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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 "main/glheader.h"
  38. #include "main/imports.h"
  39. #include "main/macros.h"
  40.  
  41. #include "m_matrix.h"
  42.  
  43.  
  44. /**
  45.  * \defgroup MatFlags MAT_FLAG_XXX-flags
  46.  *
  47.  * Bitmasks to indicate different kinds of 4x4 matrices in GLmatrix::flags
  48.  */
  49. /*@{*/
  50. #define MAT_FLAG_IDENTITY       0     /**< is an identity matrix flag.
  51.                                        *   (Not actually used - the identity
  52.                                        *   matrix is identified by the absense
  53.                                        *   of all other flags.)
  54.                                        */
  55. #define MAT_FLAG_GENERAL        0x1   /**< is a general matrix flag */
  56. #define MAT_FLAG_ROTATION       0x2   /**< is a rotation matrix flag */
  57. #define MAT_FLAG_TRANSLATION    0x4   /**< is a translation matrix flag */
  58. #define MAT_FLAG_UNIFORM_SCALE  0x8   /**< is an uniform scaling matrix flag */
  59. #define MAT_FLAG_GENERAL_SCALE  0x10  /**< is a general scaling matrix flag */
  60. #define MAT_FLAG_GENERAL_3D     0x20  /**< general 3D matrix flag */
  61. #define MAT_FLAG_PERSPECTIVE    0x40  /**< is a perspective proj matrix flag */
  62. #define MAT_FLAG_SINGULAR       0x80  /**< is a singular matrix flag */
  63. #define MAT_DIRTY_TYPE          0x100  /**< matrix type is dirty */
  64. #define MAT_DIRTY_FLAGS         0x200  /**< matrix flags are dirty */
  65. #define MAT_DIRTY_INVERSE       0x400  /**< matrix inverse is dirty */
  66.  
  67. /** angle preserving matrix flags mask */
  68. #define MAT_FLAGS_ANGLE_PRESERVING (MAT_FLAG_ROTATION | \
  69.                                     MAT_FLAG_TRANSLATION | \
  70.                                     MAT_FLAG_UNIFORM_SCALE)
  71.  
  72. /** geometry related matrix flags mask */
  73. #define MAT_FLAGS_GEOMETRY (MAT_FLAG_GENERAL | \
  74.                             MAT_FLAG_ROTATION | \
  75.                             MAT_FLAG_TRANSLATION | \
  76.                             MAT_FLAG_UNIFORM_SCALE | \
  77.                             MAT_FLAG_GENERAL_SCALE | \
  78.                             MAT_FLAG_GENERAL_3D | \
  79.                             MAT_FLAG_PERSPECTIVE | \
  80.                             MAT_FLAG_SINGULAR)
  81.  
  82. /** length preserving matrix flags mask */
  83. #define MAT_FLAGS_LENGTH_PRESERVING (MAT_FLAG_ROTATION | \
  84.                                      MAT_FLAG_TRANSLATION)
  85.  
  86.  
  87. /** 3D (non-perspective) matrix flags mask */
  88. #define MAT_FLAGS_3D (MAT_FLAG_ROTATION | \
  89.                       MAT_FLAG_TRANSLATION | \
  90.                       MAT_FLAG_UNIFORM_SCALE | \
  91.                       MAT_FLAG_GENERAL_SCALE | \
  92.                       MAT_FLAG_GENERAL_3D)
  93.  
  94. /** dirty matrix flags mask */
  95. #define MAT_DIRTY          (MAT_DIRTY_TYPE | \
  96.                             MAT_DIRTY_FLAGS | \
  97.                             MAT_DIRTY_INVERSE)
  98.  
  99. /*@}*/
  100.  
  101.  
  102. /**
  103.  * Test geometry related matrix flags.
  104.  *
  105.  * \param mat a pointer to a GLmatrix structure.
  106.  * \param a flags mask.
  107.  *
  108.  * \returns non-zero if all geometry related matrix flags are contained within
  109.  * the mask, or zero otherwise.
  110.  */
  111. #define TEST_MAT_FLAGS(mat, a)  \
  112.     ((MAT_FLAGS_GEOMETRY & (~(a)) & ((mat)->flags) ) == 0)
  113.  
  114.  
  115.  
  116. /**
  117.  * Names of the corresponding GLmatrixtype values.
  118.  */
  119. static const char *types[] = {
  120.    "MATRIX_GENERAL",
  121.    "MATRIX_IDENTITY",
  122.    "MATRIX_3D_NO_ROT",
  123.    "MATRIX_PERSPECTIVE",
  124.    "MATRIX_2D",
  125.    "MATRIX_2D_NO_ROT",
  126.    "MATRIX_3D"
  127. };
  128.  
  129.  
  130. /**
  131.  * Identity matrix.
  132.  */
  133. static GLfloat Identity[16] = {
  134.    1.0, 0.0, 0.0, 0.0,
  135.    0.0, 1.0, 0.0, 0.0,
  136.    0.0, 0.0, 1.0, 0.0,
  137.    0.0, 0.0, 0.0, 1.0
  138. };
  139.  
  140.  
  141.  
  142. /**********************************************************************/
  143. /** \name Matrix multiplication */
  144. /*@{*/
  145.  
  146. #define A(row,col)  a[(col<<2)+row]
  147. #define B(row,col)  b[(col<<2)+row]
  148. #define P(row,col)  product[(col<<2)+row]
  149.  
  150. /**
  151.  * Perform a full 4x4 matrix multiplication.
  152.  *
  153.  * \param a matrix.
  154.  * \param b matrix.
  155.  * \param product will receive the product of \p a and \p b.
  156.  *
  157.  * \warning Is assumed that \p product != \p b. \p product == \p a is allowed.
  158.  *
  159.  * \note KW: 4*16 = 64 multiplications
  160.  *
  161.  * \author This \c matmul was contributed by Thomas Malik
  162.  */
  163. static void matmul4( GLfloat *product, const GLfloat *a, const GLfloat *b )
  164. {
  165.    GLint i;
  166.    for (i = 0; i < 4; i++) {
  167.       const GLfloat ai0=A(i,0),  ai1=A(i,1),  ai2=A(i,2),  ai3=A(i,3);
  168.       P(i,0) = ai0 * B(0,0) + ai1 * B(1,0) + ai2 * B(2,0) + ai3 * B(3,0);
  169.       P(i,1) = ai0 * B(0,1) + ai1 * B(1,1) + ai2 * B(2,1) + ai3 * B(3,1);
  170.       P(i,2) = ai0 * B(0,2) + ai1 * B(1,2) + ai2 * B(2,2) + ai3 * B(3,2);
  171.       P(i,3) = ai0 * B(0,3) + ai1 * B(1,3) + ai2 * B(2,3) + ai3 * B(3,3);
  172.    }
  173. }
  174.  
  175. /**
  176.  * Multiply two matrices known to occupy only the top three rows, such
  177.  * as typical model matrices, and orthogonal matrices.
  178.  *
  179.  * \param a matrix.
  180.  * \param b matrix.
  181.  * \param product will receive the product of \p a and \p b.
  182.  */
  183. static void matmul34( GLfloat *product, const GLfloat *a, const GLfloat *b )
  184. {
  185.    GLint i;
  186.    for (i = 0; i < 3; i++) {
  187.       const GLfloat ai0=A(i,0),  ai1=A(i,1),  ai2=A(i,2),  ai3=A(i,3);
  188.       P(i,0) = ai0 * B(0,0) + ai1 * B(1,0) + ai2 * B(2,0);
  189.       P(i,1) = ai0 * B(0,1) + ai1 * B(1,1) + ai2 * B(2,1);
  190.       P(i,2) = ai0 * B(0,2) + ai1 * B(1,2) + ai2 * B(2,2);
  191.       P(i,3) = ai0 * B(0,3) + ai1 * B(1,3) + ai2 * B(2,3) + ai3;
  192.    }
  193.    P(3,0) = 0;
  194.    P(3,1) = 0;
  195.    P(3,2) = 0;
  196.    P(3,3) = 1;
  197. }
  198.  
  199. #undef A
  200. #undef B
  201. #undef P
  202.  
  203. /**
  204.  * Multiply a matrix by an array of floats with known properties.
  205.  *
  206.  * \param mat pointer to a GLmatrix structure containing the left multiplication
  207.  * matrix, and that will receive the product result.
  208.  * \param m right multiplication matrix array.
  209.  * \param flags flags of the matrix \p m.
  210.  *
  211.  * Joins both flags and marks the type and inverse as dirty.  Calls matmul34()
  212.  * if both matrices are 3D, or matmul4() otherwise.
  213.  */
  214. static void matrix_multf( GLmatrix *mat, const GLfloat *m, GLuint flags )
  215. {
  216.    mat->flags |= (flags | MAT_DIRTY_TYPE | MAT_DIRTY_INVERSE);
  217.  
  218.    if (TEST_MAT_FLAGS(mat, MAT_FLAGS_3D))
  219.       matmul34( mat->m, mat->m, m );
  220.    else
  221.       matmul4( mat->m, mat->m, m );
  222. }
  223.  
  224. /**
  225.  * Matrix multiplication.
  226.  *
  227.  * \param dest destination matrix.
  228.  * \param a left matrix.
  229.  * \param b right matrix.
  230.  *
  231.  * Joins both flags and marks the type and inverse as dirty.  Calls matmul34()
  232.  * if both matrices are 3D, or matmul4() otherwise.
  233.  */
  234. void
  235. _math_matrix_mul_matrix( GLmatrix *dest, const GLmatrix *a, const GLmatrix *b )
  236. {
  237.    dest->flags = (a->flags |
  238.                   b->flags |
  239.                   MAT_DIRTY_TYPE |
  240.                   MAT_DIRTY_INVERSE);
  241.  
  242.    if (TEST_MAT_FLAGS(dest, MAT_FLAGS_3D))
  243.       matmul34( dest->m, a->m, b->m );
  244.    else
  245.       matmul4( dest->m, a->m, b->m );
  246. }
  247.  
  248. /**
  249.  * Matrix multiplication.
  250.  *
  251.  * \param dest left and destination matrix.
  252.  * \param m right matrix array.
  253.  *
  254.  * Marks the matrix flags with general flag, and type and inverse dirty flags.
  255.  * Calls matmul4() for the multiplication.
  256.  */
  257. void
  258. _math_matrix_mul_floats( GLmatrix *dest, const GLfloat *m )
  259. {
  260.    dest->flags |= (MAT_FLAG_GENERAL |
  261.                    MAT_DIRTY_TYPE |
  262.                    MAT_DIRTY_INVERSE |
  263.                    MAT_DIRTY_FLAGS);
  264.  
  265.    matmul4( dest->m, dest->m, m );
  266. }
  267.  
  268. /*@}*/
  269.  
  270.  
  271. /**********************************************************************/
  272. /** \name Matrix output */
  273. /*@{*/
  274.  
  275. /**
  276.  * Print a matrix array.
  277.  *
  278.  * \param m matrix array.
  279.  *
  280.  * Called by _math_matrix_print() to print a matrix or its inverse.
  281.  */
  282. static void print_matrix_floats( const GLfloat m[16] )
  283. {
  284.    int i;
  285.    for (i=0;i<4;i++) {
  286.       _mesa_debug(NULL,"\t%f %f %f %f\n", m[i], m[4+i], m[8+i], m[12+i] );
  287.    }
  288. }
  289.  
  290. /**
  291.  * Dumps the contents of a GLmatrix structure.
  292.  *
  293.  * \param m pointer to the GLmatrix structure.
  294.  */
  295. void
  296. _math_matrix_print( const GLmatrix *m )
  297. {
  298.    GLfloat prod[16];
  299.  
  300.    _mesa_debug(NULL, "Matrix type: %s, flags: %x\n", types[m->type], m->flags);
  301.    print_matrix_floats(m->m);
  302.    _mesa_debug(NULL, "Inverse: \n");
  303.    print_matrix_floats(m->inv);
  304.    matmul4(prod, m->m, m->inv);
  305.    _mesa_debug(NULL, "Mat * Inverse:\n");
  306.    print_matrix_floats(prod);
  307. }
  308.  
  309. /*@}*/
  310.  
  311.  
  312. /**
  313.  * References an element of 4x4 matrix.
  314.  *
  315.  * \param m matrix array.
  316.  * \param c column of the desired element.
  317.  * \param r row of the desired element.
  318.  *
  319.  * \return value of the desired element.
  320.  *
  321.  * Calculate the linear storage index of the element and references it.
  322.  */
  323. #define MAT(m,r,c) (m)[(c)*4+(r)]
  324.  
  325.  
  326. /**********************************************************************/
  327. /** \name Matrix inversion */
  328. /*@{*/
  329.  
  330. /**
  331.  * Swaps the values of two floating point variables.
  332.  *
  333.  * Used by invert_matrix_general() to swap the row pointers.
  334.  */
  335. #define SWAP_ROWS(a, b) { GLfloat *_tmp = a; (a)=(b); (b)=_tmp; }
  336.  
  337. /**
  338.  * Compute inverse of 4x4 transformation matrix.
  339.  *
  340.  * \param mat pointer to a GLmatrix structure. The matrix inverse will be
  341.  * stored in the GLmatrix::inv attribute.
  342.  *
  343.  * \return GL_TRUE for success, GL_FALSE for failure (\p singular matrix).
  344.  *
  345.  * \author
  346.  * Code contributed by Jacques Leroy jle@star.be
  347.  *
  348.  * Calculates the inverse matrix by performing the gaussian matrix reduction
  349.  * with partial pivoting followed by back/substitution with the loops manually
  350.  * unrolled.
  351.  */
  352. static GLboolean invert_matrix_general( GLmatrix *mat )
  353. {
  354.    const GLfloat *m = mat->m;
  355.    GLfloat *out = mat->inv;
  356.    GLfloat wtmp[4][8];
  357.    GLfloat m0, m1, m2, m3, s;
  358.    GLfloat *r0, *r1, *r2, *r3;
  359.  
  360.    r0 = wtmp[0], r1 = wtmp[1], r2 = wtmp[2], r3 = wtmp[3];
  361.  
  362.    r0[0] = MAT(m,0,0), r0[1] = MAT(m,0,1),
  363.    r0[2] = MAT(m,0,2), r0[3] = MAT(m,0,3),
  364.    r0[4] = 1.0, r0[5] = r0[6] = r0[7] = 0.0,
  365.  
  366.    r1[0] = MAT(m,1,0), r1[1] = MAT(m,1,1),
  367.    r1[2] = MAT(m,1,2), r1[3] = MAT(m,1,3),
  368.    r1[5] = 1.0, r1[4] = r1[6] = r1[7] = 0.0,
  369.  
  370.    r2[0] = MAT(m,2,0), r2[1] = MAT(m,2,1),
  371.    r2[2] = MAT(m,2,2), r2[3] = MAT(m,2,3),
  372.    r2[6] = 1.0, r2[4] = r2[5] = r2[7] = 0.0,
  373.  
  374.    r3[0] = MAT(m,3,0), r3[1] = MAT(m,3,1),
  375.    r3[2] = MAT(m,3,2), r3[3] = MAT(m,3,3),
  376.    r3[7] = 1.0, r3[4] = r3[5] = r3[6] = 0.0;
  377.  
  378.    /* choose pivot - or die */
  379.    if (FABSF(r3[0])>FABSF(r2[0])) SWAP_ROWS(r3, r2);
  380.    if (FABSF(r2[0])>FABSF(r1[0])) SWAP_ROWS(r2, r1);
  381.    if (FABSF(r1[0])>FABSF(r0[0])) SWAP_ROWS(r1, r0);
  382.    if (0.0 == r0[0])  return GL_FALSE;
  383.  
  384.    /* eliminate first variable     */
  385.    m1 = r1[0]/r0[0]; m2 = r2[0]/r0[0]; m3 = r3[0]/r0[0];
  386.    s = r0[1]; r1[1] -= m1 * s; r2[1] -= m2 * s; r3[1] -= m3 * s;
  387.    s = r0[2]; r1[2] -= m1 * s; r2[2] -= m2 * s; r3[2] -= m3 * s;
  388.    s = r0[3]; r1[3] -= m1 * s; r2[3] -= m2 * s; r3[3] -= m3 * s;
  389.    s = r0[4];
  390.    if (s != 0.0) { r1[4] -= m1 * s; r2[4] -= m2 * s; r3[4] -= m3 * s; }
  391.    s = r0[5];
  392.    if (s != 0.0) { r1[5] -= m1 * s; r2[5] -= m2 * s; r3[5] -= m3 * s; }
  393.    s = r0[6];
  394.    if (s != 0.0) { r1[6] -= m1 * s; r2[6] -= m2 * s; r3[6] -= m3 * s; }
  395.    s = r0[7];
  396.    if (s != 0.0) { r1[7] -= m1 * s; r2[7] -= m2 * s; r3[7] -= m3 * s; }
  397.  
  398.    /* choose pivot - or die */
  399.    if (FABSF(r3[1])>FABSF(r2[1])) SWAP_ROWS(r3, r2);
  400.    if (FABSF(r2[1])>FABSF(r1[1])) SWAP_ROWS(r2, r1);
  401.    if (0.0 == r1[1])  return GL_FALSE;
  402.  
  403.    /* eliminate second variable */
  404.    m2 = r2[1]/r1[1]; m3 = r3[1]/r1[1];
  405.    r2[2] -= m2 * r1[2]; r3[2] -= m3 * r1[2];
  406.    r2[3] -= m2 * r1[3]; r3[3] -= m3 * r1[3];
  407.    s = r1[4]; if (0.0 != s) { r2[4] -= m2 * s; r3[4] -= m3 * s; }
  408.    s = r1[5]; if (0.0 != s) { r2[5] -= m2 * s; r3[5] -= m3 * s; }
  409.    s = r1[6]; if (0.0 != s) { r2[6] -= m2 * s; r3[6] -= m3 * s; }
  410.    s = r1[7]; if (0.0 != s) { r2[7] -= m2 * s; r3[7] -= m3 * s; }
  411.  
  412.    /* choose pivot - or die */
  413.    if (FABSF(r3[2])>FABSF(r2[2])) SWAP_ROWS(r3, r2);
  414.    if (0.0 == r2[2])  return GL_FALSE;
  415.  
  416.    /* eliminate third variable */
  417.    m3 = r3[2]/r2[2];
  418.    r3[3] -= m3 * r2[3], r3[4] -= m3 * r2[4],
  419.    r3[5] -= m3 * r2[5], r3[6] -= m3 * r2[6],
  420.    r3[7] -= m3 * r2[7];
  421.  
  422.    /* last check */
  423.    if (0.0 == r3[3]) return GL_FALSE;
  424.  
  425.    s = 1.0F/r3[3];             /* now back substitute row 3 */
  426.    r3[4] *= s; r3[5] *= s; r3[6] *= s; r3[7] *= s;
  427.  
  428.    m2 = r2[3];                 /* now back substitute row 2 */
  429.    s  = 1.0F/r2[2];
  430.    r2[4] = s * (r2[4] - r3[4] * m2), r2[5] = s * (r2[5] - r3[5] * m2),
  431.    r2[6] = s * (r2[6] - r3[6] * m2), r2[7] = s * (r2[7] - r3[7] * m2);
  432.    m1 = r1[3];
  433.    r1[4] -= r3[4] * m1, r1[5] -= r3[5] * m1,
  434.    r1[6] -= r3[6] * m1, r1[7] -= r3[7] * m1;
  435.    m0 = r0[3];
  436.    r0[4] -= r3[4] * m0, r0[5] -= r3[5] * m0,
  437.    r0[6] -= r3[6] * m0, r0[7] -= r3[7] * m0;
  438.  
  439.    m1 = r1[2];                 /* now back substitute row 1 */
  440.    s  = 1.0F/r1[1];
  441.    r1[4] = s * (r1[4] - r2[4] * m1), r1[5] = s * (r1[5] - r2[5] * m1),
  442.    r1[6] = s * (r1[6] - r2[6] * m1), r1[7] = s * (r1[7] - r2[7] * m1);
  443.    m0 = r0[2];
  444.    r0[4] -= r2[4] * m0, r0[5] -= r2[5] * m0,
  445.    r0[6] -= r2[6] * m0, r0[7] -= r2[7] * m0;
  446.  
  447.    m0 = r0[1];                 /* now back substitute row 0 */
  448.    s  = 1.0F/r0[0];
  449.    r0[4] = s * (r0[4] - r1[4] * m0), r0[5] = s * (r0[5] - r1[5] * m0),
  450.    r0[6] = s * (r0[6] - r1[6] * m0), r0[7] = s * (r0[7] - r1[7] * m0);
  451.  
  452.    MAT(out,0,0) = r0[4]; MAT(out,0,1) = r0[5],
  453.    MAT(out,0,2) = r0[6]; MAT(out,0,3) = r0[7],
  454.    MAT(out,1,0) = r1[4]; MAT(out,1,1) = r1[5],
  455.    MAT(out,1,2) = r1[6]; MAT(out,1,3) = r1[7],
  456.    MAT(out,2,0) = r2[4]; MAT(out,2,1) = r2[5],
  457.    MAT(out,2,2) = r2[6]; MAT(out,2,3) = r2[7],
  458.    MAT(out,3,0) = r3[4]; MAT(out,3,1) = r3[5],
  459.    MAT(out,3,2) = r3[6]; MAT(out,3,3) = r3[7];
  460.  
  461.    return GL_TRUE;
  462. }
  463. #undef SWAP_ROWS
  464.  
  465. /**
  466.  * Compute inverse of a general 3d transformation matrix.
  467.  *
  468.  * \param mat pointer to a GLmatrix structure. The matrix inverse will be
  469.  * stored in the GLmatrix::inv attribute.
  470.  *
  471.  * \return GL_TRUE for success, GL_FALSE for failure (\p singular matrix).
  472.  *
  473.  * \author Adapted from graphics gems II.
  474.  *
  475.  * Calculates the inverse of the upper left by first calculating its
  476.  * determinant and multiplying it to the symmetric adjust matrix of each
  477.  * element. Finally deals with the translation part by transforming the
  478.  * original translation vector using by the calculated submatrix inverse.
  479.  */
  480. static GLboolean invert_matrix_3d_general( GLmatrix *mat )
  481. {
  482.    const GLfloat *in = mat->m;
  483.    GLfloat *out = mat->inv;
  484.    GLfloat pos, neg, t;
  485.    GLfloat det;
  486.  
  487.    /* Calculate the determinant of upper left 3x3 submatrix and
  488.     * determine if the matrix is singular.
  489.     */
  490.    pos = neg = 0.0;
  491.    t =  MAT(in,0,0) * MAT(in,1,1) * MAT(in,2,2);
  492.    if (t >= 0.0) pos += t; else neg += t;
  493.  
  494.    t =  MAT(in,1,0) * MAT(in,2,1) * MAT(in,0,2);
  495.    if (t >= 0.0) pos += t; else neg += t;
  496.  
  497.    t =  MAT(in,2,0) * MAT(in,0,1) * MAT(in,1,2);
  498.    if (t >= 0.0) pos += t; else neg += t;
  499.  
  500.    t = -MAT(in,2,0) * MAT(in,1,1) * MAT(in,0,2);
  501.    if (t >= 0.0) pos += t; else neg += t;
  502.  
  503.    t = -MAT(in,1,0) * MAT(in,0,1) * MAT(in,2,2);
  504.    if (t >= 0.0) pos += t; else neg += t;
  505.  
  506.    t = -MAT(in,0,0) * MAT(in,2,1) * MAT(in,1,2);
  507.    if (t >= 0.0) pos += t; else neg += t;
  508.  
  509.    det = pos + neg;
  510.  
  511.    if (FABSF(det) < 1e-25)
  512.       return GL_FALSE;
  513.  
  514.    det = 1.0F / det;
  515.    MAT(out,0,0) = (  (MAT(in,1,1)*MAT(in,2,2) - MAT(in,2,1)*MAT(in,1,2) )*det);
  516.    MAT(out,0,1) = (- (MAT(in,0,1)*MAT(in,2,2) - MAT(in,2,1)*MAT(in,0,2) )*det);
  517.    MAT(out,0,2) = (  (MAT(in,0,1)*MAT(in,1,2) - MAT(in,1,1)*MAT(in,0,2) )*det);
  518.    MAT(out,1,0) = (- (MAT(in,1,0)*MAT(in,2,2) - MAT(in,2,0)*MAT(in,1,2) )*det);
  519.    MAT(out,1,1) = (  (MAT(in,0,0)*MAT(in,2,2) - MAT(in,2,0)*MAT(in,0,2) )*det);
  520.    MAT(out,1,2) = (- (MAT(in,0,0)*MAT(in,1,2) - MAT(in,1,0)*MAT(in,0,2) )*det);
  521.    MAT(out,2,0) = (  (MAT(in,1,0)*MAT(in,2,1) - MAT(in,2,0)*MAT(in,1,1) )*det);
  522.    MAT(out,2,1) = (- (MAT(in,0,0)*MAT(in,2,1) - MAT(in,2,0)*MAT(in,0,1) )*det);
  523.    MAT(out,2,2) = (  (MAT(in,0,0)*MAT(in,1,1) - MAT(in,1,0)*MAT(in,0,1) )*det);
  524.  
  525.    /* Do the translation part */
  526.    MAT(out,0,3) = - (MAT(in,0,3) * MAT(out,0,0) +
  527.                      MAT(in,1,3) * MAT(out,0,1) +
  528.                      MAT(in,2,3) * MAT(out,0,2) );
  529.    MAT(out,1,3) = - (MAT(in,0,3) * MAT(out,1,0) +
  530.                      MAT(in,1,3) * MAT(out,1,1) +
  531.                      MAT(in,2,3) * MAT(out,1,2) );
  532.    MAT(out,2,3) = - (MAT(in,0,3) * MAT(out,2,0) +
  533.                      MAT(in,1,3) * MAT(out,2,1) +
  534.                      MAT(in,2,3) * MAT(out,2,2) );
  535.  
  536.    return GL_TRUE;
  537. }
  538.  
  539. /**
  540.  * Compute inverse of a 3d transformation matrix.
  541.  *
  542.  * \param mat pointer to a GLmatrix structure. The matrix inverse will be
  543.  * stored in the GLmatrix::inv attribute.
  544.  *
  545.  * \return GL_TRUE for success, GL_FALSE for failure (\p singular matrix).
  546.  *
  547.  * If the matrix is not an angle preserving matrix then calls
  548.  * invert_matrix_3d_general for the actual calculation. Otherwise calculates
  549.  * the inverse matrix analyzing and inverting each of the scaling, rotation and
  550.  * translation parts.
  551.  */
  552. static GLboolean invert_matrix_3d( GLmatrix *mat )
  553. {
  554.    const GLfloat *in = mat->m;
  555.    GLfloat *out = mat->inv;
  556.  
  557.    if (!TEST_MAT_FLAGS(mat, MAT_FLAGS_ANGLE_PRESERVING)) {
  558.       return invert_matrix_3d_general( mat );
  559.    }
  560.  
  561.    if (mat->flags & MAT_FLAG_UNIFORM_SCALE) {
  562.       GLfloat scale = (MAT(in,0,0) * MAT(in,0,0) +
  563.                        MAT(in,0,1) * MAT(in,0,1) +
  564.                        MAT(in,0,2) * MAT(in,0,2));
  565.  
  566.       if (scale == 0.0)
  567.          return GL_FALSE;
  568.  
  569.       scale = 1.0F / scale;
  570.  
  571.       /* Transpose and scale the 3 by 3 upper-left submatrix. */
  572.       MAT(out,0,0) = scale * MAT(in,0,0);
  573.       MAT(out,1,0) = scale * MAT(in,0,1);
  574.       MAT(out,2,0) = scale * MAT(in,0,2);
  575.       MAT(out,0,1) = scale * MAT(in,1,0);
  576.       MAT(out,1,1) = scale * MAT(in,1,1);
  577.       MAT(out,2,1) = scale * MAT(in,1,2);
  578.       MAT(out,0,2) = scale * MAT(in,2,0);
  579.       MAT(out,1,2) = scale * MAT(in,2,1);
  580.       MAT(out,2,2) = scale * MAT(in,2,2);
  581.    }
  582.    else if (mat->flags & MAT_FLAG_ROTATION) {
  583.       /* Transpose the 3 by 3 upper-left submatrix. */
  584.       MAT(out,0,0) = MAT(in,0,0);
  585.       MAT(out,1,0) = MAT(in,0,1);
  586.       MAT(out,2,0) = MAT(in,0,2);
  587.       MAT(out,0,1) = MAT(in,1,0);
  588.       MAT(out,1,1) = MAT(in,1,1);
  589.       MAT(out,2,1) = MAT(in,1,2);
  590.       MAT(out,0,2) = MAT(in,2,0);
  591.       MAT(out,1,2) = MAT(in,2,1);
  592.       MAT(out,2,2) = MAT(in,2,2);
  593.    }
  594.    else {
  595.       /* pure translation */
  596.       memcpy( out, Identity, sizeof(Identity) );
  597.       MAT(out,0,3) = - MAT(in,0,3);
  598.       MAT(out,1,3) = - MAT(in,1,3);
  599.       MAT(out,2,3) = - MAT(in,2,3);
  600.       return GL_TRUE;
  601.    }
  602.  
  603.    if (mat->flags & MAT_FLAG_TRANSLATION) {
  604.       /* Do the translation part */
  605.       MAT(out,0,3) = - (MAT(in,0,3) * MAT(out,0,0) +
  606.                         MAT(in,1,3) * MAT(out,0,1) +
  607.                         MAT(in,2,3) * MAT(out,0,2) );
  608.       MAT(out,1,3) = - (MAT(in,0,3) * MAT(out,1,0) +
  609.                         MAT(in,1,3) * MAT(out,1,1) +
  610.                         MAT(in,2,3) * MAT(out,1,2) );
  611.       MAT(out,2,3) = - (MAT(in,0,3) * MAT(out,2,0) +
  612.                         MAT(in,1,3) * MAT(out,2,1) +
  613.                         MAT(in,2,3) * MAT(out,2,2) );
  614.    }
  615.    else {
  616.       MAT(out,0,3) = MAT(out,1,3) = MAT(out,2,3) = 0.0;
  617.    }
  618.  
  619.    return GL_TRUE;
  620. }
  621.  
  622. /**
  623.  * Compute inverse of an identity transformation matrix.
  624.  *
  625.  * \param mat pointer to a GLmatrix structure. The matrix inverse will be
  626.  * stored in the GLmatrix::inv attribute.
  627.  *
  628.  * \return always GL_TRUE.
  629.  *
  630.  * Simply copies Identity into GLmatrix::inv.
  631.  */
  632. static GLboolean invert_matrix_identity( GLmatrix *mat )
  633. {
  634.    memcpy( mat->inv, Identity, sizeof(Identity) );
  635.    return GL_TRUE;
  636. }
  637.  
  638. /**
  639.  * Compute inverse of a no-rotation 3d transformation matrix.
  640.  *
  641.  * \param mat pointer to a GLmatrix structure. The matrix inverse will be
  642.  * stored in the GLmatrix::inv attribute.
  643.  *
  644.  * \return GL_TRUE for success, GL_FALSE for failure (\p singular matrix).
  645.  *
  646.  * Calculates the
  647.  */
  648. static GLboolean invert_matrix_3d_no_rot( GLmatrix *mat )
  649. {
  650.    const GLfloat *in = mat->m;
  651.    GLfloat *out = mat->inv;
  652.  
  653.    if (MAT(in,0,0) == 0 || MAT(in,1,1) == 0 || MAT(in,2,2) == 0 )
  654.       return GL_FALSE;
  655.  
  656.    memcpy( out, Identity, 16 * sizeof(GLfloat) );
  657.    MAT(out,0,0) = 1.0F / MAT(in,0,0);
  658.    MAT(out,1,1) = 1.0F / MAT(in,1,1);
  659.    MAT(out,2,2) = 1.0F / MAT(in,2,2);
  660.  
  661.    if (mat->flags & MAT_FLAG_TRANSLATION) {
  662.       MAT(out,0,3) = - (MAT(in,0,3) * MAT(out,0,0));
  663.       MAT(out,1,3) = - (MAT(in,1,3) * MAT(out,1,1));
  664.       MAT(out,2,3) = - (MAT(in,2,3) * MAT(out,2,2));
  665.    }
  666.  
  667.    return GL_TRUE;
  668. }
  669.  
  670. /**
  671.  * Compute inverse of a no-rotation 2d transformation matrix.
  672.  *
  673.  * \param mat pointer to a GLmatrix structure. The matrix inverse will be
  674.  * stored in the GLmatrix::inv attribute.
  675.  *
  676.  * \return GL_TRUE for success, GL_FALSE for failure (\p singular matrix).
  677.  *
  678.  * Calculates the inverse matrix by applying the inverse scaling and
  679.  * translation to the identity matrix.
  680.  */
  681. static GLboolean invert_matrix_2d_no_rot( GLmatrix *mat )
  682. {
  683.    const GLfloat *in = mat->m;
  684.    GLfloat *out = mat->inv;
  685.  
  686.    if (MAT(in,0,0) == 0 || MAT(in,1,1) == 0)
  687.       return GL_FALSE;
  688.  
  689.    memcpy( out, Identity, 16 * sizeof(GLfloat) );
  690.    MAT(out,0,0) = 1.0F / MAT(in,0,0);
  691.    MAT(out,1,1) = 1.0F / MAT(in,1,1);
  692.  
  693.    if (mat->flags & MAT_FLAG_TRANSLATION) {
  694.       MAT(out,0,3) = - (MAT(in,0,3) * MAT(out,0,0));
  695.       MAT(out,1,3) = - (MAT(in,1,3) * MAT(out,1,1));
  696.    }
  697.  
  698.    return GL_TRUE;
  699. }
  700.  
  701. #if 0
  702. /* broken */
  703. static GLboolean invert_matrix_perspective( GLmatrix *mat )
  704. {
  705.    const GLfloat *in = mat->m;
  706.    GLfloat *out = mat->inv;
  707.  
  708.    if (MAT(in,2,3) == 0)
  709.       return GL_FALSE;
  710.  
  711.    memcpy( out, Identity, 16 * sizeof(GLfloat) );
  712.  
  713.    MAT(out,0,0) = 1.0F / MAT(in,0,0);
  714.    MAT(out,1,1) = 1.0F / MAT(in,1,1);
  715.  
  716.    MAT(out,0,3) = MAT(in,0,2);
  717.    MAT(out,1,3) = MAT(in,1,2);
  718.  
  719.    MAT(out,2,2) = 0;
  720.    MAT(out,2,3) = -1;
  721.  
  722.    MAT(out,3,2) = 1.0F / MAT(in,2,3);
  723.    MAT(out,3,3) = MAT(in,2,2) * MAT(out,3,2);
  724.  
  725.    return GL_TRUE;
  726. }
  727. #endif
  728.  
  729. /**
  730.  * Matrix inversion function pointer type.
  731.  */
  732. typedef GLboolean (*inv_mat_func)( GLmatrix *mat );
  733.  
  734. /**
  735.  * Table of the matrix inversion functions according to the matrix type.
  736.  */
  737. static inv_mat_func inv_mat_tab[7] = {
  738.    invert_matrix_general,
  739.    invert_matrix_identity,
  740.    invert_matrix_3d_no_rot,
  741. #if 0
  742.    /* Don't use this function for now - it fails when the projection matrix
  743.     * is premultiplied by a translation (ala Chromium's tilesort SPU).
  744.     */
  745.    invert_matrix_perspective,
  746. #else
  747.    invert_matrix_general,
  748. #endif
  749.    invert_matrix_3d,            /* lazy! */
  750.    invert_matrix_2d_no_rot,
  751.    invert_matrix_3d
  752. };
  753.  
  754. /**
  755.  * Compute inverse of a transformation matrix.
  756.  *
  757.  * \param mat pointer to a GLmatrix structure. The matrix inverse will be
  758.  * stored in the GLmatrix::inv attribute.
  759.  *
  760.  * \return GL_TRUE for success, GL_FALSE for failure (\p singular matrix).
  761.  *
  762.  * Calls the matrix inversion function in inv_mat_tab corresponding to the
  763.  * given matrix type.  In case of failure, updates the MAT_FLAG_SINGULAR flag,
  764.  * and copies the identity matrix into GLmatrix::inv.
  765.  */
  766. static GLboolean matrix_invert( GLmatrix *mat )
  767. {
  768.    if (inv_mat_tab[mat->type](mat)) {
  769.       mat->flags &= ~MAT_FLAG_SINGULAR;
  770.       return GL_TRUE;
  771.    } else {
  772.       mat->flags |= MAT_FLAG_SINGULAR;
  773.       memcpy( mat->inv, Identity, sizeof(Identity) );
  774.       return GL_FALSE;
  775.    }
  776. }
  777.  
  778. /*@}*/
  779.  
  780.  
  781. /**********************************************************************/
  782. /** \name Matrix generation */
  783. /*@{*/
  784.  
  785. /**
  786.  * Generate a 4x4 transformation matrix from glRotate parameters, and
  787.  * post-multiply the input matrix by it.
  788.  *
  789.  * \author
  790.  * This function was contributed by Erich Boleyn (erich@uruk.org).
  791.  * Optimizations contributed by Rudolf Opalla (rudi@khm.de).
  792.  */
  793. void
  794. _math_matrix_rotate( GLmatrix *mat,
  795.                      GLfloat angle, GLfloat x, GLfloat y, GLfloat z )
  796. {
  797.    GLfloat xx, yy, zz, xy, yz, zx, xs, ys, zs, one_c, s, c;
  798.    GLfloat m[16];
  799.    GLboolean optimized;
  800.  
  801.    s = (GLfloat) sin( angle * DEG2RAD );
  802.    c = (GLfloat) cos( angle * DEG2RAD );
  803.  
  804.    memcpy(m, Identity, sizeof(GLfloat)*16);
  805.    optimized = GL_FALSE;
  806.  
  807. #define M(row,col)  m[col*4+row]
  808.  
  809.    if (x == 0.0F) {
  810.       if (y == 0.0F) {
  811.          if (z != 0.0F) {
  812.             optimized = GL_TRUE;
  813.             /* rotate only around z-axis */
  814.             M(0,0) = c;
  815.             M(1,1) = c;
  816.             if (z < 0.0F) {
  817.                M(0,1) = s;
  818.                M(1,0) = -s;
  819.             }
  820.             else {
  821.                M(0,1) = -s;
  822.                M(1,0) = s;
  823.             }
  824.          }
  825.       }
  826.       else if (z == 0.0F) {
  827.          optimized = GL_TRUE;
  828.          /* rotate only around y-axis */
  829.          M(0,0) = c;
  830.          M(2,2) = c;
  831.          if (y < 0.0F) {
  832.             M(0,2) = -s;
  833.             M(2,0) = s;
  834.          }
  835.          else {
  836.             M(0,2) = s;
  837.             M(2,0) = -s;
  838.          }
  839.       }
  840.    }
  841.    else if (y == 0.0F) {
  842.       if (z == 0.0F) {
  843.          optimized = GL_TRUE;
  844.          /* rotate only around x-axis */
  845.          M(1,1) = c;
  846.          M(2,2) = c;
  847.          if (x < 0.0F) {
  848.             M(1,2) = s;
  849.             M(2,1) = -s;
  850.          }
  851.          else {
  852.             M(1,2) = -s;
  853.             M(2,1) = s;
  854.          }
  855.       }
  856.    }
  857.  
  858.    if (!optimized) {
  859.       const GLfloat mag = sqrtf(x * x + y * y + z * z);
  860.  
  861.       if (mag <= 1.0e-4) {
  862.          /* no rotation, leave mat as-is */
  863.          return;
  864.       }
  865.  
  866.       x /= mag;
  867.       y /= mag;
  868.       z /= mag;
  869.  
  870.  
  871.       /*
  872.        *     Arbitrary axis rotation matrix.
  873.        *
  874.        *  This is composed of 5 matrices, Rz, Ry, T, Ry', Rz', multiplied
  875.        *  like so:  Rz * Ry * T * Ry' * Rz'.  T is the final rotation
  876.        *  (which is about the X-axis), and the two composite transforms
  877.        *  Ry' * Rz' and Rz * Ry are (respectively) the rotations necessary
  878.        *  from the arbitrary axis to the X-axis then back.  They are
  879.        *  all elementary rotations.
  880.        *
  881.        *  Rz' is a rotation about the Z-axis, to bring the axis vector
  882.        *  into the x-z plane.  Then Ry' is applied, rotating about the
  883.        *  Y-axis to bring the axis vector parallel with the X-axis.  The
  884.        *  rotation about the X-axis is then performed.  Ry and Rz are
  885.        *  simply the respective inverse transforms to bring the arbitrary
  886.        *  axis back to its original orientation.  The first transforms
  887.        *  Rz' and Ry' are considered inverses, since the data from the
  888.        *  arbitrary axis gives you info on how to get to it, not how
  889.        *  to get away from it, and an inverse must be applied.
  890.        *
  891.        *  The basic calculation used is to recognize that the arbitrary
  892.        *  axis vector (x, y, z), since it is of unit length, actually
  893.        *  represents the sines and cosines of the angles to rotate the
  894.        *  X-axis to the same orientation, with theta being the angle about
  895.        *  Z and phi the angle about Y (in the order described above)
  896.        *  as follows:
  897.        *
  898.        *  cos ( theta ) = x / sqrt ( 1 - z^2 )
  899.        *  sin ( theta ) = y / sqrt ( 1 - z^2 )
  900.        *
  901.        *  cos ( phi ) = sqrt ( 1 - z^2 )
  902.        *  sin ( phi ) = z
  903.        *
  904.        *  Note that cos ( phi ) can further be inserted to the above
  905.        *  formulas:
  906.        *
  907.        *  cos ( theta ) = x / cos ( phi )
  908.        *  sin ( theta ) = y / sin ( phi )
  909.        *
  910.        *  ...etc.  Because of those relations and the standard trigonometric
  911.        *  relations, it is pssible to reduce the transforms down to what
  912.        *  is used below.  It may be that any primary axis chosen will give the
  913.        *  same results (modulo a sign convention) using thie method.
  914.        *
  915.        *  Particularly nice is to notice that all divisions that might
  916.        *  have caused trouble when parallel to certain planes or
  917.        *  axis go away with care paid to reducing the expressions.
  918.        *  After checking, it does perform correctly under all cases, since
  919.        *  in all the cases of division where the denominator would have
  920.        *  been zero, the numerator would have been zero as well, giving
  921.        *  the expected result.
  922.        */
  923.  
  924.       xx = x * x;
  925.       yy = y * y;
  926.       zz = z * z;
  927.       xy = x * y;
  928.       yz = y * z;
  929.       zx = z * x;
  930.       xs = x * s;
  931.       ys = y * s;
  932.       zs = z * s;
  933.       one_c = 1.0F - c;
  934.  
  935.       /* We already hold the identity-matrix so we can skip some statements */
  936.       M(0,0) = (one_c * xx) + c;
  937.       M(0,1) = (one_c * xy) - zs;
  938.       M(0,2) = (one_c * zx) + ys;
  939. /*    M(0,3) = 0.0F; */
  940.  
  941.       M(1,0) = (one_c * xy) + zs;
  942.       M(1,1) = (one_c * yy) + c;
  943.       M(1,2) = (one_c * yz) - xs;
  944. /*    M(1,3) = 0.0F; */
  945.  
  946.       M(2,0) = (one_c * zx) - ys;
  947.       M(2,1) = (one_c * yz) + xs;
  948.       M(2,2) = (one_c * zz) + c;
  949. /*    M(2,3) = 0.0F; */
  950.  
  951. /*
  952.       M(3,0) = 0.0F;
  953.       M(3,1) = 0.0F;
  954.       M(3,2) = 0.0F;
  955.       M(3,3) = 1.0F;
  956. */
  957.    }
  958. #undef M
  959.  
  960.    matrix_multf( mat, m, MAT_FLAG_ROTATION );
  961. }
  962.  
  963. /**
  964.  * Apply a perspective projection matrix.
  965.  *
  966.  * \param mat matrix to apply the projection.
  967.  * \param left left clipping plane coordinate.
  968.  * \param right right clipping plane coordinate.
  969.  * \param bottom bottom clipping plane coordinate.
  970.  * \param top top clipping plane coordinate.
  971.  * \param nearval distance to the near clipping plane.
  972.  * \param farval distance to the far clipping plane.
  973.  *
  974.  * Creates the projection matrix and multiplies it with \p mat, marking the
  975.  * MAT_FLAG_PERSPECTIVE flag.
  976.  */
  977. void
  978. _math_matrix_frustum( GLmatrix *mat,
  979.                       GLfloat left, GLfloat right,
  980.                       GLfloat bottom, GLfloat top,
  981.                       GLfloat nearval, GLfloat farval )
  982. {
  983.    GLfloat x, y, a, b, c, d;
  984.    GLfloat m[16];
  985.  
  986.    x = (2.0F*nearval) / (right-left);
  987.    y = (2.0F*nearval) / (top-bottom);
  988.    a = (right+left) / (right-left);
  989.    b = (top+bottom) / (top-bottom);
  990.    c = -(farval+nearval) / ( farval-nearval);
  991.    d = -(2.0F*farval*nearval) / (farval-nearval);  /* error? */
  992.  
  993. #define M(row,col)  m[col*4+row]
  994.    M(0,0) = x;     M(0,1) = 0.0F;  M(0,2) = a;      M(0,3) = 0.0F;
  995.    M(1,0) = 0.0F;  M(1,1) = y;     M(1,2) = b;      M(1,3) = 0.0F;
  996.    M(2,0) = 0.0F;  M(2,1) = 0.0F;  M(2,2) = c;      M(2,3) = d;
  997.    M(3,0) = 0.0F;  M(3,1) = 0.0F;  M(3,2) = -1.0F;  M(3,3) = 0.0F;
  998. #undef M
  999.  
  1000.    matrix_multf( mat, m, MAT_FLAG_PERSPECTIVE );
  1001. }
  1002.  
  1003. /**
  1004.  * Apply an orthographic projection matrix.
  1005.  *
  1006.  * \param mat matrix to apply the projection.
  1007.  * \param left left clipping plane coordinate.
  1008.  * \param right right clipping plane coordinate.
  1009.  * \param bottom bottom clipping plane coordinate.
  1010.  * \param top top clipping plane coordinate.
  1011.  * \param nearval distance to the near clipping plane.
  1012.  * \param farval distance to the far clipping plane.
  1013.  *
  1014.  * Creates the projection matrix and multiplies it with \p mat, marking the
  1015.  * MAT_FLAG_GENERAL_SCALE and MAT_FLAG_TRANSLATION flags.
  1016.  */
  1017. void
  1018. _math_matrix_ortho( GLmatrix *mat,
  1019.                     GLfloat left, GLfloat right,
  1020.                     GLfloat bottom, GLfloat top,
  1021.                     GLfloat nearval, GLfloat farval )
  1022. {
  1023.    GLfloat m[16];
  1024.  
  1025. #define M(row,col)  m[col*4+row]
  1026.    M(0,0) = 2.0F / (right-left);
  1027.    M(0,1) = 0.0F;
  1028.    M(0,2) = 0.0F;
  1029.    M(0,3) = -(right+left) / (right-left);
  1030.  
  1031.    M(1,0) = 0.0F;
  1032.    M(1,1) = 2.0F / (top-bottom);
  1033.    M(1,2) = 0.0F;
  1034.    M(1,3) = -(top+bottom) / (top-bottom);
  1035.  
  1036.    M(2,0) = 0.0F;
  1037.    M(2,1) = 0.0F;
  1038.    M(2,2) = -2.0F / (farval-nearval);
  1039.    M(2,3) = -(farval+nearval) / (farval-nearval);
  1040.  
  1041.    M(3,0) = 0.0F;
  1042.    M(3,1) = 0.0F;
  1043.    M(3,2) = 0.0F;
  1044.    M(3,3) = 1.0F;
  1045. #undef M
  1046.  
  1047.    matrix_multf( mat, m, (MAT_FLAG_GENERAL_SCALE|MAT_FLAG_TRANSLATION));
  1048. }
  1049.  
  1050. /**
  1051.  * Multiply a matrix with a general scaling matrix.
  1052.  *
  1053.  * \param mat matrix.
  1054.  * \param x x axis scale factor.
  1055.  * \param y y axis scale factor.
  1056.  * \param z z axis scale factor.
  1057.  *
  1058.  * Multiplies in-place the elements of \p mat by the scale factors. Checks if
  1059.  * the scales factors are roughly the same, marking the MAT_FLAG_UNIFORM_SCALE
  1060.  * flag, or MAT_FLAG_GENERAL_SCALE. Marks the MAT_DIRTY_TYPE and
  1061.  * MAT_DIRTY_INVERSE dirty flags.
  1062.  */
  1063. void
  1064. _math_matrix_scale( GLmatrix *mat, GLfloat x, GLfloat y, GLfloat z )
  1065. {
  1066.    GLfloat *m = mat->m;
  1067.    m[0] *= x;   m[4] *= y;   m[8]  *= z;
  1068.    m[1] *= x;   m[5] *= y;   m[9]  *= z;
  1069.    m[2] *= x;   m[6] *= y;   m[10] *= z;
  1070.    m[3] *= x;   m[7] *= y;   m[11] *= z;
  1071.  
  1072.    if (FABSF(x - y) < 1e-8 && FABSF(x - z) < 1e-8)
  1073.       mat->flags |= MAT_FLAG_UNIFORM_SCALE;
  1074.    else
  1075.       mat->flags |= MAT_FLAG_GENERAL_SCALE;
  1076.  
  1077.    mat->flags |= (MAT_DIRTY_TYPE |
  1078.                   MAT_DIRTY_INVERSE);
  1079. }
  1080.  
  1081. /**
  1082.  * Multiply a matrix with a translation matrix.
  1083.  *
  1084.  * \param mat matrix.
  1085.  * \param x translation vector x coordinate.
  1086.  * \param y translation vector y coordinate.
  1087.  * \param z translation vector z coordinate.
  1088.  *
  1089.  * Adds the translation coordinates to the elements of \p mat in-place.  Marks
  1090.  * the MAT_FLAG_TRANSLATION flag, and the MAT_DIRTY_TYPE and MAT_DIRTY_INVERSE
  1091.  * dirty flags.
  1092.  */
  1093. void
  1094. _math_matrix_translate( GLmatrix *mat, GLfloat x, GLfloat y, GLfloat z )
  1095. {
  1096.    GLfloat *m = mat->m;
  1097.    m[12] = m[0] * x + m[4] * y + m[8]  * z + m[12];
  1098.    m[13] = m[1] * x + m[5] * y + m[9]  * z + m[13];
  1099.    m[14] = m[2] * x + m[6] * y + m[10] * z + m[14];
  1100.    m[15] = m[3] * x + m[7] * y + m[11] * z + m[15];
  1101.  
  1102.    mat->flags |= (MAT_FLAG_TRANSLATION |
  1103.                   MAT_DIRTY_TYPE |
  1104.                   MAT_DIRTY_INVERSE);
  1105. }
  1106.  
  1107.  
  1108. /**
  1109.  * Set matrix to do viewport and depthrange mapping.
  1110.  * Transforms Normalized Device Coords to window/Z values.
  1111.  */
  1112. void
  1113. _math_matrix_viewport(GLmatrix *m, GLint x, GLint y, GLint width, GLint height,
  1114.                       GLfloat zNear, GLfloat zFar, GLfloat depthMax)
  1115. {
  1116.    m->m[MAT_SX] = (GLfloat) width / 2.0F;
  1117.    m->m[MAT_TX] = m->m[MAT_SX] + x;
  1118.    m->m[MAT_SY] = (GLfloat) height / 2.0F;
  1119.    m->m[MAT_TY] = m->m[MAT_SY] + y;
  1120.    m->m[MAT_SZ] = depthMax * ((zFar - zNear) / 2.0F);
  1121.    m->m[MAT_TZ] = depthMax * ((zFar - zNear) / 2.0F + zNear);
  1122.    m->flags = MAT_FLAG_GENERAL_SCALE | MAT_FLAG_TRANSLATION;
  1123.    m->type = MATRIX_3D_NO_ROT;
  1124. }
  1125.  
  1126.  
  1127. /**
  1128.  * Set a matrix to the identity matrix.
  1129.  *
  1130.  * \param mat matrix.
  1131.  *
  1132.  * Copies ::Identity into \p GLmatrix::m, and into GLmatrix::inv if not NULL.
  1133.  * Sets the matrix type to identity, and clear the dirty flags.
  1134.  */
  1135. void
  1136. _math_matrix_set_identity( GLmatrix *mat )
  1137. {
  1138.    memcpy( mat->m, Identity, 16*sizeof(GLfloat) );
  1139.    memcpy( mat->inv, Identity, 16*sizeof(GLfloat) );
  1140.  
  1141.    mat->type = MATRIX_IDENTITY;
  1142.    mat->flags &= ~(MAT_DIRTY_FLAGS|
  1143.                    MAT_DIRTY_TYPE|
  1144.                    MAT_DIRTY_INVERSE);
  1145. }
  1146.  
  1147. /*@}*/
  1148.  
  1149.  
  1150. /**********************************************************************/
  1151. /** \name Matrix analysis */
  1152. /*@{*/
  1153.  
  1154. #define ZERO(x) (1<<x)
  1155. #define ONE(x)  (1<<(x+16))
  1156.  
  1157. #define MASK_NO_TRX      (ZERO(12) | ZERO(13) | ZERO(14))
  1158. #define MASK_NO_2D_SCALE ( ONE(0)  | ONE(5))
  1159.  
  1160. #define MASK_IDENTITY    ( ONE(0)  | ZERO(4)  | ZERO(8)  | ZERO(12) |\
  1161.                           ZERO(1)  |  ONE(5)  | ZERO(9)  | ZERO(13) |\
  1162.                           ZERO(2)  | ZERO(6)  |  ONE(10) | ZERO(14) |\
  1163.                           ZERO(3)  | ZERO(7)  | ZERO(11) |  ONE(15) )
  1164.  
  1165. #define MASK_2D_NO_ROT   (           ZERO(4)  | ZERO(8)  |           \
  1166.                           ZERO(1)  |            ZERO(9)  |           \
  1167.                           ZERO(2)  | ZERO(6)  |  ONE(10) | ZERO(14) |\
  1168.                           ZERO(3)  | ZERO(7)  | ZERO(11) |  ONE(15) )
  1169.  
  1170. #define MASK_2D          (                      ZERO(8)  |           \
  1171.                                                 ZERO(9)  |           \
  1172.                           ZERO(2)  | ZERO(6)  |  ONE(10) | ZERO(14) |\
  1173.                           ZERO(3)  | ZERO(7)  | ZERO(11) |  ONE(15) )
  1174.  
  1175.  
  1176. #define MASK_3D_NO_ROT   (           ZERO(4)  | ZERO(8)  |           \
  1177.                           ZERO(1)  |            ZERO(9)  |           \
  1178.                           ZERO(2)  | ZERO(6)  |                      \
  1179.                           ZERO(3)  | ZERO(7)  | ZERO(11) |  ONE(15) )
  1180.  
  1181. #define MASK_3D          (                                           \
  1182.                                                                      \
  1183.                                                                      \
  1184.                           ZERO(3)  | ZERO(7)  | ZERO(11) |  ONE(15) )
  1185.  
  1186.  
  1187. #define MASK_PERSPECTIVE (           ZERO(4)  |            ZERO(12) |\
  1188.                           ZERO(1)  |                       ZERO(13) |\
  1189.                           ZERO(2)  | ZERO(6)  |                      \
  1190.                           ZERO(3)  | ZERO(7)  |            ZERO(15) )
  1191.  
  1192. #define SQ(x) ((x)*(x))
  1193.  
  1194. /**
  1195.  * Determine type and flags from scratch.  
  1196.  *
  1197.  * \param mat matrix.
  1198.  *
  1199.  * This is expensive enough to only want to do it once.
  1200.  */
  1201. static void analyse_from_scratch( GLmatrix *mat )
  1202. {
  1203.    const GLfloat *m = mat->m;
  1204.    GLuint mask = 0;
  1205.    GLuint i;
  1206.  
  1207.    for (i = 0 ; i < 16 ; i++) {
  1208.       if (m[i] == 0.0) mask |= (1<<i);
  1209.    }
  1210.  
  1211.    if (m[0] == 1.0F) mask |= (1<<16);
  1212.    if (m[5] == 1.0F) mask |= (1<<21);
  1213.    if (m[10] == 1.0F) mask |= (1<<26);
  1214.    if (m[15] == 1.0F) mask |= (1<<31);
  1215.  
  1216.    mat->flags &= ~MAT_FLAGS_GEOMETRY;
  1217.  
  1218.    /* Check for translation - no-one really cares
  1219.     */
  1220.    if ((mask & MASK_NO_TRX) != MASK_NO_TRX)
  1221.       mat->flags |= MAT_FLAG_TRANSLATION;
  1222.  
  1223.    /* Do the real work
  1224.     */
  1225.    if (mask == (GLuint) MASK_IDENTITY) {
  1226.       mat->type = MATRIX_IDENTITY;
  1227.    }
  1228.    else if ((mask & MASK_2D_NO_ROT) == (GLuint) MASK_2D_NO_ROT) {
  1229.       mat->type = MATRIX_2D_NO_ROT;
  1230.  
  1231.       if ((mask & MASK_NO_2D_SCALE) != MASK_NO_2D_SCALE)
  1232.          mat->flags |= MAT_FLAG_GENERAL_SCALE;
  1233.    }
  1234.    else if ((mask & MASK_2D) == (GLuint) MASK_2D) {
  1235.       GLfloat mm = DOT2(m, m);
  1236.       GLfloat m4m4 = DOT2(m+4,m+4);
  1237.       GLfloat mm4 = DOT2(m,m+4);
  1238.  
  1239.       mat->type = MATRIX_2D;
  1240.  
  1241.       /* Check for scale */
  1242.       if (SQ(mm-1) > SQ(1e-6) ||
  1243.           SQ(m4m4-1) > SQ(1e-6))
  1244.          mat->flags |= MAT_FLAG_GENERAL_SCALE;
  1245.  
  1246.       /* Check for rotation */
  1247.       if (SQ(mm4) > SQ(1e-6))
  1248.          mat->flags |= MAT_FLAG_GENERAL_3D;
  1249.       else
  1250.          mat->flags |= MAT_FLAG_ROTATION;
  1251.  
  1252.    }
  1253.    else if ((mask & MASK_3D_NO_ROT) == (GLuint) MASK_3D_NO_ROT) {
  1254.       mat->type = MATRIX_3D_NO_ROT;
  1255.  
  1256.       /* Check for scale */
  1257.       if (SQ(m[0]-m[5]) < SQ(1e-6) &&
  1258.           SQ(m[0]-m[10]) < SQ(1e-6)) {
  1259.          if (SQ(m[0]-1.0) > SQ(1e-6)) {
  1260.             mat->flags |= MAT_FLAG_UNIFORM_SCALE;
  1261.          }
  1262.       }
  1263.       else {
  1264.          mat->flags |= MAT_FLAG_GENERAL_SCALE;
  1265.       }
  1266.    }
  1267.    else if ((mask & MASK_3D) == (GLuint) MASK_3D) {
  1268.       GLfloat c1 = DOT3(m,m);
  1269.       GLfloat c2 = DOT3(m+4,m+4);
  1270.       GLfloat c3 = DOT3(m+8,m+8);
  1271.       GLfloat d1 = DOT3(m, m+4);
  1272.       GLfloat cp[3];
  1273.  
  1274.       mat->type = MATRIX_3D;
  1275.  
  1276.       /* Check for scale */
  1277.       if (SQ(c1-c2) < SQ(1e-6) && SQ(c1-c3) < SQ(1e-6)) {
  1278.          if (SQ(c1-1.0) > SQ(1e-6))
  1279.             mat->flags |= MAT_FLAG_UNIFORM_SCALE;
  1280.          /* else no scale at all */
  1281.       }
  1282.       else {
  1283.          mat->flags |= MAT_FLAG_GENERAL_SCALE;
  1284.       }
  1285.  
  1286.       /* Check for rotation */
  1287.       if (SQ(d1) < SQ(1e-6)) {
  1288.          CROSS3( cp, m, m+4 );
  1289.          SUB_3V( cp, cp, (m+8) );
  1290.          if (LEN_SQUARED_3FV(cp) < SQ(1e-6))
  1291.             mat->flags |= MAT_FLAG_ROTATION;
  1292.          else
  1293.             mat->flags |= MAT_FLAG_GENERAL_3D;
  1294.       }
  1295.       else {
  1296.          mat->flags |= MAT_FLAG_GENERAL_3D; /* shear, etc */
  1297.       }
  1298.    }
  1299.    else if ((mask & MASK_PERSPECTIVE) == MASK_PERSPECTIVE && m[11]==-1.0F) {
  1300.       mat->type = MATRIX_PERSPECTIVE;
  1301.       mat->flags |= MAT_FLAG_GENERAL;
  1302.    }
  1303.    else {
  1304.       mat->type = MATRIX_GENERAL;
  1305.       mat->flags |= MAT_FLAG_GENERAL;
  1306.    }
  1307. }
  1308.  
  1309. /**
  1310.  * Analyze a matrix given that its flags are accurate.
  1311.  *
  1312.  * This is the more common operation, hopefully.
  1313.  */
  1314. static void analyse_from_flags( GLmatrix *mat )
  1315. {
  1316.    const GLfloat *m = mat->m;
  1317.  
  1318.    if (TEST_MAT_FLAGS(mat, 0)) {
  1319.       mat->type = MATRIX_IDENTITY;
  1320.    }
  1321.    else if (TEST_MAT_FLAGS(mat, (MAT_FLAG_TRANSLATION |
  1322.                                  MAT_FLAG_UNIFORM_SCALE |
  1323.                                  MAT_FLAG_GENERAL_SCALE))) {
  1324.       if ( m[10]==1.0F && m[14]==0.0F ) {
  1325.          mat->type = MATRIX_2D_NO_ROT;
  1326.       }
  1327.       else {
  1328.          mat->type = MATRIX_3D_NO_ROT;
  1329.       }
  1330.    }
  1331.    else if (TEST_MAT_FLAGS(mat, MAT_FLAGS_3D)) {
  1332.       if (                                 m[ 8]==0.0F
  1333.             &&                             m[ 9]==0.0F
  1334.             && m[2]==0.0F && m[6]==0.0F && m[10]==1.0F && m[14]==0.0F) {
  1335.          mat->type = MATRIX_2D;
  1336.       }
  1337.       else {
  1338.          mat->type = MATRIX_3D;
  1339.       }
  1340.    }
  1341.    else if (                 m[4]==0.0F                 && m[12]==0.0F
  1342.             && m[1]==0.0F                               && m[13]==0.0F
  1343.             && m[2]==0.0F && m[6]==0.0F
  1344.             && m[3]==0.0F && m[7]==0.0F && m[11]==-1.0F && m[15]==0.0F) {
  1345.       mat->type = MATRIX_PERSPECTIVE;
  1346.    }
  1347.    else {
  1348.       mat->type = MATRIX_GENERAL;
  1349.    }
  1350. }
  1351.  
  1352. /**
  1353.  * Analyze and update a matrix.
  1354.  *
  1355.  * \param mat matrix.
  1356.  *
  1357.  * If the matrix type is dirty then calls either analyse_from_scratch() or
  1358.  * analyse_from_flags() to determine its type, according to whether the flags
  1359.  * are dirty or not, respectively. If the matrix has an inverse and it's dirty
  1360.  * then calls matrix_invert(). Finally clears the dirty flags.
  1361.  */
  1362. void
  1363. _math_matrix_analyse( GLmatrix *mat )
  1364. {
  1365.    if (mat->flags & MAT_DIRTY_TYPE) {
  1366.       if (mat->flags & MAT_DIRTY_FLAGS)
  1367.          analyse_from_scratch( mat );
  1368.       else
  1369.          analyse_from_flags( mat );
  1370.    }
  1371.  
  1372.    if (mat->inv && (mat->flags & MAT_DIRTY_INVERSE)) {
  1373.       matrix_invert( mat );
  1374.       mat->flags &= ~MAT_DIRTY_INVERSE;
  1375.    }
  1376.  
  1377.    mat->flags &= ~(MAT_DIRTY_FLAGS | MAT_DIRTY_TYPE);
  1378. }
  1379.  
  1380. /*@}*/
  1381.  
  1382.  
  1383. /**
  1384.  * Test if the given matrix preserves vector lengths.
  1385.  */
  1386. GLboolean
  1387. _math_matrix_is_length_preserving( const GLmatrix *m )
  1388. {
  1389.    return TEST_MAT_FLAGS( m, MAT_FLAGS_LENGTH_PRESERVING);
  1390. }
  1391.  
  1392.  
  1393. /**
  1394.  * Test if the given matrix does any rotation.
  1395.  * (or perhaps if the upper-left 3x3 is non-identity)
  1396.  */
  1397. GLboolean
  1398. _math_matrix_has_rotation( const GLmatrix *m )
  1399. {
  1400.    if (m->flags & (MAT_FLAG_GENERAL |
  1401.                    MAT_FLAG_ROTATION |
  1402.                    MAT_FLAG_GENERAL_3D |
  1403.                    MAT_FLAG_PERSPECTIVE))
  1404.       return GL_TRUE;
  1405.    else
  1406.       return GL_FALSE;
  1407. }
  1408.  
  1409.  
  1410. GLboolean
  1411. _math_matrix_is_general_scale( const GLmatrix *m )
  1412. {
  1413.    return (m->flags & MAT_FLAG_GENERAL_SCALE) ? GL_TRUE : GL_FALSE;
  1414. }
  1415.  
  1416.  
  1417. GLboolean
  1418. _math_matrix_is_dirty( const GLmatrix *m )
  1419. {
  1420.    return (m->flags & MAT_DIRTY) ? GL_TRUE : GL_FALSE;
  1421. }
  1422.  
  1423.  
  1424. /**********************************************************************/
  1425. /** \name Matrix setup */
  1426. /*@{*/
  1427.  
  1428. /**
  1429.  * Copy a matrix.
  1430.  *
  1431.  * \param to destination matrix.
  1432.  * \param from source matrix.
  1433.  *
  1434.  * Copies all fields in GLmatrix, creating an inverse array if necessary.
  1435.  */
  1436. void
  1437. _math_matrix_copy( GLmatrix *to, const GLmatrix *from )
  1438. {
  1439.    memcpy( to->m, from->m, sizeof(Identity) );
  1440.    memcpy(to->inv, from->inv, 16 * sizeof(GLfloat));
  1441.    to->flags = from->flags;
  1442.    to->type = from->type;
  1443. }
  1444.  
  1445. /**
  1446.  * Loads a matrix array into GLmatrix.
  1447.  *
  1448.  * \param m matrix array.
  1449.  * \param mat matrix.
  1450.  *
  1451.  * Copies \p m into GLmatrix::m and marks the MAT_FLAG_GENERAL and MAT_DIRTY
  1452.  * flags.
  1453.  */
  1454. void
  1455. _math_matrix_loadf( GLmatrix *mat, const GLfloat *m )
  1456. {
  1457.    memcpy( mat->m, m, 16*sizeof(GLfloat) );
  1458.    mat->flags = (MAT_FLAG_GENERAL | MAT_DIRTY);
  1459. }
  1460.  
  1461. /**
  1462.  * Matrix constructor.
  1463.  *
  1464.  * \param m matrix.
  1465.  *
  1466.  * Initialize the GLmatrix fields.
  1467.  */
  1468. void
  1469. _math_matrix_ctr( GLmatrix *m )
  1470. {
  1471.    m->m = _mesa_align_malloc( 16 * sizeof(GLfloat), 16 );
  1472.    if (m->m)
  1473.       memcpy( m->m, Identity, sizeof(Identity) );
  1474.    m->inv = _mesa_align_malloc( 16 * sizeof(GLfloat), 16 );
  1475.    if (m->inv)
  1476.       memcpy( m->inv, Identity, sizeof(Identity) );
  1477.    m->type = MATRIX_IDENTITY;
  1478.    m->flags = 0;
  1479. }
  1480.  
  1481. /**
  1482.  * Matrix destructor.
  1483.  *
  1484.  * \param m matrix.
  1485.  *
  1486.  * Frees the data in a GLmatrix.
  1487.  */
  1488. void
  1489. _math_matrix_dtr( GLmatrix *m )
  1490. {
  1491.    if (m->m) {
  1492.       _mesa_align_free( m->m );
  1493.       m->m = NULL;
  1494.    }
  1495.    if (m->inv) {
  1496.       _mesa_align_free( m->inv );
  1497.       m->inv = NULL;
  1498.    }
  1499. }
  1500.  
  1501. /*@}*/
  1502.  
  1503.  
  1504. /**********************************************************************/
  1505. /** \name Matrix transpose */
  1506. /*@{*/
  1507.  
  1508. /**
  1509.  * Transpose a GLfloat matrix.
  1510.  *
  1511.  * \param to destination array.
  1512.  * \param from source array.
  1513.  */
  1514. void
  1515. _math_transposef( GLfloat to[16], const GLfloat from[16] )
  1516. {
  1517.    to[0] = from[0];
  1518.    to[1] = from[4];
  1519.    to[2] = from[8];
  1520.    to[3] = from[12];
  1521.    to[4] = from[1];
  1522.    to[5] = from[5];
  1523.    to[6] = from[9];
  1524.    to[7] = from[13];
  1525.    to[8] = from[2];
  1526.    to[9] = from[6];
  1527.    to[10] = from[10];
  1528.    to[11] = from[14];
  1529.    to[12] = from[3];
  1530.    to[13] = from[7];
  1531.    to[14] = from[11];
  1532.    to[15] = from[15];
  1533. }
  1534.  
  1535. /**
  1536.  * Transpose a GLdouble matrix.
  1537.  *
  1538.  * \param to destination array.
  1539.  * \param from source array.
  1540.  */
  1541. void
  1542. _math_transposed( GLdouble to[16], const GLdouble from[16] )
  1543. {
  1544.    to[0] = from[0];
  1545.    to[1] = from[4];
  1546.    to[2] = from[8];
  1547.    to[3] = from[12];
  1548.    to[4] = from[1];
  1549.    to[5] = from[5];
  1550.    to[6] = from[9];
  1551.    to[7] = from[13];
  1552.    to[8] = from[2];
  1553.    to[9] = from[6];
  1554.    to[10] = from[10];
  1555.    to[11] = from[14];
  1556.    to[12] = from[3];
  1557.    to[13] = from[7];
  1558.    to[14] = from[11];
  1559.    to[15] = from[15];
  1560. }
  1561.  
  1562. /**
  1563.  * Transpose a GLdouble matrix and convert to GLfloat.
  1564.  *
  1565.  * \param to destination array.
  1566.  * \param from source array.
  1567.  */
  1568. void
  1569. _math_transposefd( GLfloat to[16], const GLdouble from[16] )
  1570. {
  1571.    to[0] = (GLfloat) from[0];
  1572.    to[1] = (GLfloat) from[4];
  1573.    to[2] = (GLfloat) from[8];
  1574.    to[3] = (GLfloat) from[12];
  1575.    to[4] = (GLfloat) from[1];
  1576.    to[5] = (GLfloat) from[5];
  1577.    to[6] = (GLfloat) from[9];
  1578.    to[7] = (GLfloat) from[13];
  1579.    to[8] = (GLfloat) from[2];
  1580.    to[9] = (GLfloat) from[6];
  1581.    to[10] = (GLfloat) from[10];
  1582.    to[11] = (GLfloat) from[14];
  1583.    to[12] = (GLfloat) from[3];
  1584.    to[13] = (GLfloat) from[7];
  1585.    to[14] = (GLfloat) from[11];
  1586.    to[15] = (GLfloat) from[15];
  1587. }
  1588.  
  1589. /*@}*/
  1590.  
  1591.  
  1592. /**
  1593.  * Transform a 4-element row vector (1x4 matrix) by a 4x4 matrix.  This
  1594.  * function is used for transforming clipping plane equations and spotlight
  1595.  * directions.
  1596.  * Mathematically,  u = v * m.
  1597.  * Input:  v - input vector
  1598.  *         m - transformation matrix
  1599.  * Output:  u - transformed vector
  1600.  */
  1601. void
  1602. _mesa_transform_vector( GLfloat u[4], const GLfloat v[4], const GLfloat m[16] )
  1603. {
  1604.    const GLfloat v0 = v[0], v1 = v[1], v2 = v[2], v3 = v[3];
  1605. #define M(row,col)  m[row + col*4]
  1606.    u[0] = v0 * M(0,0) + v1 * M(1,0) + v2 * M(2,0) + v3 * M(3,0);
  1607.    u[1] = v0 * M(0,1) + v1 * M(1,1) + v2 * M(2,1) + v3 * M(3,1);
  1608.    u[2] = v0 * M(0,2) + v1 * M(1,2) + v2 * M(2,2) + v3 * M(3,2);
  1609.    u[3] = v0 * M(0,3) + v1 * M(1,3) + v2 * M(2,3) + v3 * M(3,3);
  1610. #undef M
  1611. }
  1612.