xref: /external/mesa3d/test/m_matrix.c

/*
 * 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 <GLES2/gl2.h>
#include <stdio.h>
#include <math.h>
#include <assert.h>
#include <string.h>

#include "../src/mesa/main/macros.h"

#include "m_matrix.h"

#define _mesa_debug(...)
/**
 * \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 )
{
    assert(product != 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 (at) 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 (at) uruk.org).
 * Optimizations contributed by Rudolf Opalla (rudi (at) 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) sinf( angle * (M_PI / 180.0f) );
    c = (GLfloat) cosf( angle * (M_PI / 180.0f) );

    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? */

    if (0)
    {
        c /= farval; // linearize z in vs by gl_Position.z *= gl_Position.w
        d /= farval;
    }

#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));
}

// multiplies mat by a perspective transform matrix
void _math_matrix_perspective(GLmatrix * mat, GLfloat fovy, GLfloat aspect,
                              GLfloat zNear, GLfloat zFar)
{
    GLfloat xmin, xmax, ymin, ymax;

    ymax = zNear * tan(fovy * M_PI / 360.0);
    ymin = -ymax;
    xmin = ymin * aspect;
    xmax = ymax * aspect;

    _math_matrix_frustum(mat, xmin, xmax, ymin, ymax, zNear, zFar);
}

// multiplies mat by a look at matrix
void _math_matrix_lookat(GLmatrix * mat, GLfloat eyex, GLfloat eyey, GLfloat eyez,
          GLfloat centerx, GLfloat centery, GLfloat centerz,
          GLfloat upx, GLfloat upy, GLfloat upz)
{
    GLfloat m[16];
    GLfloat x[3], y[3], z[3];
    GLfloat mag;

    /* Make rotation matrix */

    /* Z vector */
    z[0] = eyex - centerx;
    z[1] = eyey - centery;
    z[2] = eyez - centerz;
    mag = sqrt(z[0] * z[0] + z[1] * z[1] + z[2] * z[2]);
    if (mag) {			/* mpichler, 19950515 */
        z[0] /= mag;
        z[1] /= mag;
        z[2] /= mag;
    }

    /* Y vector */
    y[0] = upx;
    y[1] = upy;
    y[2] = upz;

    /* X vector = Y cross Z */
    x[0] = y[1] * z[2] - y[2] * z[1];
    x[1] = -y[0] * z[2] + y[2] * z[0];
    x[2] = y[0] * z[1] - y[1] * z[0];

    /* Recompute Y = Z cross X */
    y[0] = z[1] * x[2] - z[2] * x[1];
    y[1] = -z[0] * x[2] + z[2] * x[0];
    y[2] = z[0] * x[1] - z[1] * x[0];

    /* mpichler, 19950515 */
    /* cross product gives area of parallelogram, which is < 1.0 for
     * non-perpendicular unit-length vectors; so normalize x, y here
     */

    mag = sqrt(x[0] * x[0] + x[1] * x[1] + x[2] * x[2]);
    if (mag) {
        x[0] /= mag;
        x[1] /= mag;
        x[2] /= mag;
    }

    mag = sqrt(y[0] * y[0] + y[1] * y[1] + y[2] * y[2]);
    if (mag) {
        y[0] /= mag;
        y[1] /= mag;
        y[2] /= mag;
    }

#define M(row,col)  m[col*4+row]
    M(0, 0) = x[0];
    M(0, 1) = x[1];
    M(0, 2) = x[2];
    M(0, 3) = 0.0;
    M(1, 0) = y[0];
    M(1, 1) = y[1];
    M(1, 2) = y[2];
    M(1, 3) = 0.0;
    M(2, 0) = z[0];
    M(2, 1) = z[1];
    M(2, 2) = z[2];
    M(2, 3) = 0.0;
    M(3, 0) = 0.0;
    M(3, 1) = 0.0;
    M(3, 2) = 0.0;
    M(3, 3) = 1.0;
#undef M

    GLfloat translate[16] =
    {
        1, 0, 0, 0,
        0, 1, 0, 0,
        0, 0, 1, 0,
        -eyex, -eyey, -eyez, 1,
    };

    _math_matrix_mul_floats(mat, m);

    _math_matrix_mul_floats(mat, translate);

    /* Translate Eye to Origin */
   // glTranslated(-eyex, -eyey, -eyez);

}

/**
 * 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) {
        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 *) malloc( 16 * sizeof(GLfloat));
        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];
}


/**
 * 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
}