatrix *mat )
  634. {
  635.    memcpy( mat->inv, Identity, sizeof(Identity) );
  636.    return GL_TRUE;
  637. }
  638.  
  639. /**
  640.  * Compute inverse of a no-rotation 3d transformation matrix.
  641.  *
  642.  * \param mat pointer to a GLmatrix structure. The matrix inverse will be
  643.  * stored in the GLmatrix::inv attribute.
  644.  *
  645.  * \return GL_TRUE for success, GL_FALSE for failure (\p singular matrix).
  646.  *
  647.  * Calculates the
  648.  */
  649. static GLboolean invert_matrix_3d_no_rot( GLmatrix *mat )
  650. {
  651.    const GLfloat *in = mat->m;
  652.    GLfloat *out = mat->inv;
  653.  
  654.    if (MAT(in,0,0) == 0 || MAT(in,1,1) == 0 || MAT(in,2,2) == 0 )
  655.       return GL_FALSE;
  656.  
  657.    memcpy( out, Identity, 16 * sizeof(GLfloat) );
  658.    MAT(out,0,0) = 1.0F / MAT(in,0,0);
  659.    MAT(out,1,1) = 1.0F / MAT(in,1,1);
  660.    MAT(out,2,2) = 1.0F / MAT(in,2,2);
  661.  
  662.    if (mat->flags & MAT_FLAG_TRANSLATION) {
  663.       MAT(out,0,3) = - (MAT(in,0,3) * MAT(out,0,0));
  664.       MAT(out,1,3) = - (MAT(in,1,3) * MAT(out,1,1));
  665.       MAT(out,2,3) = - (MAT(in,2,3) * MAT(out,2,2));
  666.    }
  667.  
  668.    return GL_TRUE;
  669. }
  670.  
  671. /**
  672.  * Compute inverse of a no-rotation 2d transformation matrix.
  673.  *
  674.  * \param mat pointer to a GLmatrix structure. The matrix inverse will be
  675.  * stored in the GLmatrix::inv attribute.
  676.  *
  677.  * \return GL_TRUE for success, GL_FALSE for failure (\p singular matrix).
  678.  *
  679.  * Calculates the inverse matrix by applying the inverse scaling and
  680.  * translation to the identity matrix.
  681.  */
  682. static GLboolean invert_matrix_2d_no_rot( GLmatrix *mat )
  683. {
  684.    const GLfloat *in = mat->m;
  685.    GLfloat *out = mat->inv;
  686.  
  687.    if (MAT(in,0,0) == 0 || MAT(in,1,1) == 0)
  688.       return GL_FALSE;
  689.  
  690.    memcpy( out, Identity, 16 * sizeof(GLfloat) );
  691.    MAT(out,0,0) = 1.0F / MAT(in,0,0);
  692.    MAT(out,1,1) = 1.0F / MAT(in,1,1);
  693.  
  694.    if (mat->flags & MAT_FLAG_TRANSLATION) {
  695.       MAT(out,0,3) = - (MAT(in,0,3) * MAT(out,0,0));
  696.       MAT(out,1,3) = - (MAT(in,1,3) * MAT(out,1,1));
  697.    }
  698.  
  699.    return GL_TRUE;
  700. }
  701.  
  702. #if 0
  703. /* broken */
  704. static GLboolean invert_matrix_perspective( GLmatrix *mat )
  705. {
  706.    const GLfloat *in = mat->m;
  707.    GLfloat *out = mat->inv;
  708.  
  709.    if (MAT(in,2,3) == 0)
  710.       return GL_FALSE;
  711.  
  712.    memcpy( out, Identity, 16 * sizeof(GLfloat) );
  713.  
  714.    MAT(out,0,0) = 1.0F / MAT(in,0,0);
  715.    MAT(out,1,1) = 1.0F / MAT(in,1,1);
  716.  
  717.    MAT(out,0,3) = MAT(in,0,2);
  718.    MAT(out,1,3) = MAT(in,1,2);
  719.  
  720.    MAT(out,2,2) = 0;
  721.    MAT(out,2,3) = -1;
  722.  
  723.    MAT(out,3,2) = 1.0F / MAT(in,2,3);
  724.    MAT(out,3,3) = MAT(in,2,2) * MAT(out,3,2);
  725.  
  726.    return GL_TRUE;
  727. }
  728. #endif
  729.  
  730. /**
  731.  * Matrix inversion function pointer type.
  732.  */
  733. typedef GLboolean (*inv_mat_func)( GLmatrix *mat );
  734.  
  735. /**
  736.  * Table of the matrix inversion functions according to the matrix type.
  737.  */
  738. static inv_mat_func inv_mat_tab[7] = {
  739.    invert_matrix_general,
  740.    invert_matrix_identity,
  741.    invert_matrix_3d_no_rot,
  742. #if 0
  743.    /* Don't use this function for now - it fails when the projection matrix
  744.     * is premultiplied by a translation (ala Chromium's tilesort SPU).
  745.     */
  746.    invert_matrix_perspective,
  747. #else
  748.    invert_matrix_general,
  749. #endif
  750.    invert_matrix_3d,            /* lazy! */
  751.    invert_matrix_2d_no_rot,
  752.    invert_matrix_3d
  753. };
  754.  
  755. /**
  756.  * Compute inverse of a transformation matrix.
  757.  *
  758.  * \param mat pointer to a GLmatrix structure. The matrix inverse will be
  759.  * stored in the GLmatrix::inv attribute.
  760.  *
  761.  * \return GL_TRUE for success, GL_FALSE for failure (\p singular matrix).
  762.  *
  763.  * Calls the matrix inversion function in inv_mat_tab corresponding to the
  764.  * given matrix type.  In case of failure, updates the MAT_FLAG_SINGULAR flag,
  765.  * and copies the identity matrix into GLmatrix::inv.
  766.  */
  767. static GLboolean matrix_invert( GLmatrix *mat )
  768. {
  769.    if (inv_mat_tab[mat->type](mat)) {
  770.       mat->flags &= ~MAT_FLAG_SINGULAR;
  771.       return GL_TRUE;
  772.    } else {
  773.       mat->flags |= MAT_FLAG_SINGULAR;
  774.       memcpy( mat->inv, Identity, sizeof(Identity) );
  775.       return GL_FALSE;
  776.    }
  777. }
  778.  
  779. /*@}*/
  780.  
  781.  
  782. /**********************************************************************/
  783. /** \name Matrix generation */
  784. /*@{*/
  785.  
  786. /**
  787.  * Generate a 4x4 transformation matrix from glRotate parameters, and
  788.  * post-multiply the input matrix by it.
  789.  *
  790.  * \author
  791.  * This function was contributed by Erich Boleyn (erich@uruk.org).
  792.  * Optimizations contributed by Rudolf Opalla (rudi@khm.de).
  793.  */
  794. void
  795. _math_matrix_rotate( GLmatrix *mat,
  796.                      GLfloat angle, GLfloat x, GLfloat y, GLfloat z )
  797. {
  798.    GLfloat xx, yy, zz, xy, yz, zx, xs, ys, zs, one_c, s, c;
  799.    GLfloat m[16];
  800.    GLboolean optimized;
  801.  
  802.    s = (GLfloat) sin( angle * M_PI / 180.0 );
  803.    c = (GLfloat) cos( angle * M_PI / 180.0 );
  804.  
  805.    memcpy(m, Identity, sizeof(GLfloat)*16);
  806.    optimized = GL_FALSE;
  807.  
  808. #define M(row,col)  m[col*4+row]
  809.  
  810.    if (x == 0.0F) {
  811.       if (y == 0.0F) {
  812.          if (z != 0.0F) {
  813.             optimized = GL_TRUE;
  814.             /* rotate only around z-axis */
  815.             M(0,0) = c;
  816.             M(1,1) = c;
  817.             if (z < 0.0F) {
  818.                M(0,1) = s;
  819.                M(1,0) = -s;
  820.             }
  821.             else {
  822.                M(0,1) = -s;
  823.                M(1,0) = s;
  824.             }
  825.          }
  826.       }
  827.       else if (z == 0.0F) {
  828.          optimized = GL_TRUE;
  829.          /* rotate only around y-axis */
  830.          M(0,0) = c;
  831.          M(2,2) = c;
  832.          if (y < 0.0F) {
  833.             M(0,2) = -s;
  834.             M(2,0) = s;
  835.          }
  836.          else {
  837.             M(0,2) = s;
  838.             M(2,0) = -s;
  839.          }
  840.       }
  841.    }
  842.    else if (y == 0.0F) {
  843.       if (z == 0.0F) {
  844.          optimized = GL_TRUE;
  845.          /* rotate only around x-axis */
  846.          M(1,1) = c;
  847.          M(2,2) = c;
  848.          if (x < 0.0F) {
  849.             M(1,2) = s;
  850.             M(2,1) = -s;
  851.          }
  852.          else {
  853.             M(1,2) = -s;
  854.             M(2,1) = s;
  855.          }
  856.       }
  857.    }
  858.  
  859.    if (!optimized) {
  860.       const GLfloat mag = sqrtf(x * x + y * y + z * z);
  861.  
  862.       if (mag <= 1.0e-4) {
  863.          /* no rotation, leave mat as-is */
  864.          return;
  865.       }
  866.  
  867.       x /= mag;
  868.       y /= mag;
  869.       z /= mag;
  870.  
  871.  
  872.       /*
  873.        *     Arbitrary axis rotation matrix.
  874.        *
  875.        *  This is composed of 5 matrices, Rz, Ry, T, Ry', Rz', multiplied
  876.        *  like so:  Rz * Ry * T * Ry' * Rz'.  T is the final rotation
  877.        *  (which is about the X-axis), and the two composite transforms
  878.        *  Ry' * Rz' and Rz * Ry are (respectively) the rotations necessary
  879.        *  from the arbitrary axis to the X-axis then back.  They are
  880.        *  all elementary rotations.
  881.        *
  882.        *  Rz' is a rotation about the Z-axis, to bring the axis vector
  883.        *  into the x-z plane.  Then Ry' is applied, rotating about the
  884.        *  Y-axis to bring the axis vector parallel with the X-axis.  The
  885.        *  rotation about the X-axis is then performed.  Ry and Rz are
  886.        *  simply the respective inverse transforms to bring the arbitrary
  887.        *  axis back to its original orientation.  The first transforms
  888.        *  Rz' and Ry' are considered inverses, since the data from the
  889.        *  arbitrary axis gives you info on how to get to it, not how
  890.        *  to get away from it, and an inverse must be applied.
  891.        *
  892.        *  The basic calculation used is to recognize that the arbitrary
  893.        *  axis vector (x, y, z), since it is of unit length, actually
  894.        *  represents the sines and cosines of the angles to rotate the
  895.        *  X-axis to the same orientation, with theta being the angle about
  896.        *  Z and phi the angle about Y (in the order described above)
  897.        *  as follows:
  898.        *
  899.        *  cos ( theta ) = x / sqrt ( 1 - z^2 )
  900.        *  sin ( theta ) = y / sqrt ( 1 - z^2 )
  901.        *
  902.        *  cos ( phi ) = sqrt ( 1 - z^2 )
  903.        *  sin ( phi ) = z
  904.        *
  905.        *  Note that cos ( phi ) can further be inserted to the above
  906.        *  formulas:
  907.        *
  908.        *  cos ( theta ) = x / cos ( phi )
  909.        *  sin ( theta ) = y / sin ( phi )
  910.        *
  911.        *  ...etc.  Because of those relations and the standard trigonometric
  912.        *  relations, it is pssible to reduce the transforms down to what
  913.        *  is used below.  It may be that any primary axis chosen will give the
  914.        *  same results (modulo a sign convention) using thie method.
  915.        *
  916.        *  Particularly nice is to notice that all divisions that might
  917.        *  have caused trouble when parallel to certain planes or
  918.        *  axis go away with care paid to reducing the expressions.
  919.        *  After checking, it does perform correctly under all cases, since
  920.        *  in all the cases of division where the denominator would have
  921.        *  been zero, the numerator would have been zero as well, giving
  922.        *  the expected result.
  923.        */
  924.  
  925.       xx = x * x;
  926.       yy = y * y;
  927.       zz = z * z;
  928.       xy = x * y;
  929.       yz = y * z;
  930.       zx = z * x;
  931.       xs = x * s;
  932.       ys = y * s;
  933.       zs = z * s;
  934.       one_c = 1.0F - c;
  935.  
  936.       /* We already hold the identity-matrix so we can skip some statements */
  937.       M(0,0) = (one_c * xx) + c;
  938.       M(0,1) = (one_c * xy) - zs;
  939.       M(0,2) = (one_c * zx) + ys;
  940. /*    M(0,3) = 0.0F; */
  941.  
  942.       M(1,0) = (one_c * xy) + zs;
  943.       M(1,1) = (one_c * yy) + c;
  944.       M(1,2) = (one_c * yz) - xs;
  945. /*    M(1,3) = 0.0F; */
  946.  
  947.       M(2,0) = (one_c * zx) - ys;
  948.       M(2,1) = (one_c * yz) + xs;
  949.       M(2,2) = (one_c * zz) + c;
  950. /*    M(2,3) = 0.0F; */
  951.  
  952. /*
  953.       M(3,0) = 0.0F;
  954.       M(3,1) = 0.0F;
  955.       M(3,2) = 0.0F;
  956.       M(3,3) = 1.0F;
  957. */
  958.    }
  959. #undef M
  960.  
  961.    matrix_multf( mat, m, MAT_FLAG_ROTATION );
  962. }
  963.  
  964. /**
  965.  * Apply a perspective projection matrix.
  966.  *
  967.  * \param mat matrix to apply the projection.
  968.  * \param left left clipping plane coordinate.
  969.  * \param right right clipping plane coordinate.
  970.  * \param bottom bottom clipping plane coordinate.
  971.  * \param top top clipping plane coordinate.
  972.  * \param nearval distance to the near clipping plane.
  973.  * \param farval distance to the far clipping plane.
  974.  *
  975.  * Creates the projection matrix and multiplies it with \p mat, marking the
  976.  * MAT_FLAG_PERSPECTIVE flag.
  977.  */
  978. void
  979. _math_matrix_frustum( GLmatrix *mat,
  980.                       GLfloat left, GLfloat right,
  981.                       GLfloat bottom, GLfloat top,
  982.                       GLfloat nearval, GLfloat farval )
  983. {
  984.    GLfloat x, y, a, b, c, d;
  985.    GLfloat m[16];
  986.  
  987.    x = (2.0F*nearval) / (right-left);
  988.    y = (2.0F*nearval) / (top-bottom);
  989.    a = (right+left) / (right-left);
  990.    b = (top+bottom) / (top-bottom);
  991.    c = -(farval+nearval) / ( farval-nearval);
  992.    d = -(2.0F*farval*nearval) / (farval-nearval);  /* error? */
  993.  
  994. #define M(row,col)  m[col*4+row]
  995.    M(0,0) = x;     M(0,1) = 0.0F;  M(0,2) = a;      M(0,3) = 0.0F;
  996.    M(1,0) = 0.0F;  M(1,1) = y;     M(1,2) = b;      M(1,3) = 0.0F;
  997.    M(2,0) = 0.0F;  M(2,1) = 0.0F;  M(2,2) = c;      M(2,3) = d;
  998.    M(3,0) = 0.0F;  M(3,1) = 0.0F;  M(3,2) = -1.0F;  M(3,3) = 0.0F;
  999. #undef M
  1000.  
  1001.    matrix_multf( mat, m, MAT_FLAG_PERSPECTIVE );
  1002. }
  1003.  
  1004. /**
  1005.  * Apply an orthographic projection matrix.
  1006.  *
  1007.  * \param mat matrix to apply the projection.
  1008.  * \param left left clipping plane coordinate.
  1009.  * \param right right clipping plane coordinate.
  1010.  * \param bottom bottom clipping plane coordinate.
  1011.  * \param top top clipping plane coordinate.
  1012.  * \param nearval distance to the near clipping plane.
  1013.  * \param farval distance to the far clipping plane.
  1014.  *
  1015.  * Creates the projection matrix and multiplies it with \p mat, marking the
  1016.  * MAT_FLAG_GENERAL_SCALE and MAT_FLAG_TRANSLATION flags.
  1017.  */
  1018. void
  1019. _math_matrix_ortho( GLmatrix *mat,
  1020.                     GLfloat left, GLfloat right,
  1021.                     GLfloat bottom, GLfloat top,
  1022.                     GLfloat nearval, GLfloat farval )
  1023. {
  1024.    GLfloat m[16];
  1025.  
  1026. #define M(row,col)  m[col*4+row]
  1027.    M(0,0) = 2.0F / (right-left);
  1028.    M(0,1) = 0.0F;
  1029.    M(0,2) = 0.0F;
  1030.    M(0,3) = -(right+left) / (right-left);
  1031.  
  1032.    M(1,0) = 0.0F;
  1033.    M(1,1) = 2.0F / (top-bottom);
  1034.    M(1,2) = 0.0F;
  1035.    M(1,3) = -(top+bottom) / (top-bottom);
  1036.  
  1037.    M(2,0) = 0.0F;
  1038.    M(2,1) = 0.0F;
  1039.    M(2,2) = -2.0F / (farval-nearval);
  1040.    M(2,3) = -(farval+nearval) / (farval-nearval);
  1041.  
  1042.    M(3,0) = 0.0F;
  1043.    M(3,1) = 0.0F;
  1044.    M(3,2) = 0.0F;
  1045.    M(3,3) = 1.0F;
  1046. #undef M
  1047.  
  1048.    matrix_multf( mat, m, (MAT_FLAG_GENERAL_SCALE|MAT_FLAG_TRANSLATION));
  1049. }
  1050.  
  1051. /**
  1052.  * Multiply a matrix with a general scaling matrix.
  1053.  *
  1054.  * \param mat matrix.
  1055.  * \param x x axis scale factor.
  1056.  * \param y y axis scale factor.
  1057.  * \param z z axis scale factor.
  1058.  *
  1059.  * Multiplies in-place the elements of \p mat by the scale factors. Checks if
  1060.  * the scales factors are roughly the same, marking the MAT_FLAG_UNIFORM_SCALE
  1061.  * flag, or MAT_FLAG_GENERAL_SCALE. Marks the MAT_DIRTY_TYPE and
  1062.  * MAT_DIRTY_INVERSE dirty flags.
  1063.  */
  1064. void
  1065. _math_matrix_scale( GLmatrix *mat, GLfloat x, GLfloat y, GLfloat z )
  1066. {
  1067.    GLfloat *m = mat->m;
  1068.    m[0] *= x;   m[4] *= y;   m[8]  *= z;
  1069.    m[1] *= x;   m[5] *= y;   m[9]  *= z;
  1070.    m[2] *= x;   m[6] *= y;   m[10] *= z;
  1071.    m[3] *= x;   m[7] *= y;   m[11] *= z;
  1072.  
  1073.    if (fabsf(x - y) < 1e-8 && fabsf(x - z) < 1e-8)
  1074.       mat->flags |= MAT_FLAG_UNIFORM_SCALE;
  1075.    else
  1076.       mat->flags |= MAT_FLAG_GENERAL_SCALE;
  1077.  
  1078.    mat->flags |= (MAT_DIRTY_TYPE |
  1079.                   MAT_DIRTY_INVERSE);
  1080. }
  1081.  
  1082. /**
  1083.  * Multiply a matrix with a translation matrix.
  1084.  *
  1085.  * \param mat matrix.
  1086.  * \param x translation vector x coordinate.
  1087.  * \param y translation vector y coordinate.
  1088.  * \param z translation vector z coordinate.
  1089.  *
  1090.  * Adds the translation coordinates to the elements of \p mat in-place.  Marks
  1091.  * the MAT_FLAG_TRANSLATION flag, and the MAT_DIRTY_TYPE and MAT_DIRTY_INVERSE
  1092.  * dirty flags.
  1093.  */
  1094. void
  1095. _math_matrix_translate( GLmatrix *mat, GLfloat x, GLfloat y, GLfloat z )
  1096. {
  1097.    GLfloat *m = mat->m;
  1098.    m[12] = m[0] * x + m[4] * y + m[8]  * z + m[12];
  1099.    m[13] = m[1] * x + m[5] * y + m[9]  * z + m[13];
  1100.    m[14] = m[2] * x + m[6] * y + m[10] * z + m[14];
  1101.    m[15] = m[3] * x + m[7] * y + m[11] * z + m[15];
  1102.  
  1103.    mat->flags |= (MAT_FLAG_TRANSLATION |
  1104.                   MAT_DIRTY_TYPE |
  1105.                   MAT_DIRTY_INVERSE);
  1106. }
  1107.  
  1108.  
  1109. /**
  1110.  * Set matrix to do viewport and depthrange mapping.
  1111.  * Transforms Normalized Device Coords to window/Z values.
  1112.  */
  1113. void
  1114. _math_matrix_viewport(GLmatrix *m, const double scale[3],
  1115.                       const double translate[3], double depthMax)
  1116. {
  1117.    m->m[MAT_SX] = scale[0];
  1118.    m->m[MAT_TX] = translate[0];
  1119.    m->m[MAT_SY] = scale[1];
  1120.    m->m[MAT_TY] = translate[1];
  1121.    m->m[MAT_SZ] = depthMax*scale[2];
  1122.    m->m[MAT_TZ] = depthMax*translate[2];
  1123.    m->flags = MAT_FLAG_GENERAL_SCALE | MAT_FLAG_TRANSLATION;
  1124.    m->type = MATRIX_3D_NO_ROT;
  1125. }
  1126.  
  1127.  
  1128. /**
  1129.  * Set a matrix to the identity matrix.
  1130.  *
  1131.  * \param mat matrix.
  1132.  *
  1133.  * Copies ::Identity into \p GLmatrix::m, and into GLmatrix::inv if not NULL.
  1134.  * Sets the matrix type to identity, and clear the dirty flags.
  1135.  */
  1136. void
  1137. _math_matrix_set_identity( GLmatrix *mat )
  1138. {
  1139.    memcpy( mat->m, Identity, 16*sizeof(GLfloat) );
  1140.    memcpy( mat->inv, Identity, 16*sizeof(GLfloat) );
  1141.  
  1142.    mat->type = MATRIX_IDENTITY;
  1143.    mat->flags &= ~(MAT_DIRTY_FLAGS|
  1144.                    MAT_DIRTY_TYPE|
  1145.                    MAT_DIRTY_INVERSE);
  1146. }
  1147.  
  1148. /*@}*/
  1149.  
  1150.  
  1151. /**********************************************************************/
  1152. /** \name Matrix analysis */
  1153. /*@{*/
  1154.  
  1155. #define ZERO(x) (1<<x)
  1156. #define ONE(x)  (1<<(x+16))
  1157.  
  1158. #define MASK_NO_TRX      (ZERO(12) | ZERO(13) | ZERO(14))
  1159. #define MASK_NO_2D_SCALE ( ONE(0)  | ONE(5))
  1160.  
  1161. #define MASK_IDENTITY    ( ONE(0)  | ZERO(4)  | ZERO(8)  | ZERO(12) |\
  1162.                           ZERO(1)  |  ONE(5)  | ZERO(9)  | ZERO(13) |\
  1163.                           ZERO(2)  | ZERO(6)  |  ONE(10) | ZERO(14) |\
  1164.                           ZERO(3)  | ZERO(7)  | ZERO(11) |  ONE(15) )
  1165.  
  1166. #define MASK_2D_NO_ROT   (           ZERO(4)  | ZERO(8)  |           \
  1167.                           ZERO(1)  |            ZERO(9)  |           \
  1168.                           ZERO(2)  | ZERO(6)  |  ONE(10) | ZERO(14) |\
  1169.                           ZERO(3)  | ZERO(7)  | ZERO(11) |  ONE(15) )
  1170.  
  1171. #define MASK_2D          (                      ZERO(8)  |           \
  1172.                                                 ZERO(9)  |           \
  1173.                           ZERO(2)  | ZERO(6)  |  ONE(10) | ZERO(14) |\
  1174.                           ZERO(3)  | ZERO(7)  | ZERO(11) |  ONE(15) )
  1175.  
  1176.  
  1177. #define MASK_3D_NO_ROT   (           ZERO(4)  | ZERO(8)  |           \
  1178.                           ZERO(1)  |            ZERO(9)  |           \
  1179.                           ZERO(2)  | ZERO(6)  |                      \
  1180.                           ZERO(3)  | ZERO(7)  | ZERO(11) |  ONE(15) )
  1181.  
  1182. #define MASK_3D          (                                           \
  1183.                                                                      \
  1184.                                                                      \
  1185.                           ZERO(3)  | ZERO(7)  | ZERO(11) |  ONE(15) )
  1186.  
  1187.  
  1188. #define MASK_PERSPECTIVE (           ZERO(4)  |            ZERO(12) |\
  1189.                           ZERO(1)  |                       ZERO(13) |\
  1190.                           ZERO(2)  | ZERO(6)  |                      \
  1191.                           ZERO(3)  | ZERO(7)  |            ZERO(15) )
  1192.  
  1193. #define SQ(x) ((x)*(x))
  1194.  
  1195. /**
  1196.  * Determine type and flags from scratch.  
  1197.  *
  1198.  * \param mat matrix.
  1199.  *
  1200.  * This is expensive enough to only want to do it once.
  1201.  */
  1202. static void analyse_from_scratch( GLmatrix *mat )
  1203. {
  1204.    const GLfloat *m = mat->m;
  1205.    GLuint mask = 0;
  1206.    GLuint i;
  1207.  
  1208.    for (i = 0 ; i < 16 ; i++) {
  1209.       if (m[i] == 0.0) mask |= (1<<i);
  1210.    }
  1211.  
  1212.    if (m[0] == 1.0F) mask |= (1<<16);
  1213.    if (m[5] == 1.0F) mask |= (1<<21);
  1214.    if (m[10] == 1.0F) mask |= (1<<26);
  1215.    if (m[15] == 1.0F) mask |= (1<<31);
  1216.  
  1217.    mat->flags &= ~MAT_FLAGS_GEOMETRY;
  1218.  
  1219.    /* Check for translation - no-one really cares
  1220.     */
  1221.    if ((mask & MASK_NO_TRX) != MASK_NO_TRX)
  1222.       mat->flags |= MAT_FLAG_TRANSLATION;
  1223.  
  1224.    /* Do the real work
  1225.     */
  1226.    if (mask == (GLuint) MASK_IDENTITY) {
  1227.       mat->type = MATRIX_IDENTITY;
  1228.    }
  1229.    else if ((mask & MASK_2D_NO_ROT) == (GLuint) MASK_2D_NO_ROT) {
  1230.       mat->type = MATRIX_2D_NO_ROT;
  1231.  
  1232.       if ((mask & MASK_NO_2D_SCALE) != MASK_NO_2D_SCALE)
  1233.          mat->flags |= MAT_FLAG_GENERAL_SCALE;
  1234.    }
  1235.    else if ((mask & MASK_2D) == (GLuint) MASK_2D) {
  1236.       GLfloat mm = DOT2(m, m);
  1237.       GLfloat m4m4 = DOT2(m+4,m+4);
  1238.       GLfloat mm4 = DOT2(m,m+4);
  1239.  
  1240.       mat->type = MATRIX_2D;
  1241.  
  1242.       /* Check for scale */
  1243.       if (SQ(mm-1) > SQ(1e-6) ||
  1244.           SQ(m4m4-1) > SQ(1e-6))
  1245.          mat->flags |= MAT_FLAG_GENERAL_SCALE;
  1246.  
  1247.       /* Check for rotation */
  1248.       if (SQ(mm4) > SQ(1e-6))
  1249.          mat->flags |= MAT_FLAG_GENERAL_3D;
  1250.       else
  1251.          mat->flags |= MAT_FLAG_ROTATION;
  1252.  
  1253.    }
  1254.    else if ((mask & MASK_3D_NO_ROT) == (GLuint) MASK_3D_NO_ROT) {
  1255.       mat->type = MATRIX_3D_NO_ROT;
  1256.  
  1257.       /* Check for scale */
  1258.       if (SQ(m[0]-m[5]) < SQ(1e-6) &&
  1259.           SQ(m[0]-m[10]) < SQ(1e-6)) {
  1260.          if (SQ(m[0]-1.0) > SQ(1e-6)) {
  1261.             mat->flags |= MAT_FLAG_UNIFORM_SCALE;
  1262.          }
  1263.       }
  1264.       else {
  1265.          mat->flags |= MAT_FLAG_GENERAL_SCALE;
  1266.       }
  1267.    }
  1268.    else if ((mask & MASK_3D) == (GLuint) MASK_3D) {
  1269.       GLfloat c1 = DOT3(m,m);
  1270.       GLfloat c2 = DOT3(m+4,m+4);
  1271.       GLfloat c3 = DOT3(m+8,m+8);
  1272.       GLfloat d1 = DOT3(m, m+4);
  1273.       GLfloat cp[3];
  1274.  
  1275.       mat->type = MATRIX_3D;
  1276.  
  1277.       /* Check for scale */
  1278.       if (SQ(c1-c2) < SQ(1e-6) && SQ(c1-c3) < SQ(1e-6)) {
  1279.          if (SQ(c1-1.0) > SQ(1e-6))
  1280.             mat->flags |= MAT_FLAG_UNIFORM_SCALE;
  1281.          /* else no scale at all */
  1282.       }
  1283.       else {
  1284.          mat->flags |= MAT_FLAG_GENERAL_SCALE;
  1285.       }
  1286.  
  1287.       /* Check for rotation */
  1288.       if (SQ(d1) < SQ(1e-6)) {
  1289.          CROSS3( cp, m, m+4 );
  1290.          SUB_3V( cp, cp, (m+8) );
  1291.          if (LEN_SQUARED_3FV(cp) < SQ(1e-6))
  1292.             mat->flags |= MAT_FLAG_ROTATION;
  1293.          else
  1294.             mat->flags |= MAT_FLAG_GENERAL_3D;
  1295.       }
  1296.       else {
  1297.          mat->flags |= MAT_FLAG_GENERAL_3D; /* shear, etc */
  1298.       }
  1299.    }
  1300.    else if ((mask & MASK_PERSPECTIVE) == MASK_PERSPECTIVE && m[11]==-1.0F) {
  1301.       mat->type = MATRIX_PERSPECTIVE;
  1302.       mat->flags |= MAT_FLAG_GENERAL;
  1303.    }
  1304.    else {
  1305.       mat->type = MATRIX_GENERAL;
  1306.       mat->flags |= MAT_FLAG_GENERAL;
  1307.    }
  1308. }
  1309.  
  1310. /**
  1311.  * Analyze a matrix given that its flags are accurate.
  1312.  *
  1313.  * This is the more common operation, hopefully.
  1314.  */
  1315. static void analyse_from_flags( GLmatrix *mat )
  1316. {
  1317.    const GLfloat *m = mat->m;
  1318.  
  1319.    if (TEST_MAT_FLAGS(mat, 0)) {
  1320.       mat->type = MATRIX_IDENTITY;
  1321.    }
  1322.    else if (TEST_MAT_FLAGS(mat, (MAT_FLAG_TRANSLATION |
  1323.                                  MAT_FLAG_UNIFORM_SCALE |
  1324.                                  MAT_FLAG_GENERAL_SCALE))) {
  1325.       if ( m[10]==1.0F && m[14]==0.0F ) {
  1326.          mat->type = MATRIX_2D_NO_ROT;
  1327.       }
  1328.       else {
  1329.          mat->type = MATRIX_3D_NO_ROT;
  1330.       }
  1331.    }
  1332.    else if (TEST_MAT_FLAGS(mat, MAT_FLAGS_3D)) {
  1333.       if (                                 m[ 8]==0.0F
  1334.             &&                             m[ 9]==0.0F
  1335.             && m[2]==0.0F && m[6]==0.0F && m[10]==1.0F && m[14]==0.0F) {
  1336.          mat->type = MATRIX_2D;
  1337.       }
  1338.       else {
  1339.          mat->type = MATRIX_3D;
  1340.       }
  1341.    }
  1342.    else if (                 m[4]==0.0F                 && m[12]==0.0F
  1343.             && m[1]==0.0F                               && m[13]==0.0F
  1344.             && m[2]==0.0F && m[6]==0.0F
  1345.             && m[3]==0.0F && m[7]==0.0F && m[11]==-1.0F && m[15]==0.0F) {
  1346.       mat->type = MATRIX_PERSPECTIVE;
  1347.    }
  1348.    else {
  1349.       mat->type = MATRIX_GENERAL;
  1350.    }
  1351. }
  1352.  
  1353. /**
  1354.  * Analyze and update a matrix.
  1355.  *
  1356.  * \param mat matrix.
  1357.  *
  1358.  * If the matrix type is dirty then calls either analyse_from_scratch() or
  1359.  * analyse_from_flags() to determine its type, according to whether the flags
  1360.  * are dirty or not, respectively. If the matrix has an inverse and it's dirty
  1361.  * then calls matrix_invert(). Finally clears the dirty flags.
  1362.  */
  1363. void
  1364. _math_matrix_analyse( GLmatrix *mat )
  1365. {
  1366.    if (mat->flags & MAT_DIRTY_TYPE) {
  1367.       if (mat->flags & MAT_DIRTY_FLAGS)
  1368.          analyse_from_scratch( mat );
  1369.       else
  1370.          analyse_from_flags( mat );
  1371.    }
  1372.  
  1373.    if (mat->inv && (mat->flags & MAT_DIRTY_INVERSE)) {
  1374.       matrix_invert( mat );
  1375.       mat->flags &= ~MAT_DIRTY_INVERSE;
  1376.    }
  1377.  
  1378.    mat->flags &= ~(MAT_DIRTY_FLAGS | MAT_DIRTY_TYPE);
  1379. }
  1380.  
  1381. /*@}*/
  1382.  
  1383.  
  1384. /**
  1385.  * Test if the given matrix preserves vector lengths.
  1386.  */
  1387. GLboolean
  1388. _math_matrix_is_length_preserving( const GLmatrix *m )
  1389. {
  1390.    return TEST_MAT_FLAGS( m, MAT_FLAGS_LENGTH_PRESERVING);
  1391. }
  1392.  
  1393.  
  1394. /**
  1395.  * Test if the given matrix does any rotation.
  1396.  * (or perhaps if the upper-left 3x3 is non-identity)
  1397.  */
  1398. GLboolean
  1399. _math_matrix_has_rotation( const GLmatrix *m )
  1400. {
  1401.    if (m->flags & (MAT_FLAG_GENERAL |
  1402.                    MAT_FLAG_ROTATION |
  1403.                    MAT_FLAG_GENERAL_3D |
  1404.                    MAT_FLAG_PERSPECTIVE))
  1405.       return GL_TRUE;
  1406.    else
  1407.       return GL_FALSE;
  1408. }
  1409.  
  1410.  
  1411. GLboolean
  1412. _math_matrix_is_general_scale( const GLmatrix *m )
  1413. {
  1414.    return (m->flags & MAT_FLAG_GENERAL_SCALE) ? GL_TRUE : GL_FALSE;
  1415. }
  1416.  
  1417.  
  1418. GLboolean
  1419. _math_matrix_is_dirty( const GLmatrix *m )
  1420. {
  1421.    return (m->flags & MAT_DIRTY) ? GL_TRUE : GL_FALSE;
  1422. }
  1423.  
  1424.  
  1425. /**********************************************************************/
  1426. /** \name Matrix setup */
  1427. /*@{*/
  1428.  
  1429. /**
  1430.  * Copy a matrix.
  1431.  *
  1432.  * \param to destination matrix.
  1433.  * \param from source matrix.
  1434.  *
  1435.  * Copies all fields in GLmatrix, creating an inverse array if necessary.
  1436.  */
  1437. void
  1438. _math_matrix_copy( GLmatrix *to, const GLmatrix *from )
  1439. {
  1440.    memcpy( to->m, from->m, sizeof(Identity) );
  1441.    memcpy(to->inv, from->inv, 16 * sizeof(GLfloat));
  1442.    to->flags = from->flags;
  1443.    to->type = from->type;
  1444. }
  1445.  
  1446. /**
  1447.  * Loads a matrix array into GLmatrix.
  1448.  *
  1449.  * \param m matrix array.
  1450.  * \param mat matrix.
  1451.  *
  1452.  * Copies \p m into GLmatrix::m and marks the MAT_FLAG_GENERAL and MAT_DIRTY
  1453.  * flags.
  1454.  */
  1455. void
  1456. _math_matrix_loadf( GLmatrix *mat, const GLfloat *m )
  1457. {
  1458.    memcpy( mat->m, m, 16*sizeof(GLfloat) );
  1459.    mat->flags = (MAT_FLAG_GENERAL | MAT_DIRTY);
  1460. }
  1461.  
  1462. /**
  1463.  * Matrix constructor.
  1464.  *
  1465.  * \param m matrix.
  1466.  *
  1467.  * Initialize the GLmatrix fields.
  1468.  */
  1469. void
  1470. _math_matrix_ctr( GLmatrix *m )
  1471. {
  1472.    m->m = _mesa_align_malloc( 16 * sizeof(GLfloat), 16 );
  1473.    if (m->m)
  1474.       memcpy( m->m, Identity, sizeof(Identity) );
  1475.    m->inv = _mesa_align_malloc( 16 * sizeof(GLfloat), 16 );
  1476.    if (m->inv)
  1477.       memcpy( m->inv, Identity, sizeof(Identity) );
  1478.    m->type = MATRIX_IDENTITY;
  1479.    m->flags = 0;
  1480. }
  1481.  
  1482. /**
  1483.  * Matrix destructor.
  1484.  *
  1485.  * \param m matrix.
  1486.  *
  1487.  * Frees the data in a GLmatrix.
  1488.  */
  1489. void
  1490. _math_matrix_dtr( GLmatrix *m )
  1491. {
  1492.    _mesa_align_free( m->m );
  1493.    m->m = NULL;
  1494.  
  1495.    _mesa_align_free( m->inv );
  1496.    m->inv = NULL;
  1497. }
  1498.  
  1499. /*@}*/
  1500.  
  1501.  
  1502. /**********************************************************************/
  1503. /** \name Matrix transpose */
  1504. /*@{*/
  1505.  
  1506. /**
  1507.  * Transpose a GLfloat matrix.
  1508.  *
  1509.  * \param to destination array.
  1510.  * \param from source array.
  1511.  */
  1512. void
  1513. _math_transposef( GLfloat to[16], const GLfloat from[16] )
  1514. {
  1515.    to[0] = from[0];
  1516.    to[1] = from[4];
  1517.    to[2] = from[8];
  1518.    to[3] = from[12];
  1519.    to[4] = from[1];
  1520.    to[5] = from[5];
  1521.    to[6] = from[9];
  1522.    to[7] = from[13];
  1523.    to[8] = from[2];
  1524.    to[9] = from[6];
  1525.    to[10] = from[10];
  1526.    to[11] = from[14];
  1527.    to[12] = from[3];
  1528.    to[13] = from[7];
  1529.    to[14] = from[11];
  1530.    to[15] = from[15];
  1531. }
  1532.  
  1533. /**
  1534.  * Transpose a GLdouble matrix.
  1535.  *
  1536.  * \param to destination array.
  1537.  * \param from source array.
  1538.  */
  1539. void
  1540. _math_transposed( GLdouble to[16], const GLdouble from[16] )
  1541. {
  1542.    to[0] = from[0];
  1543.    to[1] = from[4];
  1544.    to[2] = from[8];
  1545.    to[3] = from[12];
  1546.    to[4] = from[1];
  1547.    to[5] = from[5];
  1548.    to[6] = from[9];
  1549.    to[7] = from[13];
  1550.    to[8] = from[2];
  1551.    to[9] = from[6];
  1552.    to[10] = from[10];
  1553.    to[11] = from[14];
  1554.    to[12] = from[3];
  1555.    to[13] = from[7];
  1556.    to[14] = from[11];
  1557.    to[15] = from[15];
  1558. }
  1559.  
  1560. /**
  1561.  * Transpose a GLdouble matrix and convert to GLfloat.
  1562.  *
  1563.  * \param to destination array.
  1564.  * \param from source array.
  1565.  */
  1566. void
  1567. _math_transposefd( GLfloat to[16], const GLdouble from[16] )
  1568. {
  1569.    to[0] = (GLfloat) from[0];
  1570.    to[1] = (GLfloat) from[4];
  1571.    to[2] = (GLfloat) from[8];
  1572.    to[3] = (GLfloat) from[12];
  1573.    to[4] = (GLfloat) from[1];
  1574.    to[5] = (GLfloat) from[5];
  1575.    to[6] = (GLfloat) from[9];
  1576.    to[7] = (GLfloat) from[13];
  1577.    to[8] = (GLfloat) from[2];
  1578.    to[9] = (GLfloat) from[6];
  1579.    to[10] = (GLfloat) from[10];
  1580.    to[11] = (GLfloat) from[14];
  1581.    to[12] = (GLfloat) from[3];
  1582.    to[13] = (GLfloat) from[7];
  1583.    to[14] = (GLfloat) from[11];
  1584.    to[15] = (GLfloat) from[15];
  1585. }
  1586.  
  1587. /*@}*/
  1588.  
  1589.  
  1590. /**
  1591.  * Transform a 4-element row vector (1x4 matrix) by a 4x4 matrix.  This
  1592.  * function is used for transforming clipping plane equations and spotlight
  1593.  * directions.
  1594.  * Mathematically,  u = v * m.
  1595.  * Input:  v - input vector
  1596.  *         m - transformation matrix
  1597.  * Output:  u - transformed vector
  1598.  */
  1599. void
  1600. _mesa_transform_vector( GLfloat u[4], const GLfloat v[4], const GLfloat m[16] )
  1601. {
  1602.    const GLfloat v0 = v[0], v1 = v[1], v2 = v[2], v3 = v[3];
  1603. #define M(row,col)  m[row + col*4]
  1604.    u[0] = v0 * M(0,0) + v1 * M(1,0) + v2 * M(2,0) + v3 * M(3,0);
  1605.    u[1] = v0 * M(0,1) + v1 * M(1,1) + v2 * M(2,1) + v3 * M(3,1);
  1606.    u[2] = v0 * M(0,2) + v1 * M(1,2) + v2 * M(2,2) + v3 * M(3,2);
  1607.    u[3] = v0 * M(0,3) + v1 * M(1,3) + v2 * M(2,3) + v3 * M(3,3);
  1608. #undef M
  1609. }
  1610.