+/* Copyright 1988, Brown Computer Graphics Group. All Rights Reserved. */
+
+/* --------------------------------------------------------------------------
+ * This file contains routines that operate solely on matrices.
+ * -------------------------------------------------------------------------*/
+
+#include "mat3defs.h"
+
+/* -------------------------- Static Routines ---------------------------- */
+
+#define SMALL 1e-20 /* Small enough to be considered zero */
+
+/*
+ * Shuffles rows in inverse of 3x3. See comment in MAT3_inv3_second_col().
+ */
+
+static void
+MAT3_inv3_swap( register double inv[3][3], int row0, int row1, int row2)
+{
+ register int i, tempi;
+ double temp;
+
+#define SWAP_ROWS(a, b) \
+ for (i = 0; i < 3; i++) SWAP(inv[a][i], inv[b][i], temp); \
+ SWAP(a, b, tempi)
+
+ if (row0 != 0){
+ if (row1 == 0) {
+ SWAP_ROWS(row0, row1);
+ }
+ else {
+ SWAP_ROWS(row0, row2);
+ }
+ }
+
+ if (row1 != 1) {
+ SWAP_ROWS(row1, row2);
+ }
+}
+
+/*
+ * Does Gaussian elimination on second column.
+ */
+
+static int
+MAT3_inv3_second_col (register double source[3][3], register double inv[3][3], int row0)
+{
+ register int row1, row2, i1, i2, i;
+ double temp;
+ double a, b;
+
+ /* Find which row to use */
+ if (row0 == 0) i1 = 1, i2 = 2;
+ else if (row0 == 1) i1 = 0, i2 = 2;
+ else i1 = 0, i2 = 1;
+
+ /* Find which is larger in abs. val.:the entry in [i1][1] or [i2][1] */
+ /* and use that value for pivoting. */
+
+ a = source[i1][1]; if (a < 0) a = -a;
+ b = source[i2][1]; if (b < 0) b = -b;
+ if (a > b) row1 = i1;
+ else row1 = i2;
+ row2 = (row1 == i1 ? i2 : i1);
+
+ /* Scale row1 in source */
+ if ((source[row1][1] < SMALL) && (source[row1][1] > -SMALL)) return(FALSE);
+ temp = 1.0 / source[row1][1];
+ source[row1][1] = 1.0;
+ source[row1][2] *= temp; /* source[row1][0] is zero already */
+
+ /* Scale row1 in inv */
+ inv[row1][row1] = temp; /* it used to be a 1.0 */
+ inv[row1][row0] *= temp;
+
+ /* Clear column one, source, and make corresponding changes in inv */
+
+ for (i = 0; i < 3; i++) if (i != row1) { /* for i = all rows but row1 */
+ temp = -source[i][1];
+ source[i][1] = 0.0;
+ source[i][2] += temp * source[row1][2];
+
+ inv[i][row1] = temp * inv[row1][row1];
+ inv[i][row0] += temp * inv[row1][row0];
+ }
+
+ /* Scale row2 in source */
+ if ((source[row2][2] < SMALL) && (source[row2][2] > -SMALL)) return(FALSE);
+ temp = 1.0 / source[row2][2];
+ source[row2][2] = 1.0; /* source[row2][*] is zero already */
+
+ /* Scale row2 in inv */
+ inv[row2][row2] = temp; /* it used to be a 1.0 */
+ inv[row2][row0] *= temp;
+ inv[row2][row1] *= temp;
+
+ /* Clear column one, source, and make corresponding changes in inv */
+ for (i = 0; i < 3; i++) if (i != row2) { /* for i = all rows but row2 */
+ temp = -source[i][2];
+ source[i][2] = 0.0;
+ inv[i][row0] += temp * inv[row2][row0];
+ inv[i][row1] += temp * inv[row2][row1];
+ inv[i][row2] += temp * inv[row2][row2];
+ }
+
+ /*
+ * Now all is done except that the inverse needs to have its rows shuffled.
+ * row0 needs to be moved to inv[0][*], row1 to inv[1][*], etc.
+ *
+ * We *didn't* do the swapping before the elimination so that we could more
+ * easily keep track of what ops are needed to be done in the inverse.
+ */
+ MAT3_inv3_swap(inv, row0, row1, row2);
+
+ return(TRUE);
+}
+
+/*
+ * Fast inversion routine for 3 x 3 matrices. - Written by jfh.
+ *
+ * This takes 30 multiplies/divides, as opposed to 39 for Cramer's Rule.
+ * The algorithm consists of performing fast gaussian elimination, by never
+ * doing any operations where the result is guaranteed to be zero, or where
+ * one operand is guaranteed to be zero. This is done at the cost of clarity,
+ * alas.
+ *
+ * Returns 1 if the inverse was successful, 0 if it failed.
+ */
+
+static int
+MAT3_invert3 (register double source[3][3], register double inv[3][3])
+{
+ register int i, row0;
+ double temp;
+ double a, b, c;
+
+ inv[0][0] = inv[1][1] = inv[2][2] = 1.0;
+ inv[0][1] = inv[0][2] = inv[1][0] = inv[1][2] = inv[2][0] = inv[2][1] = 0.0;
+
+ /* attempt to find the largest entry in first column to use as pivot */
+ a = source[0][0]; if (a < 0) a = -a;
+ b = source[1][0]; if (b < 0) b = -b;
+ c = source[2][0]; if (c < 0) c = -c;
+
+ if (a > b) {
+ if (a > c) row0 = 0;
+ else row0 = 2;
+ }
+ else {
+ if (b > c) row0 = 1;
+ else row0 = 2;
+ }
+
+ /* Scale row0 of source */
+ if ((source[row0][0] < SMALL) && (source[row0][0] > -SMALL)) return(FALSE);
+ temp = 1.0 / source[row0][0];
+ source[row0][0] = 1.0;
+ source[row0][1] *= temp;
+ source[row0][2] *= temp;
+
+ /* Scale row0 of inverse */
+ inv[row0][row0] = temp; /* other entries are zero -- no effort */
+
+ /* Clear column zero of source, and make corresponding changes in inverse */
+
+ for (i = 0; i < 3; i++) if (i != row0) { /* for i = all rows but row0 */
+ temp = -source[i][0];
+ source[i][0] = 0.0;
+ source[i][1] += temp * source[row0][1];
+ source[i][2] += temp * source[row0][2];
+ inv[i][row0] = temp * inv[row0][row0];
+ }
+
+ /*
+ * We've now done gaussian elimination so that the source and
+ * inverse look like this:
+ *
+ * 1 * * * 0 0
+ * 0 * * * 1 0
+ * 0 * * * 0 1
+ *
+ * We now proceed to do elimination on the second column.
+ */
+ if (! MAT3_inv3_second_col(source, inv, row0)) return(FALSE);
+
+ return(TRUE);
+}
+
+/*
+ * Finds a new pivot for a non-simple 4x4. See comments in MAT3invert().
+ */
+
+static int
+MAT3_inv4_pivot (register MAT3mat src, MAT3vec r, double *s, int *swap)
+{
+ register int i, j;
+ double temp, max;
+
+ *swap = -1;
+
+ if (MAT3_IS_ZERO(src[3][3])) {
+
+ /* Look for a different pivot element: one with largest abs value */
+ max = 0.0;
+
+ for (i = 0; i < 4; i++) {
+ if (src[i][3] > max) max = src[*swap = i][3];
+ else if (src[i][3] < -max) max = -src[*swap = i][3];
+ }
+
+ /* No pivot element available ! */
+ if (*swap < 0) return(FALSE);
+
+ else for (j = 0; j < 4; j++) SWAP(src[*swap][j], src[3][j], temp);
+ }
+
+ MAT3_SET_VEC (r, -src[0][3], -src[1][3], -src[2][3]);
+
+ *s = 1.0 / src[3][3];
+
+ src[0][3] = src[1][3] = src[2][3] = 0.0;
+ src[3][3] = 1.0;
+
+ MAT3_SCALE_VEC(src[3], src[3], *s);
+
+ for (i = 0; i < 3; i++) {
+ src[0][i] += r[0] * src[3][i];
+ src[1][i] += r[1] * src[3][i];
+ src[2][i] += r[2] * src[3][i];
+ }
+
+ return(TRUE);
+}
+
+/* ------------------------- Internal Routines --------------------------- */
+
+/* -------------------------- Public Routines ---------------------------- */
+
+/*
+ * This returns the inverse of the given matrix. The result matrix
+ * may be the same as the one to invert.
+ *
+ * Fast inversion routine for 4 x 4 matrices, written by jfh.
+ *
+ * Returns 1 if the inverse was successful, 0 if it failed.
+ *
+ * This routine has been specially tweaked to notice the following:
+ * If the matrix has the form
+ * * * * 0
+ * * * * 0
+ * * * * 0
+ * * * * 1
+ *
+ * (as do many matrices in graphics), then we compute the inverse of
+ * the upper left 3x3 matrix and use this to find the general inverse.
+ *
+ * In the event that the right column is not 0-0-0-1, we do gaussian
+ * elimination to make it so, then use the 3x3 inverse, and then do
+ * our gaussian elimination.
+ */
+
+int
+MAT3invert(result_mat, mat)
+MAT3mat result_mat, mat;
+{
+ MAT3mat src, inv;
+ register int i, j, simple;
+ double m[3][3], inv3[3][3], s, temp;
+ MAT3vec r, t;
+ int swap;
+
+ MAT3copy(src, mat);
+ MAT3identity(inv);
+
+ /* If last column is not (0,0,0,1), use special code */
+ simple = (mat[0][3] == 0.0 && mat[1][3] == 0.0 &&
+ mat[2][3] == 0.0 && mat[3][3] == 1.0);
+
+ if (! simple && ! MAT3_inv4_pivot(src, r, &s, &swap)) return(FALSE);
+
+ MAT3_COPY_VEC(t, src[3]); /* Translation vector */
+
+ /* Copy upper-left 3x3 matrix */
+ for (i = 0; i < 3; i++) for (j = 0; j < 3; j++) m[i][j] = src[i][j];
+
+ if (! MAT3_invert3(m, inv3)) return(FALSE);
+
+ for (i = 0; i < 3; i++) for (j = 0; j < 3; j++) inv[i][j] = inv3[i][j];
+
+ for (i = 0; i < 3; i++) for (j = 0; j < 3; j++)
+ inv[3][i] -= t[j] * inv3[j][i];
+
+ if (! simple) {
+
+ /* We still have to undo our gaussian elimination from earlier on */
+ /* add r0 * first col to last col */
+ /* add r1 * 2nd col to last col */
+ /* add r2 * 3rd col to last col */
+
+ for (i = 0; i < 4; i++) {
+ inv[i][3] += r[0] * inv[i][0] + r[1] * inv[i][1] + r[2] * inv[i][2];
+ inv[i][3] *= s;
+ }
+
+ if (swap >= 0)
+ for (i = 0; i < 4; i++) SWAP(inv[i][swap], inv[i][3], temp);
+ }
+
+ MAT3copy(result_mat, inv);
+
+ return(TRUE);
+}