[mutter/gbsneto/graphene-matrix: 14/57] cogl/matrix: Calculate inverse using graphene matrices




commit ea2bb2ab138da92e1f4c74fcf9e411141cde1f96
Author: Georges Basile Stavracas Neto <georges stavracas gmail com>
Date:   Thu Sep 10 14:28:04 2020 -0300

    cogl/matrix: Calculate inverse using graphene matrices
    
    Turns out inverting a matrix was the largest chunk of the CoglMatrix
    code. By switching to Graphene, a lot of it can go away. The inverse
    is still cached in the CoglMatrix struct itself, to preserve the
    optimization.
    
    However, switching to graphene_matrix_t to calculate the inverse has
    a challenge: float precision. We had to work around it here, and it
    needs an explanation.
    
    The way to detect whether a matrix is invertible or not (i.e.
    whether it's not a "singular" matrix, or not) is by checking
    if the determinant equals 0. So far, so good.
    
    Both graphene_matrix_t and CoglMatrix use single-precision
    floats to store their 4x4 matrices. Graphene uses vectorized
    operations to optimize determinant calculation, while Cogl
    tries to keep track of the matrix type and has special-purpose
    determinant functions for different matrix types (the most
    common one being a 3D matrix).
    
    Cogl, however, has a fundamentally flawed check for whether
    the matrix is invertible or not. Have a look:
    
    ```
    float det;
    
    …
    
    if (det*det < 1e-25)
       return FALSE;
    ```
    
    Notice that 1e-25 is *way* smaller than FLT_EPSILON. This
    check is fundamentally flawed.
    
    "In practice, what does it break?", the reader might ask.
    Well, in this case, the answer is opposite of that: Cogl
    inverts matrices that should not be invertible. Let's see
    an example: the model-view-projection of a 4K monitor. It
    looks like this:
    
    ```
    | +0,002693 +0,000000 +0,000000 +0,000000 |
    | +0,000000 -0,002693 +0,000000 +0,000000 |
    | +0,000000 +0,000000 +0,002693 +0,000000 |
    | -5,169809 +2,908017 -5,036834 +1,000000 |
    ```
    
    The determinant of this matrix is -0.000000019530306557.
    It evidently is smaller than FLT_EPSILON. In this situation,
    Cogl would happily calculate the inverse matrix, whereas
    Graphene (correctly) bails out and thinks it's a singular
    matrix.
    
    This commit works around that by exploiting the maths around
    it. The basis of it is:
    
      inverse(scalar * M) = (1/scalar) * M'
    
    which can be extrapolated to:
    
      inverse(M) = scalar * inverse(scalar * M) = M'
    
    In other words, scaling the to-be-inversed matrix, then
    scaling the inverse matrix by the same factor, gives us
    the desired inverse.
    
    I'm sorry for everyone that has to read through this :(
    
    https://gitlab.gnome.org/GNOME/mutter/-/merge_requests/1439

 cogl/cogl/cogl-matrix.c | 728 ++----------------------------------------------
 1 file changed, 26 insertions(+), 702 deletions(-)
---
diff --git a/cogl/cogl/cogl-matrix.c b/cogl/cogl/cogl-matrix.c
index 914bda0d87..f9687256e1 100644
--- a/cogl/cogl/cogl-matrix.c
+++ b/cogl/cogl/cogl-matrix.c
@@ -112,28 +112,6 @@ enum CoglMatrixType {
    COGL_MATRIX_N_TYPES
 } ;
 
-#define DEG2RAD (G_PI/180.0)
-
-/* Dot product of two 2-element vectors */
-#define DOT2(A,B)  ( (A)[0]*(B)[0] + (A)[1]*(B)[1] )
-
-/* Dot product of two 3-element vectors */
-#define DOT3(A,B)  ( (A)[0]*(B)[0] + (A)[1]*(B)[1] + (A)[2]*(B)[2] )
-
-#define CROSS3(N, U, V) \
-do { \
-    (N)[0] = (U)[1]*(V)[2] - (U)[2]*(V)[1]; \
-    (N)[1] = (U)[2]*(V)[0] - (U)[0]*(V)[2]; \
-    (N)[2] = (U)[0]*(V)[1] - (U)[1]*(V)[0]; \
-} while (0)
-
-#define SUB_3V(DST, SRCA, SRCB) \
-do { \
-    (DST)[0] = (SRCA)[0] - (SRCB)[0]; \
-    (DST)[1] = (SRCA)[1] - (SRCB)[1]; \
-    (DST)[2] = (SRCA)[2] - (SRCB)[2]; \
-} while (0)
-
 #define LEN_SQUARED_3FV( V ) ((V)[0]*(V)[0]+(V)[1]*(V)[1]+(V)[2]*(V)[2])
 
 /*
@@ -338,714 +316,60 @@ cogl_debug_matrix_print (const CoglMatrix *matrix)
 #define MAT(m,r,c) (m)[(c)*4+(r)]
 
 /*
- * Swaps the values of two floating pointer variables.
- *
- * Used by invert_matrix_general() to swap the row pointers.
- */
-#define SWAP_ROWS(a, b) { float *_tmp = a; (a)=(b); (b)=_tmp; }
-
-/*
- * Compute inverse of 4x4 transformation matrix.
- *
- * @mat pointer to a CoglMatrix structure. The matrix inverse will be
- * stored in the CoglMatrix::inv attribute.
- *
- * Returns: %TRUE for success, %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 gboolean
-invert_matrix_general (CoglMatrix *matrix)
-{
-  const float *m = (float *)matrix;
-  float *out = matrix->inv;
-  float wtmp[4][8];
-  float m0, m1, m2, m3, s;
-  float *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 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 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 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 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 TRUE;
-}
-#undef SWAP_ROWS
-
-/*
- * Compute inverse of a general 3d transformation matrix.
- *
- * @mat pointer to a CoglMatrix structure. The matrix inverse will be
- * stored in the CoglMatrix::inv attribute.
- *
- * Returns: %TRUE for success, %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 gboolean
-invert_matrix_3d_general (CoglMatrix *matrix)
-{
-  const float *in = (float *)matrix;
-  float *out = matrix->inv;
-  float pos, neg, t;
-  float 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 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 TRUE;
-}
-
-/*
- * Compute inverse of a 3d transformation matrix.
- *
- * @mat pointer to a CoglMatrix structure. The matrix inverse will be
- * stored in the CoglMatrix::inv attribute.
- *
- * Returns: %TRUE for success, %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 gboolean
-invert_matrix_3d (CoglMatrix *matrix)
-{
-  const float *in = (float *)matrix;
-  float *out = matrix->inv;
-
-  memcpy (out, identity, 16 * sizeof (float));
-
-  if (!TEST_MAT_FLAGS(matrix, MAT_FLAGS_ANGLE_PRESERVING))
-    return invert_matrix_3d_general (matrix);
-
-  if (matrix->flags & MAT_FLAG_UNIFORM_SCALE)
-    {
-      float 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 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 (matrix->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, 16 * sizeof (float));
-      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 TRUE;
-    }
-
-  if (matrix->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 TRUE;
-}
-
-/*
- * Compute inverse of an identity transformation matrix.
- *
- * @mat pointer to a CoglMatrix structure. The matrix inverse will be
- * stored in the CoglMatrix::inv attribute.
- *
- * Returns: always %TRUE.
- *
- * Simply copies identity into CoglMatrix::inv.
- */
-static gboolean
-invert_matrix_identity (CoglMatrix *matrix)
-{
-  memcpy (matrix->inv, identity, 16 * sizeof (float));
-  return TRUE;
-}
-
-/*
- * Compute inverse of a no-rotation 3d transformation matrix.
- *
- * @mat pointer to a CoglMatrix structure. The matrix inverse will be
- * stored in the CoglMatrix::inv attribute.
- *
- * Returns: %TRUE for success, %FALSE for failure (\p singular matrix).
- *
- * Calculates the
- */
-static gboolean
-invert_matrix_3d_no_rotation (CoglMatrix *matrix)
-{
-  const float *in = (float *)matrix;
-  float *out = matrix->inv;
-
-  if (MAT (in,0,0) == 0 || MAT (in,1,1) == 0 || MAT (in,2,2) == 0)
-    return FALSE;
-
-  memcpy (out, identity, 16 * sizeof (float));
-  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 (matrix->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 TRUE;
-}
-
-/*
- * Compute inverse of a no-rotation 2d transformation matrix.
+ * Compute inverse of a transformation matrix.
  *
  * @mat pointer to a CoglMatrix structure. The matrix inverse will be
  * stored in the CoglMatrix::inv attribute.
  *
  * Returns: %TRUE for success, %FALSE for failure (\p singular matrix).
  *
- * Calculates the inverse matrix by applying the inverse scaling and
- * translation to the identity matrix.
- */
-static gboolean
-invert_matrix_2d_no_rotation (CoglMatrix *matrix)
-{
-  const float *in = (float *)matrix;
-  float *out = matrix->inv;
-
-  if (MAT (in, 0, 0) == 0 || MAT (in, 1, 1) == 0)
-    return FALSE;
-
-  memcpy (out, identity, 16 * sizeof (float));
-  MAT (out, 0, 0) = 1.0f / MAT (in, 0, 0);
-  MAT (out, 1, 1) = 1.0f / MAT (in, 1, 1);
-
-  if (matrix->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 TRUE;
-}
-
-#if 0
-/* broken */
-static gboolean
-invert_matrix_perspective (CoglMatrix *matrix)
-{
-  const float *in = matrix;
-  float *out = matrix->inv;
-
-  if (MAT (in,2,3) == 0)
-    return FALSE;
-
-  memcpy( out, identity, 16 * sizeof(float) );
-
-  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 TRUE;
-}
-#endif
-
-/*
- * Matrix inversion function pointer type.
- */
-typedef gboolean (*inv_mat_func)(CoglMatrix *matrix);
-
-/*
- * Table of the matrix inversion functions according to the matrix type.
+ * 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 CoglMatrix::inv.
  */
-static inv_mat_func inv_mat_tab[7] = {
-    invert_matrix_general,
-    invert_matrix_identity,
-    invert_matrix_3d_no_rotation,
-#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_rotation,
-    invert_matrix_3d
-};
-
-#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 SCALE 1000.F
 
-#define SQ(x) ((x)*(x))
-
-/*
- * Determine type and flags from scratch.
- *
- * This is expensive enough to only want to do it once.
- */
-static void
-analyse_from_scratch (CoglMatrix *matrix)
+static inline gboolean
+calculate_inverse (CoglMatrix *matrix)
 {
-  const float *m = (float *)matrix;
-  unsigned int mask = 0;
-  unsigned int 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);
-
-  matrix->flags &= ~MAT_FLAGS_GEOMETRY;
-
-  /* Check for translation - no-one really cares
-  */
-  if ((mask & MASK_NO_TRX) != MASK_NO_TRX)
-    matrix->flags |= MAT_FLAG_TRANSLATION;
-
-  /* Do the real work
-  */
-  if (mask == (unsigned int) MASK_IDENTITY)
-    matrix->type = COGL_MATRIX_TYPE_IDENTITY;
-  else if ((mask & MASK_2D_NO_ROT) == (unsigned int) MASK_2D_NO_ROT)
-    {
-      matrix->type = COGL_MATRIX_TYPE_2D_NO_ROT;
-
-      if ((mask & MASK_NO_2D_SCALE) != MASK_NO_2D_SCALE)
-        matrix->flags |= MAT_FLAG_GENERAL_SCALE;
-    }
-  else if ((mask & MASK_2D) == (unsigned int) MASK_2D)
-    {
-      float mm = DOT2 (m, m);
-      float m4m4 = DOT2 (m+4,m+4);
-      float mm4 = DOT2 (m,m+4);
+  graphene_matrix_t inverse;
+  graphene_matrix_t scaled;
+  graphene_matrix_t m;
+  gboolean invertible;
 
-      matrix->type = COGL_MATRIX_TYPE_2D;
+  graphene_matrix_init_scale (&scaled, SCALE, SCALE, SCALE);
 
-      /* Check for scale */
-      if (SQ (mm-1) > SQ (1e-6) ||
-          SQ (m4m4-1) > SQ (1e-6))
-        matrix->flags |= MAT_FLAG_GENERAL_SCALE;
+  /* Float precision is a limiting factor */
+  cogl_matrix_to_graphene_matrix (matrix, &m);
+  graphene_matrix_multiply (&m, &scaled, &m);
 
-      /* Check for rotation */
-      if (SQ (mm4) > SQ (1e-6))
-        matrix->flags |= MAT_FLAG_GENERAL_3D;
-      else
-        matrix->flags |= MAT_FLAG_ROTATION;
+  invertible = graphene_matrix_inverse (&m, &inverse);
 
-    }
-  else if ((mask & MASK_3D_NO_ROT) == (unsigned int) MASK_3D_NO_ROT)
-    {
-      matrix->type = COGL_MATRIX_TYPE_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))
-            matrix->flags |= MAT_FLAG_UNIFORM_SCALE;
-        }
-      else
-        matrix->flags |= MAT_FLAG_GENERAL_SCALE;
-    }
-  else if ((mask & MASK_3D) == (unsigned int) MASK_3D)
-    {
-      float c1 = DOT3 (m,m);
-      float c2 = DOT3 (m+4,m+4);
-      float c3 = DOT3 (m+8,m+8);
-      float d1 = DOT3 (m, m+4);
-      float cp[3];
-
-      matrix->type = COGL_MATRIX_TYPE_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))
-            matrix->flags |= MAT_FLAG_UNIFORM_SCALE;
-          /* else no scale at all */
-        }
-      else
-        matrix->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))
-            matrix->flags |= MAT_FLAG_ROTATION;
-          else
-            matrix->flags |= MAT_FLAG_GENERAL_3D;
-        }
-      else
-        matrix->flags |= MAT_FLAG_GENERAL_3D; /* shear, etc */
-    }
-  else if ((mask & MASK_PERSPECTIVE) == MASK_PERSPECTIVE && m[11]==-1.0f)
-    {
-      matrix->type = COGL_MATRIX_TYPE_PERSPECTIVE;
-      matrix->flags |= MAT_FLAG_GENERAL;
-    }
+  if (invertible)
+    graphene_matrix_multiply (&scaled, &inverse, &inverse);
   else
-    {
-      matrix->type = COGL_MATRIX_TYPE_GENERAL;
-      matrix->flags |= MAT_FLAG_GENERAL;
-    }
-}
+    graphene_matrix_init_identity (&inverse);
 
-/*
- * Analyze a matrix given that its flags are accurate.
- *
- * This is the more common operation, hopefully.
- */
-static void
-analyse_from_flags (CoglMatrix *matrix)
-{
-  const float *m = (float *)matrix;
+  graphene_matrix_to_float (&inverse, matrix->inv);
 
-  if (TEST_MAT_FLAGS(matrix, 0))
-    matrix->type = COGL_MATRIX_TYPE_IDENTITY;
-  else if (TEST_MAT_FLAGS(matrix, (MAT_FLAG_TRANSLATION |
-                                   MAT_FLAG_UNIFORM_SCALE |
-                                   MAT_FLAG_GENERAL_SCALE)))
-    {
-      if ( m[10] == 1.0f && m[14] == 0.0f )
-        matrix->type = COGL_MATRIX_TYPE_2D_NO_ROT;
-      else
-        matrix->type = COGL_MATRIX_TYPE_3D_NO_ROT;
-    }
-  else if (TEST_MAT_FLAGS (matrix, 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)
-        {
-          matrix->type = COGL_MATRIX_TYPE_2D;
-        }
-      else
-        matrix->type = COGL_MATRIX_TYPE_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)
-    {
-      matrix->type = COGL_MATRIX_TYPE_PERSPECTIVE;
-    }
-  else
-    matrix->type = COGL_MATRIX_TYPE_GENERAL;
-}
-
-/*
- * Analyze and update the type and flags of a 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.
- */
-static void
-_cogl_matrix_update_type_and_flags (CoglMatrix *matrix)
-{
-  if (matrix->flags & MAT_DIRTY_TYPE)
-    {
-      if (matrix->flags & MAT_DIRTY_FLAGS)
-        analyse_from_scratch (matrix);
-      else
-        analyse_from_flags (matrix);
-    }
-
-  matrix->flags &= ~(MAT_DIRTY_FLAGS | MAT_DIRTY_TYPE);
+  return invertible;
 }
 
-/*
- * Compute inverse of a transformation matrix.
- *
- * @mat pointer to a CoglMatrix structure. The matrix inverse will be
- * stored in the CoglMatrix::inv attribute.
- *
- * Returns: %TRUE for success, %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 CoglMatrix::inv.
- */
 static gboolean
 _cogl_matrix_update_inverse (CoglMatrix *matrix)
 {
   if (matrix->flags & MAT_DIRTY_FLAGS ||
       matrix->flags & MAT_DIRTY_INVERSE)
     {
-      _cogl_matrix_update_type_and_flags (matrix);
-
-      if (inv_mat_tab[matrix->type](matrix))
+      if (calculate_inverse (matrix))
         matrix->flags &= ~MAT_FLAG_SINGULAR;
       else
-        {
-          matrix->flags |= MAT_FLAG_SINGULAR;
-          memcpy (matrix->inv, identity, 16 * sizeof (float));
-        }
+        matrix->flags |= MAT_FLAG_SINGULAR;
 
-      matrix->flags &= ~MAT_DIRTY_INVERSE;
+      matrix->flags &= ~(MAT_DIRTY_FLAGS |
+                         MAT_DIRTY_TYPE |
+                         MAT_DIRTY_INVERSE);
     }
 
   if (matrix->flags & MAT_FLAG_SINGULAR)


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