summaryrefslogtreecommitdiffstats
path: root/libdimension/math/matrix.c
blob: 25590d8e00f27b5bed45c162c7de191773417574 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
/*************************************************************************
 * Copyright (C) 2009-2014 Tavian Barnes <tavianator@tavianator.com>     *
 *                                                                       *
 * This file is part of The Dimension Library.                           *
 *                                                                       *
 * The Dimension Library is free software; you can redistribute it and/  *
 * or modify it under the terms of the GNU Lesser General Public License *
 * as published by the Free Software Foundation; either version 3 of the *
 * License, or (at your option) any later version.                       *
 *                                                                       *
 * The Dimension Library is distributed in the hope that it will be      *
 * useful, but WITHOUT ANY WARRANTY; without even the implied warranty   *
 * of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU  *
 * Lesser General Public License for more details.                       *
 *                                                                       *
 * You should have received a copy of the GNU Lesser General Public      *
 * License along with this program.  If not, see                         *
 * <http://www.gnu.org/licenses/>.                                       *
 *************************************************************************/

/**
 * @file
 * Matrix function implementations.
 */

#include "internal.h"
#include "dimension/math.h"
#include <math.h>

// Identity matrix
dmnsn_matrix
dmnsn_identity_matrix(void)
{
  return dmnsn_new_matrix(1.0, 0.0, 0.0, 0.0,
                          0.0, 1.0, 0.0, 0.0,
                          0.0, 0.0, 1.0, 0.0);
}

// Scaling matrix
dmnsn_matrix
dmnsn_scale_matrix(dmnsn_vector s)
{
  return dmnsn_new_matrix(s.x, 0.0, 0.0, 0.0,
                          0.0, s.y, 0.0, 0.0,
                          0.0, 0.0, s.z, 0.0);
}

// Translation matrix
dmnsn_matrix
dmnsn_translation_matrix(dmnsn_vector d)
{
  return dmnsn_new_matrix(1.0, 0.0, 0.0, d.x,
                          0.0, 1.0, 0.0, d.y,
                          0.0, 0.0, 1.0, d.z);
}

// Left-handed rotation matrix; theta/|theta| = axis, |theta| = angle
dmnsn_matrix
dmnsn_rotation_matrix(dmnsn_vector theta)
{
  // Two trig calls, 25 multiplications, 13 additions

  double angle = dmnsn_vector_norm(theta);
  if (fabs(angle) < dmnsn_epsilon) {
    return dmnsn_identity_matrix();
  }
  dmnsn_vector axis = dmnsn_vector_div(theta, angle);

  // Shorthand to make dmnsn_new_matrix() call legible

  double s = sin(angle);
  double t = 1.0 - cos(angle);

  double x = axis.x;
  double y = axis.y;
  double z = axis.z;

  return dmnsn_new_matrix(
    1.0 + t*(x*x - 1.0), -z*s + t*x*y,        y*s + t*x*z,         0.0,
    z*s + t*x*y,         1.0 + t*(y*y - 1.0), -x*s + t*y*z,        0.0,
    -y*s + t*x*z,        x*s + t*y*z,         1.0 + t*(z*z - 1.0), 0.0
  );
}

// Find the angle between two vectors with respect to an axis
static double
dmnsn_axis_angle(dmnsn_vector from, dmnsn_vector to, dmnsn_vector axis)
{
  from = dmnsn_vector_sub(from, dmnsn_vector_proj(from, axis));
  to   = dmnsn_vector_sub(to,   dmnsn_vector_proj(to, axis));

  double fromnorm = dmnsn_vector_norm(from);
  double tonorm   = dmnsn_vector_norm(to);
  if (fromnorm < dmnsn_epsilon || tonorm < dmnsn_epsilon) {
    return 0.0;
  }

  from = dmnsn_vector_div(from, fromnorm);
  to   = dmnsn_vector_div(to,   tonorm);

  double angle = acos(dmnsn_vector_dot(from, to));

  if (dmnsn_vector_dot(dmnsn_vector_cross(from, to), axis) > 0.0) {
    return angle;
  } else {
    return -angle;
  }
}

// Alignment matrix
dmnsn_matrix
dmnsn_alignment_matrix(dmnsn_vector from, dmnsn_vector to,
                       dmnsn_vector axis1, dmnsn_vector axis2)
{
  double theta1 = dmnsn_axis_angle(from, to, axis1);
  dmnsn_matrix align1 = dmnsn_rotation_matrix(dmnsn_vector_mul(theta1, axis1));
  from  = dmnsn_transform_direction(align1, from);
  axis2 = dmnsn_transform_direction(align1, axis2);

  double theta2 = dmnsn_axis_angle(from, to, axis2);
  dmnsn_matrix align2 = dmnsn_rotation_matrix(dmnsn_vector_mul(theta2, axis2));

  return dmnsn_matrix_mul(align2, align1);
}

// Matrix inversion helper functions

/// A 2x2 matrix for inversion by partitioning.
typedef struct { double n[2][2]; } dmnsn_matrix2;

/// Construct a 2x2 matrix.
static dmnsn_matrix2 dmnsn_new_matrix2(double a1, double a2,
                                       double b1, double b2);
/// Invert a 2x2 matrix.
static dmnsn_matrix2 dmnsn_matrix2_inverse(dmnsn_matrix2 A);
/// Negate a 2x2 matrix.
static dmnsn_matrix2 dmnsn_matrix2_negate(dmnsn_matrix2 A);
/// Subtract two 2x2 matricies.
static dmnsn_matrix2 dmnsn_matrix2_sub(dmnsn_matrix2 lhs, dmnsn_matrix2 rhs);
/// Add two 2x2 matricies.
static dmnsn_matrix2 dmnsn_matrix2_mul(dmnsn_matrix2 lhs, dmnsn_matrix2 rhs);

/// Invert a matrix with the slower cofactor algorithm, if partitioning failed.
static dmnsn_matrix dmnsn_matrix_inverse_generic(dmnsn_matrix A);
/// Get the [\p row, \p col] cofactor of A.
static double dmnsn_matrix_cofactor(dmnsn_matrix A, size_t row, size_t col);

// Invert a matrix, by partitioning
dmnsn_matrix
dmnsn_matrix_inverse(dmnsn_matrix A)
{
  // Use partitioning to invert a matrix:
  //
  //     [ P Q ] -1
  //     [ R S ]
  //
  //   = [ PP QQ ]
  //     [ RR SS ],
  //
  // with PP = inv(P) - inv(P)*Q*RR,
  //      QQ = -inv(P)*Q*SS,
  //      RR = -SS*R*inv(P), and
  //      SS = inv(S - R*inv(P)*Q).

  // The algorithm uses 2 inversions, 6 multiplications, and 2 subtractions,
  // giving 52 multiplications, 34 additions, and 8 divisions.

  dmnsn_matrix2 P, Q, R, S, Pi, RPi, PiQ, RPiQ, PP, QQ, RR, SS;
  double Pdet = A.n[0][0]*A.n[1][1] - A.n[0][1]*A.n[1][0];

  if (dmnsn_unlikely(fabs(Pdet) < dmnsn_epsilon)) {
    // If P is close to singular, try a more generic algorithm; this is very
    // unlikely, but not impossible, eg.
    //   [ 1 1 0 0 ]
    //   [ 1 1 1 0 ]
    //   [ 0 1 1 0 ]
    //   [ 0 0 0 1 ]
    return dmnsn_matrix_inverse_generic(A);
  }

  // Partition the matrix
  P = dmnsn_new_matrix2(A.n[0][0], A.n[0][1],
                        A.n[1][0], A.n[1][1]);
  Q = dmnsn_new_matrix2(A.n[0][2], A.n[0][3],
                        A.n[1][2], A.n[1][3]);
  R = dmnsn_new_matrix2(A.n[2][0], A.n[2][1],
                        0.0,       0.0);
  S = dmnsn_new_matrix2(A.n[2][2], A.n[2][3],
                        0.0,       1.0);

  // Do this inversion ourselves, since we already have the determinant
  Pi = dmnsn_new_matrix2( P.n[1][1]/Pdet, -P.n[0][1]/Pdet,
                         -P.n[1][0]/Pdet,  P.n[0][0]/Pdet);

  // Calculate R*inv(P), inv(P)*Q, and R*inv(P)*Q
  RPi  = dmnsn_matrix2_mul(R, Pi);
  PiQ  = dmnsn_matrix2_mul(Pi, Q);
  RPiQ = dmnsn_matrix2_mul(R, PiQ);

  // Calculate the partitioned inverse
  SS = dmnsn_matrix2_inverse(dmnsn_matrix2_sub(S, RPiQ));
  RR = dmnsn_matrix2_negate(dmnsn_matrix2_mul(SS, RPi));
  QQ = dmnsn_matrix2_negate(dmnsn_matrix2_mul(PiQ, SS));
  PP = dmnsn_matrix2_sub(Pi, dmnsn_matrix2_mul(PiQ, RR));

  // Reconstruct the matrix
  return dmnsn_new_matrix(PP.n[0][0], PP.n[0][1], QQ.n[0][0], QQ.n[0][1],
                          PP.n[1][0], PP.n[1][1], QQ.n[1][0], QQ.n[1][1],
                          RR.n[0][0], RR.n[0][1], SS.n[0][0], SS.n[0][1]);
}

// For nice shorthand
static dmnsn_matrix2
dmnsn_new_matrix2(double a1, double a2, double b1, double b2)
{
  dmnsn_matrix2 m = { { { a1, a2 },
                        { b1, b2 } } };
  return m;
}

// Invert a 2x2 matrix
static dmnsn_matrix2
dmnsn_matrix2_inverse(dmnsn_matrix2 A)
{
  // 4 divisions, 2 multiplications, 1 addition
  double det = A.n[0][0]*A.n[1][1] - A.n[0][1]*A.n[1][0];
  return dmnsn_new_matrix2( A.n[1][1]/det, -A.n[0][1]/det,
                           -A.n[1][0]/det,  A.n[0][0]/det);
}

// Also basically a shorthand
static dmnsn_matrix2
dmnsn_matrix2_negate(dmnsn_matrix2 A)
{
  return dmnsn_new_matrix2(-A.n[0][0], -A.n[0][1],
                           -A.n[1][0], -A.n[1][1]);
}

// 2x2 matrix subtraction
static dmnsn_matrix2
dmnsn_matrix2_sub(dmnsn_matrix2 lhs, dmnsn_matrix2 rhs)
{
  // 4 additions
  return dmnsn_new_matrix2(
    lhs.n[0][0] - rhs.n[0][0], lhs.n[0][1] - rhs.n[0][1],
    lhs.n[1][0] - rhs.n[1][0], lhs.n[1][1] - rhs.n[1][1]
  );
}

// 2x2 matrix multiplication
static dmnsn_matrix2
dmnsn_matrix2_mul(dmnsn_matrix2 lhs, dmnsn_matrix2 rhs)
{
  // 8 multiplications, 4 additions
  return dmnsn_new_matrix2(
    lhs.n[0][0]*rhs.n[0][0] + lhs.n[0][1]*rhs.n[1][0],
      lhs.n[0][0]*rhs.n[0][1] + lhs.n[0][1]*rhs.n[1][1],
    lhs.n[1][0]*rhs.n[0][0] + lhs.n[1][1]*rhs.n[1][0],
      lhs.n[1][0]*rhs.n[0][1] + lhs.n[1][1]*rhs.n[1][1]
  );
}

// Invert a matrix, if partitioning failed (|P| == 0)
static dmnsn_matrix
dmnsn_matrix_inverse_generic(dmnsn_matrix A)
{
  // For A = [ A'      b ]  A^-1 = [ A'^-1   -(A'^-1)*b ]
  //         [ 0 ... 0 1 ],        [ 0 ... 0      1     ].
  //
  // Invert A' by calculating its adjucate.
  dmnsn_matrix inv;
  double det = 0.0, C;

  // Perform a Laplace expansion along the first row to give us the adjugate's
  // first column and the determinant
  for (size_t j = 0; j < 3; ++j) {
    C = dmnsn_matrix_cofactor(A, 0, j);
    det += A.n[0][j]*C;
    inv.n[j][0] = C;
  }

  // Divide the first column by the determinant
  for (size_t j = 0; j < 3; ++j) {
    inv.n[j][0] /= det;
  }

  // Find the rest of A'
  for (size_t j = 0; j < 3; ++j) {
    for (size_t i = 1; i < 3; ++i) {
      inv.n[j][i] = dmnsn_matrix_cofactor(A, i, j)/det;
    }
    inv.n[j][3] = 0.0;
  }

  // Find the translational component of the inverse
  for (size_t i = 0; i < 3; ++i) {
    for (size_t j = 0; j < 3; ++j) {
      inv.n[i][3] -= inv.n[i][j]*A.n[j][3];
    }
  }

  return inv;
}

// Gives the cofactor at row, col; the determinant of the matrix formed from the
// upper-left 3x3 corner of A by ignoring row `row' and column `col',
// times (-1)^(row + col)
static double
dmnsn_matrix_cofactor(dmnsn_matrix A, size_t row, size_t col)
{
  // 2 multiplications, 1 addition
  double n[4];
  size_t k = 0;
  for (size_t i = 0; i < 3; ++i) {
    for (size_t j = 0; j < 3; ++j) {
      if (i != row && j != col) {
        n[k] = A.n[i][j];
        ++k;
      }
    }
  }

  double C = n[0]*n[3] - n[1]*n[2];
  if ((row + col)%2 == 0) {
    return C;
  } else {
    return -C;
  }
}

// 4x4 matrix multiplication
dmnsn_matrix
dmnsn_matrix_mul(dmnsn_matrix lhs, dmnsn_matrix rhs)
{
  // 36 multiplications, 27 additions
  dmnsn_matrix r;

  r.n[0][0] = lhs.n[0][0]*rhs.n[0][0] + lhs.n[0][1]*rhs.n[1][0] + lhs.n[0][2]*rhs.n[2][0];
  r.n[0][1] = lhs.n[0][0]*rhs.n[0][1] + lhs.n[0][1]*rhs.n[1][1] + lhs.n[0][2]*rhs.n[2][1];
  r.n[0][2] = lhs.n[0][0]*rhs.n[0][2] + lhs.n[0][1]*rhs.n[1][2] + lhs.n[0][2]*rhs.n[2][2];
  r.n[0][3] = lhs.n[0][0]*rhs.n[0][3] + lhs.n[0][1]*rhs.n[1][3] + lhs.n[0][2]*rhs.n[2][3] + lhs.n[0][3];

  r.n[1][0] = lhs.n[1][0]*rhs.n[0][0] + lhs.n[1][1]*rhs.n[1][0] + lhs.n[1][2]*rhs.n[2][0];
  r.n[1][1] = lhs.n[1][0]*rhs.n[0][1] + lhs.n[1][1]*rhs.n[1][1] + lhs.n[1][2]*rhs.n[2][1];
  r.n[1][2] = lhs.n[1][0]*rhs.n[0][2] + lhs.n[1][1]*rhs.n[1][2] + lhs.n[1][2]*rhs.n[2][2];
  r.n[1][3] = lhs.n[1][0]*rhs.n[0][3] + lhs.n[1][1]*rhs.n[1][3] + lhs.n[1][2]*rhs.n[2][3] + lhs.n[1][3];

  r.n[2][0] = lhs.n[2][0]*rhs.n[0][0] + lhs.n[2][1]*rhs.n[1][0] + lhs.n[2][2]*rhs.n[2][0];
  r.n[2][1] = lhs.n[2][0]*rhs.n[0][1] + lhs.n[2][1]*rhs.n[1][1] + lhs.n[2][2]*rhs.n[2][1];
  r.n[2][2] = lhs.n[2][0]*rhs.n[0][2] + lhs.n[2][1]*rhs.n[1][2] + lhs.n[2][2]*rhs.n[2][2];
  r.n[2][3] = lhs.n[2][0]*rhs.n[0][3] + lhs.n[2][1]*rhs.n[1][3] + lhs.n[2][2]*rhs.n[2][3] + lhs.n[2][3];

  return r;
}

// Give an axis-aligned box that contains the given box transformed by `lhs'
dmnsn_aabb
dmnsn_transform_aabb(dmnsn_matrix trans, dmnsn_aabb box)
{
  // Infinite/zero bounding box support
  if (isinf(box.min.x)) {
    return box;
  }

  // Taking the "absolute value" of the matrix saves some min/max calculations
  for (int i = 0; i < 3; ++i) {
    for (int j = 0; j < 3; ++j) {
      trans.n[i][j] = fabs(trans.n[i][j]);
    }
  }

  dmnsn_vector Mt = dmnsn_matrix_column(trans, 3);
  dmnsn_aabb ret = { Mt, Mt };

  dmnsn_vector Mz = dmnsn_matrix_column(trans, 2);
  ret.min = dmnsn_vector_add(ret.min, dmnsn_vector_mul(box.min.z, Mz));
  ret.max = dmnsn_vector_add(ret.max, dmnsn_vector_mul(box.max.z, Mz));

  dmnsn_vector My = dmnsn_matrix_column(trans, 1);
  ret.min = dmnsn_vector_add(ret.min, dmnsn_vector_mul(box.min.y, My));
  ret.max = dmnsn_vector_add(ret.max, dmnsn_vector_mul(box.max.y, My));

  dmnsn_vector Mx = dmnsn_matrix_column(trans, 0);
  ret.min = dmnsn_vector_add(ret.min, dmnsn_vector_mul(box.min.x, Mx));
  ret.max = dmnsn_vector_add(ret.max, dmnsn_vector_mul(box.max.x, Mx));

  return ret;
}