svn merge ^/branches/released . --username marha

1 files changed, 1642 insertions, 0 deletions

diff --git a/mesalib/src/mesa/math/m_matrix.c b/mesalib/src/mesa/math/m_matrix.c
new file mode 100644
index 000000000..da6956efe
--- /dev/null
+++ b/mesalib/src/mesa/math/m_matrix.c
@@ -0,0 +1,1642 @@
+/*
+ * Mesa 3-D graphics library
+ * Version:  6.3
+ *
+ * Copyright (C) 1999-2005  Brian Paul   All Rights Reserved.
+ *
+ * Permission is hereby granted, free of charge, to any person obtaining a
+ * copy of this software and associated documentation files (the "Software"),
+ * to deal in the Software without restriction, including without limitation
+ * the rights to use, copy, modify, merge, publish, distribute, sublicense,
+ * and/or sell copies of the Software, and to permit persons to whom the
+ * Software is furnished to do so, subject to the following conditions:
+ *
+ * The above copyright notice and this permission notice shall be included
+ * in all copies or substantial portions of the Software.
+ *
+ * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
+ * OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+ * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL
+ * BRIAN PAUL BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN
+ * AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
+ * CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
+ */
+
+
+/**
+ * \file m_matrix.c
+ * Matrix operations.
+ *
+ * \note
+ * -# 4x4 transformation matrices are stored in memory in column major order.
+ * -# Points/vertices are to be thought of as column vectors.
+ * -# Transformation of a point p by a matrix M is: p' = M * p
+ */
+
+
+#include "main/glheader.h"
+#include "main/imports.h"
+#include "main/macros.h"
+#include "main/imports.h"
+
+#include "m_matrix.h"
+
+
+/**
+ * \defgroup MatFlags MAT_FLAG_XXX-flags
+ *
+ * Bitmasks to indicate different kinds of 4x4 matrices in GLmatrix::flags
+ * It would be nice to make all these flags private to m_matrix.c
+ */
+/*@{*/
+#define MAT_FLAG_IDENTITY       0     /**< is an identity matrix flag.
+                                       *   (Not actually used - the identity
+                                       *   matrix is identified by the absense
+                                       *   of all other flags.)
+                                       */
+#define MAT_FLAG_GENERAL        0x1   /**< is a general matrix flag */
+#define MAT_FLAG_ROTATION       0x2   /**< is a rotation matrix flag */
+#define MAT_FLAG_TRANSLATION    0x4   /**< is a translation matrix flag */
+#define MAT_FLAG_UNIFORM_SCALE  0x8   /**< is an uniform scaling matrix flag */
+#define MAT_FLAG_GENERAL_SCALE  0x10  /**< is a general scaling matrix flag */
+#define MAT_FLAG_GENERAL_3D     0x20  /**< general 3D matrix flag */
+#define MAT_FLAG_PERSPECTIVE    0x40  /**< is a perspective proj matrix flag */
+#define MAT_FLAG_SINGULAR       0x80  /**< is a singular matrix flag */
+#define MAT_DIRTY_TYPE          0x100  /**< matrix type is dirty */
+#define MAT_DIRTY_FLAGS         0x200  /**< matrix flags are dirty */
+#define MAT_DIRTY_INVERSE       0x400  /**< matrix inverse is dirty */
+
+/** angle preserving matrix flags mask */
+#define MAT_FLAGS_ANGLE_PRESERVING (MAT_FLAG_ROTATION | \
+				    MAT_FLAG_TRANSLATION | \
+				    MAT_FLAG_UNIFORM_SCALE)
+
+/** geometry related matrix flags mask */
+#define MAT_FLAGS_GEOMETRY (MAT_FLAG_GENERAL | \
+			    MAT_FLAG_ROTATION | \
+			    MAT_FLAG_TRANSLATION | \
+			    MAT_FLAG_UNIFORM_SCALE | \
+			    MAT_FLAG_GENERAL_SCALE | \
+			    MAT_FLAG_GENERAL_3D | \
+			    MAT_FLAG_PERSPECTIVE | \
+	                    MAT_FLAG_SINGULAR)
+
+/** length preserving matrix flags mask */
+#define MAT_FLAGS_LENGTH_PRESERVING (MAT_FLAG_ROTATION | \
+				     MAT_FLAG_TRANSLATION)
+
+
+/** 3D (non-perspective) matrix flags mask */
+#define MAT_FLAGS_3D (MAT_FLAG_ROTATION | \
+		      MAT_FLAG_TRANSLATION | \
+		      MAT_FLAG_UNIFORM_SCALE | \
+		      MAT_FLAG_GENERAL_SCALE | \
+		      MAT_FLAG_GENERAL_3D)
+
+/** dirty matrix flags mask */
+#define MAT_DIRTY          (MAT_DIRTY_TYPE | \
+			    MAT_DIRTY_FLAGS | \
+			    MAT_DIRTY_INVERSE)
+
+/*@}*/
+
+
+/** 
+ * Test geometry related matrix flags.
+ * 
+ * \param mat a pointer to a GLmatrix structure.
+ * \param a flags mask.
+ *
+ * \returns non-zero if all geometry related matrix flags are contained within
+ * the mask, or zero otherwise.
+ */ 
+#define TEST_MAT_FLAGS(mat, a)  \
+    ((MAT_FLAGS_GEOMETRY & (~(a)) & ((mat)->flags) ) == 0)
+
+
+
+/**
+ * Names of the corresponding GLmatrixtype values.
+ */
+static const char *types[] = {
+   "MATRIX_GENERAL",
+   "MATRIX_IDENTITY",
+   "MATRIX_3D_NO_ROT",
+   "MATRIX_PERSPECTIVE",
+   "MATRIX_2D",
+   "MATRIX_2D_NO_ROT",
+   "MATRIX_3D"
+};
+
+
+/**
+ * Identity matrix.
+ */
+static GLfloat Identity[16] = {
+   1.0, 0.0, 0.0, 0.0,
+   0.0, 1.0, 0.0, 0.0,
+   0.0, 0.0, 1.0, 0.0,
+   0.0, 0.0, 0.0, 1.0
+};
+
+
+
+/**********************************************************************/
+/** \name Matrix multiplication */
+/*@{*/
+
+#define A(row,col)  a[(col<<2)+row]
+#define B(row,col)  b[(col<<2)+row]
+#define P(row,col)  product[(col<<2)+row]
+
+/**
+ * Perform a full 4x4 matrix multiplication.
+ *
+ * \param a matrix.
+ * \param b matrix.
+ * \param product will receive the product of \p a and \p b.
+ *
+ * \warning Is assumed that \p product != \p b. \p product == \p a is allowed.
+ *
+ * \note KW: 4*16 = 64 multiplications
+ * 
+ * \author This \c matmul was contributed by Thomas Malik
+ */
+static void matmul4( GLfloat *product, const GLfloat *a, const GLfloat *b )
+{
+   GLint i;
+   for (i = 0; i < 4; i++) {
+      const GLfloat ai0=A(i,0),  ai1=A(i,1),  ai2=A(i,2),  ai3=A(i,3);
+      P(i,0) = ai0 * B(0,0) + ai1 * B(1,0) + ai2 * B(2,0) + ai3 * B(3,0);
+      P(i,1) = ai0 * B(0,1) + ai1 * B(1,1) + ai2 * B(2,1) + ai3 * B(3,1);
+      P(i,2) = ai0 * B(0,2) + ai1 * B(1,2) + ai2 * B(2,2) + ai3 * B(3,2);
+      P(i,3) = ai0 * B(0,3) + ai1 * B(1,3) + ai2 * B(2,3) + ai3 * B(3,3);
+   }
+}
+
+/**
+ * Multiply two matrices known to occupy only the top three rows, such
+ * as typical model matrices, and orthogonal matrices.
+ *
+ * \param a matrix.
+ * \param b matrix.
+ * \param product will receive the product of \p a and \p b.
+ */
+static void matmul34( GLfloat *product, const GLfloat *a, const GLfloat *b )
+{
+   GLint i;
+   for (i = 0; i < 3; i++) {
+      const GLfloat ai0=A(i,0),  ai1=A(i,1),  ai2=A(i,2),  ai3=A(i,3);
+      P(i,0) = ai0 * B(0,0) + ai1 * B(1,0) + ai2 * B(2,0);
+      P(i,1) = ai0 * B(0,1) + ai1 * B(1,1) + ai2 * B(2,1);
+      P(i,2) = ai0 * B(0,2) + ai1 * B(1,2) + ai2 * B(2,2);
+      P(i,3) = ai0 * B(0,3) + ai1 * B(1,3) + ai2 * B(2,3) + ai3;
+   }
+   P(3,0) = 0;
+   P(3,1) = 0;
+   P(3,2) = 0;
+   P(3,3) = 1;
+}
+
+#undef A
+#undef B
+#undef P
+
+/**
+ * Multiply a matrix by an array of floats with known properties.
+ *
+ * \param mat pointer to a GLmatrix structure containing the left multiplication
+ * matrix, and that will receive the product result.
+ * \param m right multiplication matrix array.
+ * \param flags flags of the matrix \p m.
+ * 
+ * Joins both flags and marks the type and inverse as dirty.  Calls matmul34()
+ * if both matrices are 3D, or matmul4() otherwise.
+ */
+static void matrix_multf( GLmatrix *mat, const GLfloat *m, GLuint flags )
+{
+   mat->flags |= (flags | MAT_DIRTY_TYPE | MAT_DIRTY_INVERSE);
+
+   if (TEST_MAT_FLAGS(mat, MAT_FLAGS_3D))
+      matmul34( mat->m, mat->m, m );
+   else
+      matmul4( mat->m, mat->m, m );
+}
+
+/**
+ * Matrix multiplication.
+ *
+ * \param dest destination matrix.
+ * \param a left matrix.
+ * \param b right matrix.
+ * 
+ * Joins both flags and marks the type and inverse as dirty.  Calls matmul34()
+ * if both matrices are 3D, or matmul4() otherwise.
+ */
+void
+_math_matrix_mul_matrix( GLmatrix *dest, const GLmatrix *a, const GLmatrix *b )
+{
+   dest->flags = (a->flags |
+		  b->flags |
+		  MAT_DIRTY_TYPE |
+		  MAT_DIRTY_INVERSE);
+
+   if (TEST_MAT_FLAGS(dest, MAT_FLAGS_3D))
+      matmul34( dest->m, a->m, b->m );
+   else
+      matmul4( dest->m, a->m, b->m );
+}
+
+/**
+ * Matrix multiplication.
+ *
+ * \param dest left and destination matrix.
+ * \param m right matrix array.
+ * 
+ * Marks the matrix flags with general flag, and type and inverse dirty flags.
+ * Calls matmul4() for the multiplication.
+ */
+void
+_math_matrix_mul_floats( GLmatrix *dest, const GLfloat *m )
+{
+   dest->flags |= (MAT_FLAG_GENERAL |
+		   MAT_DIRTY_TYPE |
+		   MAT_DIRTY_INVERSE |
+                   MAT_DIRTY_FLAGS);
+
+   matmul4( dest->m, dest->m, m );
+}
+
+/*@}*/
+
+
+/**********************************************************************/
+/** \name Matrix output */
+/*@{*/
+
+/**
+ * Print a matrix array.
+ *
+ * \param m matrix array.
+ *
+ * Called by _math_matrix_print() to print a matrix or its inverse.
+ */
+static void print_matrix_floats( const GLfloat m[16] )
+{
+   int i;
+   for (i=0;i<4;i++) {
+      _mesa_debug(NULL,"\t%f %f %f %f\n", m[i], m[4+i], m[8+i], m[12+i] );
+   }
+}
+
+/**
+ * Dumps the contents of a GLmatrix structure.
+ * 
+ * \param m pointer to the GLmatrix structure.
+ */
+void
+_math_matrix_print( const GLmatrix *m )
+{
+   _mesa_debug(NULL, "Matrix type: %s, flags: %x\n", types[m->type], m->flags);
+   print_matrix_floats(m->m);
+   _mesa_debug(NULL, "Inverse: \n");
+   if (m->inv) {
+      GLfloat prod[16];
+      print_matrix_floats(m->inv);
+      matmul4(prod, m->m, m->inv);
+      _mesa_debug(NULL, "Mat * Inverse:\n");
+      print_matrix_floats(prod);
+   }
+   else {
+      _mesa_debug(NULL, "  - not available\n");
+   }
+}
+
+/*@}*/
+
+
+/**
+ * References an element of 4x4 matrix.
+ *
+ * \param m matrix array.
+ * \param c column of the desired element.
+ * \param r row of the desired element.
+ * 
+ * \return value of the desired element.
+ *
+ * Calculate the linear storage index of the element and references it. 
+ */
+#define MAT(m,r,c) (m)[(c)*4+(r)]
+
+
+/**********************************************************************/
+/** \name Matrix inversion */
+/*@{*/
+
+/**
+ * Swaps the values of two floating pointer variables.
+ *
+ * Used by invert_matrix_general() to swap the row pointers.
+ */
+#define SWAP_ROWS(a, b) { GLfloat *_tmp = a; (a)=(b); (b)=_tmp; }
+
+/**
+ * Compute inverse of 4x4 transformation matrix.
+ * 
+ * \param mat pointer to a GLmatrix structure. The matrix inverse will be
+ * stored in the GLmatrix::inv attribute.
+ * 
+ * \return GL_TRUE for success, GL_FALSE for failure (\p singular matrix).
+ * 
+ * \author
+ * Code contributed by Jacques Leroy jle@star.be
+ *
+ * Calculates the inverse matrix by performing the gaussian matrix reduction
+ * with partial pivoting followed by back/substitution with the loops manually
+ * unrolled.
+ */
+static GLboolean invert_matrix_general( GLmatrix *mat )
+{
+   const GLfloat *m = mat->m;
+   GLfloat *out = mat->inv;
+   GLfloat wtmp[4][8];
+   GLfloat m0, m1, m2, m3, s;
+   GLfloat *r0, *r1, *r2, *r3;
+
+   r0 = wtmp[0], r1 = wtmp[1], r2 = wtmp[2], r3 = wtmp[3];
+
+   r0[0] = MAT(m,0,0), r0[1] = MAT(m,0,1),
+   r0[2] = MAT(m,0,2), r0[3] = MAT(m,0,3),
+   r0[4] = 1.0, r0[5] = r0[6] = r0[7] = 0.0,
+
+   r1[0] = MAT(m,1,0), r1[1] = MAT(m,1,1),
+   r1[2] = MAT(m,1,2), r1[3] = MAT(m,1,3),
+   r1[5] = 1.0, r1[4] = r1[6] = r1[7] = 0.0,
+
+   r2[0] = MAT(m,2,0), r2[1] = MAT(m,2,1),
+   r2[2] = MAT(m,2,2), r2[3] = MAT(m,2,3),
+   r2[6] = 1.0, r2[4] = r2[5] = r2[7] = 0.0,
+
+   r3[0] = MAT(m,3,0), r3[1] = MAT(m,3,1),
+   r3[2] = MAT(m,3,2), r3[3] = MAT(m,3,3),
+   r3[7] = 1.0, r3[4] = r3[5] = r3[6] = 0.0;
+
+   /* choose pivot - or die */
+   if (FABSF(r3[0])>FABSF(r2[0])) SWAP_ROWS(r3, r2);
+   if (FABSF(r2[0])>FABSF(r1[0])) SWAP_ROWS(r2, r1);
+   if (FABSF(r1[0])>FABSF(r0[0])) SWAP_ROWS(r1, r0);
+   if (0.0 == r0[0])  return GL_FALSE;
+
+   /* eliminate first variable     */
+   m1 = r1[0]/r0[0]; m2 = r2[0]/r0[0]; m3 = r3[0]/r0[0];
+   s = r0[1]; r1[1] -= m1 * s; r2[1] -= m2 * s; r3[1] -= m3 * s;
+   s = r0[2]; r1[2] -= m1 * s; r2[2] -= m2 * s; r3[2] -= m3 * s;
+   s = r0[3]; r1[3] -= m1 * s; r2[3] -= m2 * s; r3[3] -= m3 * s;
+   s = r0[4];
+   if (s != 0.0) { r1[4] -= m1 * s; r2[4] -= m2 * s; r3[4] -= m3 * s; }
+   s = r0[5];
+   if (s != 0.0) { r1[5] -= m1 * s; r2[5] -= m2 * s; r3[5] -= m3 * s; }
+   s = r0[6];
+   if (s != 0.0) { r1[6] -= m1 * s; r2[6] -= m2 * s; r3[6] -= m3 * s; }
+   s = r0[7];
+   if (s != 0.0) { r1[7] -= m1 * s; r2[7] -= m2 * s; r3[7] -= m3 * s; }
+
+   /* choose pivot - or die */
+   if (FABSF(r3[1])>FABSF(r2[1])) SWAP_ROWS(r3, r2);
+   if (FABSF(r2[1])>FABSF(r1[1])) SWAP_ROWS(r2, r1);
+   if (0.0 == r1[1])  return GL_FALSE;
+
+   /* eliminate second variable */
+   m2 = r2[1]/r1[1]; m3 = r3[1]/r1[1];
+   r2[2] -= m2 * r1[2]; r3[2] -= m3 * r1[2];
+   r2[3] -= m2 * r1[3]; r3[3] -= m3 * r1[3];
+   s = r1[4]; if (0.0 != s) { r2[4] -= m2 * s; r3[4] -= m3 * s; }
+   s = r1[5]; if (0.0 != s) { r2[5] -= m2 * s; r3[5] -= m3 * s; }
+   s = r1[6]; if (0.0 != s) { r2[6] -= m2 * s; r3[6] -= m3 * s; }
+   s = r1[7]; if (0.0 != s) { r2[7] -= m2 * s; r3[7] -= m3 * s; }
+
+   /* choose pivot - or die */
+   if (FABSF(r3[2])>FABSF(r2[2])) SWAP_ROWS(r3, r2);
+   if (0.0 == r2[2])  return GL_FALSE;
+
+   /* eliminate third variable */
+   m3 = r3[2]/r2[2];
+   r3[3] -= m3 * r2[3], r3[4] -= m3 * r2[4],
+   r3[5] -= m3 * r2[5], r3[6] -= m3 * r2[6],
+   r3[7] -= m3 * r2[7];
+
+   /* last check */
+   if (0.0 == r3[3]) return GL_FALSE;
+
+   s = 1.0F/r3[3];             /* now back substitute row 3 */
+   r3[4] *= s; r3[5] *= s; r3[6] *= s; r3[7] *= s;
+
+   m2 = r2[3];                 /* now back substitute row 2 */
+   s  = 1.0F/r2[2];
+   r2[4] = s * (r2[4] - r3[4] * m2), r2[5] = s * (r2[5] - r3[5] * m2),
+   r2[6] = s * (r2[6] - r3[6] * m2), r2[7] = s * (r2[7] - r3[7] * m2);
+   m1 = r1[3];
+   r1[4] -= r3[4] * m1, r1[5] -= r3[5] * m1,
+   r1[6] -= r3[6] * m1, r1[7] -= r3[7] * m1;
+   m0 = r0[3];
+   r0[4] -= r3[4] * m0, r0[5] -= r3[5] * m0,
+   r0[6] -= r3[6] * m0, r0[7] -= r3[7] * m0;
+
+   m1 = r1[2];                 /* now back substitute row 1 */
+   s  = 1.0F/r1[1];
+   r1[4] = s * (r1[4] - r2[4] * m1), r1[5] = s * (r1[5] - r2[5] * m1),
+   r1[6] = s * (r1[6] - r2[6] * m1), r1[7] = s * (r1[7] - r2[7] * m1);
+   m0 = r0[2];
+   r0[4] -= r2[4] * m0, r0[5] -= r2[5] * m0,
+   r0[6] -= r2[6] * m0, r0[7] -= r2[7] * m0;
+
+   m0 = r0[1];                 /* now back substitute row 0 */
+   s  = 1.0F/r0[0];
+   r0[4] = s * (r0[4] - r1[4] * m0), r0[5] = s * (r0[5] - r1[5] * m0),
+   r0[6] = s * (r0[6] - r1[6] * m0), r0[7] = s * (r0[7] - r1[7] * m0);
+
+   MAT(out,0,0) = r0[4]; MAT(out,0,1) = r0[5],
+   MAT(out,0,2) = r0[6]; MAT(out,0,3) = r0[7],
+   MAT(out,1,0) = r1[4]; MAT(out,1,1) = r1[5],
+   MAT(out,1,2) = r1[6]; MAT(out,1,3) = r1[7],
+   MAT(out,2,0) = r2[4]; MAT(out,2,1) = r2[5],
+   MAT(out,2,2) = r2[6]; MAT(out,2,3) = r2[7],
+   MAT(out,3,0) = r3[4]; MAT(out,3,1) = r3[5],
+   MAT(out,3,2) = r3[6]; MAT(out,3,3) = r3[7];
+
+   return GL_TRUE;
+}
+#undef SWAP_ROWS
+
+/**
+ * Compute inverse of a general 3d transformation matrix.
+ * 
+ * \param mat pointer to a GLmatrix structure. The matrix inverse will be
+ * stored in the GLmatrix::inv attribute.
+ * 
+ * \return GL_TRUE for success, GL_FALSE for failure (\p singular matrix).
+ *
+ * \author Adapted from graphics gems II.
+ *
+ * Calculates the inverse of the upper left by first calculating its
+ * determinant and multiplying it to the symmetric adjust matrix of each
+ * element. Finally deals with the translation part by transforming the
+ * original translation vector using by the calculated submatrix inverse.
+ */
+static GLboolean invert_matrix_3d_general( GLmatrix *mat )
+{
+   const GLfloat *in = mat->m;
+   GLfloat *out = mat->inv;
+   GLfloat pos, neg, t;
+   GLfloat det;
+
+   /* Calculate the determinant of upper left 3x3 submatrix and
+    * determine if the matrix is singular.
+    */
+   pos = neg = 0.0;
+   t =  MAT(in,0,0) * MAT(in,1,1) * MAT(in,2,2);
+   if (t >= 0.0) pos += t; else neg += t;
+
+   t =  MAT(in,1,0) * MAT(in,2,1) * MAT(in,0,2);
+   if (t >= 0.0) pos += t; else neg += t;
+
+   t =  MAT(in,2,0) * MAT(in,0,1) * MAT(in,1,2);
+   if (t >= 0.0) pos += t; else neg += t;
+
+   t = -MAT(in,2,0) * MAT(in,1,1) * MAT(in,0,2);
+   if (t >= 0.0) pos += t; else neg += t;
+
+   t = -MAT(in,1,0) * MAT(in,0,1) * MAT(in,2,2);
+   if (t >= 0.0) pos += t; else neg += t;
+
+   t = -MAT(in,0,0) * MAT(in,2,1) * MAT(in,1,2);
+   if (t >= 0.0) pos += t; else neg += t;
+
+   det = pos + neg;
+
+   if (det*det < 1e-25)
+      return GL_FALSE;
+
+   det = 1.0F / det;
+   MAT(out,0,0) = (  (MAT(in,1,1)*MAT(in,2,2) - MAT(in,2,1)*MAT(in,1,2) )*det);
+   MAT(out,0,1) = (- (MAT(in,0,1)*MAT(in,2,2) - MAT(in,2,1)*MAT(in,0,2) )*det);
+   MAT(out,0,2) = (  (MAT(in,0,1)*MAT(in,1,2) - MAT(in,1,1)*MAT(in,0,2) )*det);
+   MAT(out,1,0) = (- (MAT(in,1,0)*MAT(in,2,2) - MAT(in,2,0)*MAT(in,1,2) )*det);
+   MAT(out,1,1) = (  (MAT(in,0,0)*MAT(in,2,2) - MAT(in,2,0)*MAT(in,0,2) )*det);
+   MAT(out,1,2) = (- (MAT(in,0,0)*MAT(in,1,2) - MAT(in,1,0)*MAT(in,0,2) )*det);
+   MAT(out,2,0) = (  (MAT(in,1,0)*MAT(in,2,1) - MAT(in,2,0)*MAT(in,1,1) )*det);
+   MAT(out,2,1) = (- (MAT(in,0,0)*MAT(in,2,1) - MAT(in,2,0)*MAT(in,0,1) )*det);
+   MAT(out,2,2) = (  (MAT(in,0,0)*MAT(in,1,1) - MAT(in,1,0)*MAT(in,0,1) )*det);
+
+   /* Do the translation part */
+   MAT(out,0,3) = - (MAT(in,0,3) * MAT(out,0,0) +
+		     MAT(in,1,3) * MAT(out,0,1) +
+		     MAT(in,2,3) * MAT(out,0,2) );
+   MAT(out,1,3) = - (MAT(in,0,3) * MAT(out,1,0) +
+		     MAT(in,1,3) * MAT(out,1,1) +
+		     MAT(in,2,3) * MAT(out,1,2) );
+   MAT(out,2,3) = - (MAT(in,0,3) * MAT(out,2,0) +
+		     MAT(in,1,3) * MAT(out,2,1) +
+		     MAT(in,2,3) * MAT(out,2,2) );
+
+   return GL_TRUE;
+}
+
+/**
+ * Compute inverse of a 3d transformation matrix.
+ * 
+ * \param mat pointer to a GLmatrix structure. The matrix inverse will be
+ * stored in the GLmatrix::inv attribute.
+ * 
+ * \return GL_TRUE for success, GL_FALSE for failure (\p singular matrix).
+ *
+ * If the matrix is not an angle preserving matrix then calls
+ * invert_matrix_3d_general for the actual calculation. Otherwise calculates
+ * the inverse matrix analyzing and inverting each of the scaling, rotation and
+ * translation parts.
+ */
+static GLboolean invert_matrix_3d( GLmatrix *mat )
+{
+   const GLfloat *in = mat->m;
+   GLfloat *out = mat->inv;
+
+   if (!TEST_MAT_FLAGS(mat, MAT_FLAGS_ANGLE_PRESERVING)) {
+      return invert_matrix_3d_general( mat );
+   }
+
+   if (mat->flags & MAT_FLAG_UNIFORM_SCALE) {
+      GLfloat scale = (MAT(in,0,0) * MAT(in,0,0) +
+                       MAT(in,0,1) * MAT(in,0,1) +
+                       MAT(in,0,2) * MAT(in,0,2));
+
+      if (scale == 0.0)
+         return GL_FALSE;
+
+      scale = 1.0F / scale;
+
+      /* Transpose and scale the 3 by 3 upper-left submatrix. */
+      MAT(out,0,0) = scale * MAT(in,0,0);
+      MAT(out,1,0) = scale * MAT(in,0,1);
+      MAT(out,2,0) = scale * MAT(in,0,2);
+      MAT(out,0,1) = scale * MAT(in,1,0);
+      MAT(out,1,1) = scale * MAT(in,1,1);
+      MAT(out,2,1) = scale * MAT(in,1,2);
+      MAT(out,0,2) = scale * MAT(in,2,0);
+      MAT(out,1,2) = scale * MAT(in,2,1);
+      MAT(out,2,2) = scale * MAT(in,2,2);
+   }
+   else if (mat->flags & MAT_FLAG_ROTATION) {
+      /* Transpose the 3 by 3 upper-left submatrix. */
+      MAT(out,0,0) = MAT(in,0,0);
+      MAT(out,1,0) = MAT(in,0,1);
+      MAT(out,2,0) = MAT(in,0,2);
+      MAT(out,0,1) = MAT(in,1,0);
+      MAT(out,1,1) = MAT(in,1,1);
+      MAT(out,2,1) = MAT(in,1,2);
+      MAT(out,0,2) = MAT(in,2,0);
+      MAT(out,1,2) = MAT(in,2,1);
+      MAT(out,2,2) = MAT(in,2,2);
+   }
+   else {
+      /* pure translation */
+      MEMCPY( out, Identity, sizeof(Identity) );
+      MAT(out,0,3) = - MAT(in,0,3);
+      MAT(out,1,3) = - MAT(in,1,3);
+      MAT(out,2,3) = - MAT(in,2,3);
+      return GL_TRUE;
+   }
+
+   if (mat->flags & MAT_FLAG_TRANSLATION) {
+      /* Do the translation part */
+      MAT(out,0,3) = - (MAT(in,0,3) * MAT(out,0,0) +
+			MAT(in,1,3) * MAT(out,0,1) +
+			MAT(in,2,3) * MAT(out,0,2) );
+      MAT(out,1,3) = - (MAT(in,0,3) * MAT(out,1,0) +
+			MAT(in,1,3) * MAT(out,1,1) +
+			MAT(in,2,3) * MAT(out,1,2) );
+      MAT(out,2,3) = - (MAT(in,0,3) * MAT(out,2,0) +
+			MAT(in,1,3) * MAT(out,2,1) +
+			MAT(in,2,3) * MAT(out,2,2) );
+   }
+   else {
+      MAT(out,0,3) = MAT(out,1,3) = MAT(out,2,3) = 0.0;
+   }
+
+   return GL_TRUE;
+}
+
+/**
+ * Compute inverse of an identity transformation matrix.
+ * 
+ * \param mat pointer to a GLmatrix structure. The matrix inverse will be
+ * stored in the GLmatrix::inv attribute.
+ * 
+ * \return always GL_TRUE.
+ *
+ * Simply copies Identity into GLmatrix::inv.
+ */
+static GLboolean invert_matrix_identity( GLmatrix *mat )
+{
+   MEMCPY( mat->inv, Identity, sizeof(Identity) );
+   return GL_TRUE;
+}
+
+/**
+ * Compute inverse of a no-rotation 3d transformation matrix.
+ * 
+ * \param mat pointer to a GLmatrix structure. The matrix inverse will be
+ * stored in the GLmatrix::inv attribute.
+ * 
+ * \return GL_TRUE for success, GL_FALSE for failure (\p singular matrix).
+ *
+ * Calculates the 
+ */
+static GLboolean invert_matrix_3d_no_rot( GLmatrix *mat )
+{
+   const GLfloat *in = mat->m;
+   GLfloat *out = mat->inv;
+
+   if (MAT(in,0,0) == 0 || MAT(in,1,1) == 0 || MAT(in,2,2) == 0 )
+      return GL_FALSE;
+
+   MEMCPY( out, Identity, 16 * sizeof(GLfloat) );
+   MAT(out,0,0) = 1.0F / MAT(in,0,0);
+   MAT(out,1,1) = 1.0F / MAT(in,1,1);
+   MAT(out,2,2) = 1.0F / MAT(in,2,2);
+
+   if (mat->flags & MAT_FLAG_TRANSLATION) {
+      MAT(out,0,3) = - (MAT(in,0,3) * MAT(out,0,0));
+      MAT(out,1,3) = - (MAT(in,1,3) * MAT(out,1,1));
+      MAT(out,2,3) = - (MAT(in,2,3) * MAT(out,2,2));
+   }
+
+   return GL_TRUE;
+}
+
+/**
+ * Compute inverse of a no-rotation 2d transformation matrix.
+ * 
+ * \param mat pointer to a GLmatrix structure. The matrix inverse will be
+ * stored in the GLmatrix::inv attribute.
+ * 
+ * \return GL_TRUE for success, GL_FALSE for failure (\p singular matrix).
+ *
+ * Calculates the inverse matrix by applying the inverse scaling and
+ * translation to the identity matrix.
+ */
+static GLboolean invert_matrix_2d_no_rot( GLmatrix *mat )
+{
+   const GLfloat *in = mat->m;
+   GLfloat *out = mat->inv;
+
+   if (MAT(in,0,0) == 0 || MAT(in,1,1) == 0)
+      return GL_FALSE;
+
+   MEMCPY( out, Identity, 16 * sizeof(GLfloat) );
+   MAT(out,0,0) = 1.0F / MAT(in,0,0);
+   MAT(out,1,1) = 1.0F / MAT(in,1,1);
+
+   if (mat->flags & MAT_FLAG_TRANSLATION) {
+      MAT(out,0,3) = - (MAT(in,0,3) * MAT(out,0,0));
+      MAT(out,1,3) = - (MAT(in,1,3) * MAT(out,1,1));
+   }
+
+   return GL_TRUE;
+}
+
+#if 0
+/* broken */
+static GLboolean invert_matrix_perspective( GLmatrix *mat )
+{
+   const GLfloat *in = mat->m;
+   GLfloat *out = mat->inv;
+
+   if (MAT(in,2,3) == 0)
+      return GL_FALSE;
+
+   MEMCPY( out, Identity, 16 * sizeof(GLfloat) );
+
+   MAT(out,0,0) = 1.0F / MAT(in,0,0);
+   MAT(out,1,1) = 1.0F / MAT(in,1,1);
+
+   MAT(out,0,3) = MAT(in,0,2);
+   MAT(out,1,3) = MAT(in,1,2);
+
+   MAT(out,2,2) = 0;
+   MAT(out,2,3) = -1;
+
+   MAT(out,3,2) = 1.0F / MAT(in,2,3);
+   MAT(out,3,3) = MAT(in,2,2) * MAT(out,3,2);
+
+   return GL_TRUE;
+}
+#endif
+
+/**
+ * Matrix inversion function pointer type.
+ */
+typedef GLboolean (*inv_mat_func)( GLmatrix *mat );
+
+/**
+ * Table of the matrix inversion functions according to the matrix type.
+ */
+static inv_mat_func inv_mat_tab[7] = {
+   invert_matrix_general,
+   invert_matrix_identity,
+   invert_matrix_3d_no_rot,
+#if 0
+   /* Don't use this function for now - it fails when the projection matrix
+    * is premultiplied by a translation (ala Chromium's tilesort SPU).
+    */
+   invert_matrix_perspective,
+#else
+   invert_matrix_general,
+#endif
+   invert_matrix_3d,		/* lazy! */
+   invert_matrix_2d_no_rot,
+   invert_matrix_3d
+};
+
+/**
+ * Compute inverse of a transformation matrix.
+ * 
+ * \param mat pointer to a GLmatrix structure. The matrix inverse will be
+ * stored in the GLmatrix::inv attribute.
+ * 
+ * \return GL_TRUE for success, GL_FALSE for failure (\p singular matrix).
+ *
+ * Calls the matrix inversion function in inv_mat_tab corresponding to the
+ * given matrix type.  In case of failure, updates the MAT_FLAG_SINGULAR flag,
+ * and copies the identity matrix into GLmatrix::inv.
+ */
+static GLboolean matrix_invert( GLmatrix *mat )
+{
+   if (inv_mat_tab[mat->type](mat)) {
+      mat->flags &= ~MAT_FLAG_SINGULAR;
+      return GL_TRUE;
+   } else {
+      mat->flags |= MAT_FLAG_SINGULAR;
+      MEMCPY( mat->inv, Identity, sizeof(Identity) );
+      return GL_FALSE;
+   }
+}
+
+/*@}*/
+
+
+/**********************************************************************/
+/** \name Matrix generation */
+/*@{*/
+
+/**
+ * Generate a 4x4 transformation matrix from glRotate parameters, and
+ * post-multiply the input matrix by it.
+ *
+ * \author
+ * This function was contributed by Erich Boleyn (erich@uruk.org).
+ * Optimizations contributed by Rudolf Opalla (rudi@khm.de).
+ */
+void
+_math_matrix_rotate( GLmatrix *mat,
+		     GLfloat angle, GLfloat x, GLfloat y, GLfloat z )
+{
+   GLfloat xx, yy, zz, xy, yz, zx, xs, ys, zs, one_c, s, c;
+   GLfloat m[16];
+   GLboolean optimized;
+
+   s = (GLfloat) _mesa_sin( angle * DEG2RAD );
+   c = (GLfloat) _mesa_cos( angle * DEG2RAD );
+
+   MEMCPY(m, Identity, sizeof(GLfloat)*16);
+   optimized = GL_FALSE;
+
+#define M(row,col)  m[col*4+row]
+
+   if (x == 0.0F) {
+      if (y == 0.0F) {
+         if (z != 0.0F) {
+            optimized = GL_TRUE;
+            /* rotate only around z-axis */
+            M(0,0) = c;
+            M(1,1) = c;
+            if (z < 0.0F) {
+               M(0,1) = s;
+               M(1,0) = -s;
+            }
+            else {
+               M(0,1) = -s;
+               M(1,0) = s;
+            }
+         }
+      }
+      else if (z == 0.0F) {
+         optimized = GL_TRUE;
+         /* rotate only around y-axis */
+         M(0,0) = c;
+         M(2,2) = c;
+         if (y < 0.0F) {
+            M(0,2) = -s;
+            M(2,0) = s;
+         }
+         else {
+            M(0,2) = s;
+            M(2,0) = -s;
+         }
+      }
+   }
+   else if (y == 0.0F) {
+      if (z == 0.0F) {
+         optimized = GL_TRUE;
+         /* rotate only around x-axis */
+         M(1,1) = c;
+         M(2,2) = c;
+         if (x < 0.0F) {
+            M(1,2) = s;
+            M(2,1) = -s;
+         }
+         else {
+            M(1,2) = -s;
+            M(2,1) = s;
+         }
+      }
+   }
+
+   if (!optimized) {
+      const GLfloat mag = SQRTF(x * x + y * y + z * z);
+
+      if (mag <= 1.0e-4) {
+         /* no rotation, leave mat as-is */
+         return;
+      }
+
+      x /= mag;
+      y /= mag;
+      z /= mag;
+
+
+      /*
+       *     Arbitrary axis rotation matrix.
+       *
+       *  This is composed of 5 matrices, Rz, Ry, T, Ry', Rz', multiplied
+       *  like so:  Rz * Ry * T * Ry' * Rz'.  T is the final rotation
+       *  (which is about the X-axis), and the two composite transforms
+       *  Ry' * Rz' and Rz * Ry are (respectively) the rotations necessary
+       *  from the arbitrary axis to the X-axis then back.  They are
+       *  all elementary rotations.
+       *
+       *  Rz' is a rotation about the Z-axis, to bring the axis vector
+       *  into the x-z plane.  Then Ry' is applied, rotating about the
+       *  Y-axis to bring the axis vector parallel with the X-axis.  The
+       *  rotation about the X-axis is then performed.  Ry and Rz are
+       *  simply the respective inverse transforms to bring the arbitrary
+       *  axis back to it's original orientation.  The first transforms
+       *  Rz' and Ry' are considered inverses, since the data from the
+       *  arbitrary axis gives you info on how to get to it, not how
+       *  to get away from it, and an inverse must be applied.
+       *
+       *  The basic calculation used is to recognize that the arbitrary
+       *  axis vector (x, y, z), since it is of unit length, actually
+       *  represents the sines and cosines of the angles to rotate the
+       *  X-axis to the same orientation, with theta being the angle about
+       *  Z and phi the angle about Y (in the order described above)
+       *  as follows:
+       *
+       *  cos ( theta ) = x / sqrt ( 1 - z^2 )
+       *  sin ( theta ) = y / sqrt ( 1 - z^2 )
+       *
+       *  cos ( phi ) = sqrt ( 1 - z^2 )
+       *  sin ( phi ) = z
+       *
+       *  Note that cos ( phi ) can further be inserted to the above
+       *  formulas:
+       *
+       *  cos ( theta ) = x / cos ( phi )
+       *  sin ( theta ) = y / sin ( phi )
+       *
+       *  ...etc.  Because of those relations and the standard trigonometric
+       *  relations, it is pssible to reduce the transforms down to what
+       *  is used below.  It may be that any primary axis chosen will give the
+       *  same results (modulo a sign convention) using thie method.
+       *
+       *  Particularly nice is to notice that all divisions that might
+       *  have caused trouble when parallel to certain planes or
+       *  axis go away with care paid to reducing the expressions.
+       *  After checking, it does perform correctly under all cases, since
+       *  in all the cases of division where the denominator would have
+       *  been zero, the numerator would have been zero as well, giving
+       *  the expected result.
+       */
+
+      xx = x * x;
+      yy = y * y;
+      zz = z * z;
+      xy = x * y;
+      yz = y * z;
+      zx = z * x;
+      xs = x * s;
+      ys = y * s;
+      zs = z * s;
+      one_c = 1.0F - c;
+
+      /* We already hold the identity-matrix so we can skip some statements */
+      M(0,0) = (one_c * xx) + c;
+      M(0,1) = (one_c * xy) - zs;
+      M(0,2) = (one_c * zx) + ys;
+/*    M(0,3) = 0.0F; */
+
+      M(1,0) = (one_c * xy) + zs;
+      M(1,1) = (one_c * yy) + c;
+      M(1,2) = (one_c * yz) - xs;
+/*    M(1,3) = 0.0F; */
+
+      M(2,0) = (one_c * zx) - ys;
+      M(2,1) = (one_c * yz) + xs;
+      M(2,2) = (one_c * zz) + c;
+/*    M(2,3) = 0.0F; */
+
+/*
+      M(3,0) = 0.0F;
+      M(3,1) = 0.0F;
+      M(3,2) = 0.0F;
+      M(3,3) = 1.0F;
+*/
+   }
+#undef M
+
+   matrix_multf( mat, m, MAT_FLAG_ROTATION );
+}
+
+/**
+ * Apply a perspective projection matrix.
+ *
+ * \param mat matrix to apply the projection.
+ * \param left left clipping plane coordinate.
+ * \param right right clipping plane coordinate.
+ * \param bottom bottom clipping plane coordinate.
+ * \param top top clipping plane coordinate.
+ * \param nearval distance to the near clipping plane.
+ * \param farval distance to the far clipping plane.
+ *
+ * Creates the projection matrix and multiplies it with \p mat, marking the
+ * MAT_FLAG_PERSPECTIVE flag.
+ */
+void
+_math_matrix_frustum( GLmatrix *mat,
+		      GLfloat left, GLfloat right,
+		      GLfloat bottom, GLfloat top,
+		      GLfloat nearval, GLfloat farval )
+{
+   GLfloat x, y, a, b, c, d;
+   GLfloat m[16];
+
+   x = (2.0F*nearval) / (right-left);
+   y = (2.0F*nearval) / (top-bottom);
+   a = (right+left) / (right-left);
+   b = (top+bottom) / (top-bottom);
+   c = -(farval+nearval) / ( farval-nearval);
+   d = -(2.0F*farval*nearval) / (farval-nearval);  /* error? */
+
+#define M(row,col)  m[col*4+row]
+   M(0,0) = x;     M(0,1) = 0.0F;  M(0,2) = a;      M(0,3) = 0.0F;
+   M(1,0) = 0.0F;  M(1,1) = y;     M(1,2) = b;      M(1,3) = 0.0F;
+   M(2,0) = 0.0F;  M(2,1) = 0.0F;  M(2,2) = c;      M(2,3) = d;
+   M(3,0) = 0.0F;  M(3,1) = 0.0F;  M(3,2) = -1.0F;  M(3,3) = 0.0F;
+#undef M
+
+   matrix_multf( mat, m, MAT_FLAG_PERSPECTIVE );
+}
+
+/**
+ * Apply an orthographic projection matrix.
+ *
+ * \param mat matrix to apply the projection.
+ * \param left left clipping plane coordinate.
+ * \param right right clipping plane coordinate.
+ * \param bottom bottom clipping plane coordinate.
+ * \param top top clipping plane coordinate.
+ * \param nearval distance to the near clipping plane.
+ * \param farval distance to the far clipping plane.
+ *
+ * Creates the projection matrix and multiplies it with \p mat, marking the
+ * MAT_FLAG_GENERAL_SCALE and MAT_FLAG_TRANSLATION flags.
+ */
+void
+_math_matrix_ortho( GLmatrix *mat,
+		    GLfloat left, GLfloat right,
+		    GLfloat bottom, GLfloat top,
+		    GLfloat nearval, GLfloat farval )
+{
+   GLfloat m[16];
+
+#define M(row,col)  m[col*4+row]
+   M(0,0) = 2.0F / (right-left);
+   M(0,1) = 0.0F;
+   M(0,2) = 0.0F;
+   M(0,3) = -(right+left) / (right-left);
+
+   M(1,0) = 0.0F;
+   M(1,1) = 2.0F / (top-bottom);
+   M(1,2) = 0.0F;
+   M(1,3) = -(top+bottom) / (top-bottom);
+
+   M(2,0) = 0.0F;
+   M(2,1) = 0.0F;
+   M(2,2) = -2.0F / (farval-nearval);
+   M(2,3) = -(farval+nearval) / (farval-nearval);
+
+   M(3,0) = 0.0F;
+   M(3,1) = 0.0F;
+   M(3,2) = 0.0F;
+   M(3,3) = 1.0F;
+#undef M
+
+   matrix_multf( mat, m, (MAT_FLAG_GENERAL_SCALE|MAT_FLAG_TRANSLATION));
+}
+
+/**
+ * Multiply a matrix with a general scaling matrix.
+ *
+ * \param mat matrix.
+ * \param x x axis scale factor.
+ * \param y y axis scale factor.
+ * \param z z axis scale factor.
+ *
+ * Multiplies in-place the elements of \p mat by the scale factors. Checks if
+ * the scales factors are roughly the same, marking the MAT_FLAG_UNIFORM_SCALE
+ * flag, or MAT_FLAG_GENERAL_SCALE. Marks the MAT_DIRTY_TYPE and
+ * MAT_DIRTY_INVERSE dirty flags.
+ */
+void
+_math_matrix_scale( GLmatrix *mat, GLfloat x, GLfloat y, GLfloat z )
+{
+   GLfloat *m = mat->m;
+   m[0] *= x;   m[4] *= y;   m[8]  *= z;
+   m[1] *= x;   m[5] *= y;   m[9]  *= z;
+   m[2] *= x;   m[6] *= y;   m[10] *= z;
+   m[3] *= x;   m[7] *= y;   m[11] *= z;
+
+   if (FABSF(x - y) < 1e-8 && FABSF(x - z) < 1e-8)
+      mat->flags |= MAT_FLAG_UNIFORM_SCALE;
+   else
+      mat->flags |= MAT_FLAG_GENERAL_SCALE;
+
+   mat->flags |= (MAT_DIRTY_TYPE |
+		  MAT_DIRTY_INVERSE);
+}
+
+/**
+ * Multiply a matrix with a translation matrix.
+ *
+ * \param mat matrix.
+ * \param x translation vector x coordinate.
+ * \param y translation vector y coordinate.
+ * \param z translation vector z coordinate.
+ *
+ * Adds the translation coordinates to the elements of \p mat in-place.  Marks
+ * the MAT_FLAG_TRANSLATION flag, and the MAT_DIRTY_TYPE and MAT_DIRTY_INVERSE
+ * dirty flags.
+ */
+void
+_math_matrix_translate( GLmatrix *mat, GLfloat x, GLfloat y, GLfloat z )
+{
+   GLfloat *m = mat->m;
+   m[12] = m[0] * x + m[4] * y + m[8]  * z + m[12];
+   m[13] = m[1] * x + m[5] * y + m[9]  * z + m[13];
+   m[14] = m[2] * x + m[6] * y + m[10] * z + m[14];
+   m[15] = m[3] * x + m[7] * y + m[11] * z + m[15];
+
+   mat->flags |= (MAT_FLAG_TRANSLATION |
+		  MAT_DIRTY_TYPE |
+		  MAT_DIRTY_INVERSE);
+}
+
+
+/**
+ * Set matrix to do viewport and depthrange mapping.
+ * Transforms Normalized Device Coords to window/Z values.
+ */
+void
+_math_matrix_viewport(GLmatrix *m, GLint x, GLint y, GLint width, GLint height,
+                      GLfloat zNear, GLfloat zFar, GLfloat depthMax)
+{
+   m->m[MAT_SX] = (GLfloat) width / 2.0F;
+   m->m[MAT_TX] = m->m[MAT_SX] + x;
+   m->m[MAT_SY] = (GLfloat) height / 2.0F;
+   m->m[MAT_TY] = m->m[MAT_SY] + y;
+   m->m[MAT_SZ] = depthMax * ((zFar - zNear) / 2.0F);
+   m->m[MAT_TZ] = depthMax * ((zFar - zNear) / 2.0F + zNear);
+   m->flags = MAT_FLAG_GENERAL_SCALE | MAT_FLAG_TRANSLATION;
+   m->type = MATRIX_3D_NO_ROT;
+}
+
+
+/**
+ * Set a matrix to the identity matrix.
+ *
+ * \param mat matrix.
+ *
+ * Copies ::Identity into \p GLmatrix::m, and into GLmatrix::inv if not NULL.
+ * Sets the matrix type to identity, and clear the dirty flags.
+ */
+void
+_math_matrix_set_identity( GLmatrix *mat )
+{
+   MEMCPY( mat->m, Identity, 16*sizeof(GLfloat) );
+
+   if (mat->inv)
+      MEMCPY( mat->inv, Identity, 16*sizeof(GLfloat) );
+
+   mat->type = MATRIX_IDENTITY;
+   mat->flags &= ~(MAT_DIRTY_FLAGS|
+		   MAT_DIRTY_TYPE|
+		   MAT_DIRTY_INVERSE);
+}
+
+/*@}*/
+
+
+/**********************************************************************/
+/** \name Matrix analysis */
+/*@{*/
+
+#define ZERO(x) (1<<x)
+#define ONE(x)  (1<<(x+16))
+
+#define MASK_NO_TRX      (ZERO(12) | ZERO(13) | ZERO(14))
+#define MASK_NO_2D_SCALE ( ONE(0)  | ONE(5))
+
+#define MASK_IDENTITY    ( ONE(0)  | ZERO(4)  | ZERO(8)  | ZERO(12) |\
+			  ZERO(1)  |  ONE(5)  | ZERO(9)  | ZERO(13) |\
+			  ZERO(2)  | ZERO(6)  |  ONE(10) | ZERO(14) |\
+			  ZERO(3)  | ZERO(7)  | ZERO(11) |  ONE(15) )
+
+#define MASK_2D_NO_ROT   (           ZERO(4)  | ZERO(8)  |           \
+			  ZERO(1)  |            ZERO(9)  |           \
+			  ZERO(2)  | ZERO(6)  |  ONE(10) | ZERO(14) |\
+			  ZERO(3)  | ZERO(7)  | ZERO(11) |  ONE(15) )
+
+#define MASK_2D          (                      ZERO(8)  |           \
+			                        ZERO(9)  |           \
+			  ZERO(2)  | ZERO(6)  |  ONE(10) | ZERO(14) |\
+			  ZERO(3)  | ZERO(7)  | ZERO(11) |  ONE(15) )
+
+
+#define MASK_3D_NO_ROT   (           ZERO(4)  | ZERO(8)  |           \
+			  ZERO(1)  |            ZERO(9)  |           \
+			  ZERO(2)  | ZERO(6)  |                      \
+			  ZERO(3)  | ZERO(7)  | ZERO(11) |  ONE(15) )
+
+#define MASK_3D          (                                           \
+			                                             \
+			                                             \
+			  ZERO(3)  | ZERO(7)  | ZERO(11) |  ONE(15) )
+
+
+#define MASK_PERSPECTIVE (           ZERO(4)  |            ZERO(12) |\
+			  ZERO(1)  |                       ZERO(13) |\
+			  ZERO(2)  | ZERO(6)  |                      \
+			  ZERO(3)  | ZERO(7)  |            ZERO(15) )
+
+#define SQ(x) ((x)*(x))
+
+/**
+ * Determine type and flags from scratch.  
+ *
+ * \param mat matrix.
+ * 
+ * This is expensive enough to only want to do it once.
+ */
+static void analyse_from_scratch( GLmatrix *mat )
+{
+   const GLfloat *m = mat->m;
+   GLuint mask = 0;
+   GLuint i;
+
+   for (i = 0 ; i < 16 ; i++) {
+      if (m[i] == 0.0) mask |= (1<<i);
+   }
+
+   if (m[0] == 1.0F) mask |= (1<<16);
+   if (m[5] == 1.0F) mask |= (1<<21);
+   if (m[10] == 1.0F) mask |= (1<<26);
+   if (m[15] == 1.0F) mask |= (1<<31);
+
+   mat->flags &= ~MAT_FLAGS_GEOMETRY;
+
+   /* Check for translation - no-one really cares
+    */
+   if ((mask & MASK_NO_TRX) != MASK_NO_TRX)
+      mat->flags |= MAT_FLAG_TRANSLATION;
+
+   /* Do the real work
+    */
+   if (mask == (GLuint) MASK_IDENTITY) {
+      mat->type = MATRIX_IDENTITY;
+   }
+   else if ((mask & MASK_2D_NO_ROT) == (GLuint) MASK_2D_NO_ROT) {
+      mat->type = MATRIX_2D_NO_ROT;
+
+      if ((mask & MASK_NO_2D_SCALE) != MASK_NO_2D_SCALE)
+	 mat->flags |= MAT_FLAG_GENERAL_SCALE;
+   }
+   else if ((mask & MASK_2D) == (GLuint) MASK_2D) {
+      GLfloat mm = DOT2(m, m);
+      GLfloat m4m4 = DOT2(m+4,m+4);
+      GLfloat mm4 = DOT2(m,m+4);
+
+      mat->type = MATRIX_2D;
+
+      /* Check for scale */
+      if (SQ(mm-1) > SQ(1e-6) ||
+	  SQ(m4m4-1) > SQ(1e-6))
+	 mat->flags |= MAT_FLAG_GENERAL_SCALE;
+
+      /* Check for rotation */
+      if (SQ(mm4) > SQ(1e-6))
+	 mat->flags |= MAT_FLAG_GENERAL_3D;
+      else
+	 mat->flags |= MAT_FLAG_ROTATION;
+
+   }
+   else if ((mask & MASK_3D_NO_ROT) == (GLuint) MASK_3D_NO_ROT) {
+      mat->type = MATRIX_3D_NO_ROT;
+
+      /* Check for scale */
+      if (SQ(m[0]-m[5]) < SQ(1e-6) &&
+	  SQ(m[0]-m[10]) < SQ(1e-6)) {
+	 if (SQ(m[0]-1.0) > SQ(1e-6)) {
+	    mat->flags |= MAT_FLAG_UNIFORM_SCALE;
+         }
+      }
+      else {
+	 mat->flags |= MAT_FLAG_GENERAL_SCALE;
+      }
+   }
+   else if ((mask & MASK_3D) == (GLuint) MASK_3D) {
+      GLfloat c1 = DOT3(m,m);
+      GLfloat c2 = DOT3(m+4,m+4);
+      GLfloat c3 = DOT3(m+8,m+8);
+      GLfloat d1 = DOT3(m, m+4);
+      GLfloat cp[3];
+
+      mat->type = MATRIX_3D;
+
+      /* Check for scale */
+      if (SQ(c1-c2) < SQ(1e-6) && SQ(c1-c3) < SQ(1e-6)) {
+	 if (SQ(c1-1.0) > SQ(1e-6))
+	    mat->flags |= MAT_FLAG_UNIFORM_SCALE;
+	 /* else no scale at all */
+      }
+      else {
+	 mat->flags |= MAT_FLAG_GENERAL_SCALE;
+      }
+
+      /* Check for rotation */
+      if (SQ(d1) < SQ(1e-6)) {
+	 CROSS3( cp, m, m+4 );
+	 SUB_3V( cp, cp, (m+8) );
+	 if (LEN_SQUARED_3FV(cp) < SQ(1e-6))
+	    mat->flags |= MAT_FLAG_ROTATION;
+	 else
+	    mat->flags |= MAT_FLAG_GENERAL_3D;
+      }
+      else {
+	 mat->flags |= MAT_FLAG_GENERAL_3D; /* shear, etc */
+      }
+   }
+   else if ((mask & MASK_PERSPECTIVE) == MASK_PERSPECTIVE && m[11]==-1.0F) {
+      mat->type = MATRIX_PERSPECTIVE;
+      mat->flags |= MAT_FLAG_GENERAL;
+   }
+   else {
+      mat->type = MATRIX_GENERAL;
+      mat->flags |= MAT_FLAG_GENERAL;
+   }
+}
+
+/**
+ * Analyze a matrix given that its flags are accurate.
+ * 
+ * This is the more common operation, hopefully.
+ */
+static void analyse_from_flags( GLmatrix *mat )
+{
+   const GLfloat *m = mat->m;
+
+   if (TEST_MAT_FLAGS(mat, 0)) {
+      mat->type = MATRIX_IDENTITY;
+   }
+   else if (TEST_MAT_FLAGS(mat, (MAT_FLAG_TRANSLATION |
+				 MAT_FLAG_UNIFORM_SCALE |
+				 MAT_FLAG_GENERAL_SCALE))) {
+      if ( m[10]==1.0F && m[14]==0.0F ) {
+	 mat->type = MATRIX_2D_NO_ROT;
+      }
+      else {
+	 mat->type = MATRIX_3D_NO_ROT;
+      }
+   }
+   else if (TEST_MAT_FLAGS(mat, MAT_FLAGS_3D)) {
+      if (                                 m[ 8]==0.0F
+            &&                             m[ 9]==0.0F
+            && m[2]==0.0F && m[6]==0.0F && m[10]==1.0F && m[14]==0.0F) {
+	 mat->type = MATRIX_2D;
+      }
+      else {
+	 mat->type = MATRIX_3D;
+      }
+   }
+   else if (                 m[4]==0.0F                 && m[12]==0.0F
+            && m[1]==0.0F                               && m[13]==0.0F
+            && m[2]==0.0F && m[6]==0.0F
+            && m[3]==0.0F && m[7]==0.0F && m[11]==-1.0F && m[15]==0.0F) {
+      mat->type = MATRIX_PERSPECTIVE;
+   }
+   else {
+      mat->type = MATRIX_GENERAL;
+   }
+}
+
+/**
+ * Analyze and update a matrix.
+ *
+ * \param mat matrix.
+ *
+ * If the matrix type is dirty then calls either analyse_from_scratch() or
+ * analyse_from_flags() to determine its type, according to whether the flags
+ * are dirty or not, respectively. If the matrix has an inverse and it's dirty
+ * then calls matrix_invert(). Finally clears the dirty flags.
+ */
+void
+_math_matrix_analyse( GLmatrix *mat )
+{
+   if (mat->flags & MAT_DIRTY_TYPE) {
+      if (mat->flags & MAT_DIRTY_FLAGS)
+	 analyse_from_scratch( mat );
+      else
+	 analyse_from_flags( mat );
+   }
+
+   if (mat->inv && (mat->flags & MAT_DIRTY_INVERSE)) {
+      matrix_invert( mat );
+      mat->flags &= ~MAT_DIRTY_INVERSE;
+   }
+
+   mat->flags &= ~(MAT_DIRTY_FLAGS | MAT_DIRTY_TYPE);
+}
+
+/*@}*/
+
+
+/**
+ * Test if the given matrix preserves vector lengths.
+ */
+GLboolean
+_math_matrix_is_length_preserving( const GLmatrix *m )
+{
+   return TEST_MAT_FLAGS( m, MAT_FLAGS_LENGTH_PRESERVING);
+}
+
+
+/**
+ * Test if the given matrix does any rotation.
+ * (or perhaps if the upper-left 3x3 is non-identity)
+ */
+GLboolean
+_math_matrix_has_rotation( const GLmatrix *m )
+{
+   if (m->flags & (MAT_FLAG_GENERAL |
+                   MAT_FLAG_ROTATION |
+                   MAT_FLAG_GENERAL_3D |
+                   MAT_FLAG_PERSPECTIVE))
+      return GL_TRUE;
+   else
+      return GL_FALSE;
+}
+
+
+GLboolean
+_math_matrix_is_general_scale( const GLmatrix *m )
+{
+   return (m->flags & MAT_FLAG_GENERAL_SCALE) ? GL_TRUE : GL_FALSE;
+}
+
+
+GLboolean
+_math_matrix_is_dirty( const GLmatrix *m )
+{
+   return (m->flags & MAT_DIRTY) ? GL_TRUE : GL_FALSE;
+}
+
+
+/**********************************************************************/
+/** \name Matrix setup */
+/*@{*/
+
+/**
+ * Copy a matrix.
+ *
+ * \param to destination matrix.
+ * \param from source matrix.
+ *
+ * Copies all fields in GLmatrix, creating an inverse array if necessary.
+ */
+void
+_math_matrix_copy( GLmatrix *to, const GLmatrix *from )
+{
+   MEMCPY( to->m, from->m, sizeof(Identity) );
+   to->flags = from->flags;
+   to->type = from->type;
+
+   if (to->inv != 0) {
+      if (from->inv == 0) {
+	 matrix_invert( to );
+      }
+      else {
+	 MEMCPY(to->inv, from->inv, sizeof(GLfloat)*16);
+      }
+   }
+}
+
+/**
+ * Loads a matrix array into GLmatrix.
+ * 
+ * \param m matrix array.
+ * \param mat matrix.
+ *
+ * Copies \p m into GLmatrix::m and marks the MAT_FLAG_GENERAL and MAT_DIRTY
+ * flags.
+ */
+void
+_math_matrix_loadf( GLmatrix *mat, const GLfloat *m )
+{
+   MEMCPY( mat->m, m, 16*sizeof(GLfloat) );
+   mat->flags = (MAT_FLAG_GENERAL | MAT_DIRTY);
+}
+
+/**
+ * Matrix constructor.
+ *
+ * \param m matrix.
+ *
+ * Initialize the GLmatrix fields.
+ */
+void
+_math_matrix_ctr( GLmatrix *m )
+{
+   m->m = (GLfloat *) ALIGN_MALLOC( 16 * sizeof(GLfloat), 16 );
+   if (m->m)
+      MEMCPY( m->m, Identity, sizeof(Identity) );
+   m->inv = NULL;
+   m->type = MATRIX_IDENTITY;
+   m->flags = 0;
+}
+
+/**
+ * Matrix destructor.
+ *
+ * \param m matrix.
+ *
+ * Frees the data in a GLmatrix.
+ */
+void
+_math_matrix_dtr( GLmatrix *m )
+{
+   if (m->m) {
+      ALIGN_FREE( m->m );
+      m->m = NULL;
+   }
+   if (m->inv) {
+      ALIGN_FREE( m->inv );
+      m->inv = NULL;
+   }
+}
+
+/**
+ * Allocate a matrix inverse.
+ *
+ * \param m matrix.
+ *
+ * Allocates the matrix inverse, GLmatrix::inv, and sets it to Identity.
+ */
+void
+_math_matrix_alloc_inv( GLmatrix *m )
+{
+   if (!m->inv) {
+      m->inv = (GLfloat *) ALIGN_MALLOC( 16 * sizeof(GLfloat), 16 );
+      if (m->inv)
+         MEMCPY( m->inv, Identity, 16 * sizeof(GLfloat) );
+   }
+}
+
+/*@}*/
+
+
+/**********************************************************************/
+/** \name Matrix transpose */
+/*@{*/
+
+/**
+ * Transpose a GLfloat matrix.
+ *
+ * \param to destination array.
+ * \param from source array.
+ */
+void
+_math_transposef( GLfloat to[16], const GLfloat from[16] )
+{
+   to[0] = from[0];
+   to[1] = from[4];
+   to[2] = from[8];
+   to[3] = from[12];
+   to[4] = from[1];
+   to[5] = from[5];
+   to[6] = from[9];
+   to[7] = from[13];
+   to[8] = from[2];
+   to[9] = from[6];
+   to[10] = from[10];
+   to[11] = from[14];
+   to[12] = from[3];
+   to[13] = from[7];
+   to[14] = from[11];
+   to[15] = from[15];
+}
+
+/**
+ * Transpose a GLdouble matrix.
+ *
+ * \param to destination array.
+ * \param from source array.
+ */
+void
+_math_transposed( GLdouble to[16], const GLdouble from[16] )
+{
+   to[0] = from[0];
+   to[1] = from[4];
+   to[2] = from[8];
+   to[3] = from[12];
+   to[4] = from[1];
+   to[5] = from[5];
+   to[6] = from[9];
+   to[7] = from[13];
+   to[8] = from[2];
+   to[9] = from[6];
+   to[10] = from[10];
+   to[11] = from[14];
+   to[12] = from[3];
+   to[13] = from[7];
+   to[14] = from[11];
+   to[15] = from[15];
+}
+
+/**
+ * Transpose a GLdouble matrix and convert to GLfloat.
+ *
+ * \param to destination array.
+ * \param from source array.
+ */
+void
+_math_transposefd( GLfloat to[16], const GLdouble from[16] )
+{
+   to[0] = (GLfloat) from[0];
+   to[1] = (GLfloat) from[4];
+   to[2] = (GLfloat) from[8];
+   to[3] = (GLfloat) from[12];
+   to[4] = (GLfloat) from[1];
+   to[5] = (GLfloat) from[5];
+   to[6] = (GLfloat) from[9];
+   to[7] = (GLfloat) from[13];
+   to[8] = (GLfloat) from[2];
+   to[9] = (GLfloat) from[6];
+   to[10] = (GLfloat) from[10];
+   to[11] = (GLfloat) from[14];
+   to[12] = (GLfloat) from[3];
+   to[13] = (GLfloat) from[7];
+   to[14] = (GLfloat) from[11];
+   to[15] = (GLfloat) from[15];
+}
+
+/*@}*/
+
+
+/**
+ * Transform a 4-element row vector (1x4 matrix) by a 4x4 matrix.  This
+ * function is used for transforming clipping plane equations and spotlight
+ * directions.
+ * Mathematically,  u = v * m.
+ * Input:  v - input vector
+ *         m - transformation matrix
+ * Output:  u - transformed vector
+ */
+void
+_mesa_transform_vector( GLfloat u[4], const GLfloat v[4], const GLfloat m[16] )
+{
+   const GLfloat v0 = v[0], v1 = v[1], v2 = v[2], v3 = v[3];
+#define M(row,col)  m[row + col*4]
+   u[0] = v0 * M(0,0) + v1 * M(1,0) + v2 * M(2,0) + v3 * M(3,0);
+   u[1] = v0 * M(0,1) + v1 * M(1,1) + v2 * M(2,1) + v3 * M(3,1);
+   u[2] = v0 * M(0,2) + v1 * M(1,2) + v2 * M(2,2) + v3 * M(3,2);
+   u[3] = v0 * M(0,3) + v1 * M(1,3) + v2 * M(2,3) + v3 * M(3,3);
+#undef M
+}