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
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
|
/*************************************************************************
* Copyright (C) 2009-2010 Tavian Barnes <tavianator@gmail.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
* Core geometric types like vectors, matricies, and rays.
*/
#ifndef DIMENSION_GEOMETRY_H
#define DIMENSION_GEOMETRY_H
#include <math.h>
#include <stdbool.h>
/** A vector in 3 dimensions. */
typedef struct dmnsn_vector {
double x; /**< The x component. */
double y; /**< The y component. */
double z; /**< The z component. */
} dmnsn_vector;
/** A standard format string for vectors. */
#define DMNSN_VECTOR_FORMAT "<%g, %g, %g>"
/** The appropriate arguements to printf() a vector. */
#define DMNSN_VECTOR_PRINTF(v) (v).x, (v).y, (v).z
/** A 4x4 affine transformation matrix. */
typedef struct dmnsn_matrix {
double n[4][4]; /**< The matrix elements in row-major order. */
} dmnsn_matrix;
/** A standard format string for matricies. */
#define DMNSN_MATRIX_FORMAT \
"[%g\t%g\t%g\t%g]\n" \
"[%g\t%g\t%g\t%g]\n" \
"[%g\t%g\t%g\t%g]\n" \
"[%g\t%g\t%g\t%g]"
/** The appropriate arguements to printf() a matrix. */
#define DMNSN_MATRIX_PRINTF(m) \
(m).n[0][0], (m).n[0][1], (m).n[0][2], (m).n[0][3], \
(m).n[1][0], (m).n[1][1], (m).n[1][2], (m).n[1][3], \
(m).n[2][0], (m).n[2][1], (m).n[2][2], (m).n[2][3], \
(m).n[3][0], (m).n[3][1], (m).n[3][2], (m).n[3][3]
/** A line, or ray. */
typedef struct dmnsn_line {
dmnsn_vector x0; /**< A point on the line. */
dmnsn_vector n; /**< A normal vector; the direction of the line. */
} dmnsn_line;
/** A standard format string for lines. */
#define DMNSN_LINE_FORMAT "(<%g, %g, %g> + t*<%g, %g, %g>)"
/** The appropriate arguements to printf() a line. */
#define DMNSN_LINE_PRINTF(l) \
DMNSN_VECTOR_PRINTF((l).x0), DMNSN_VECTOR_PRINTF((l).n)
/** An axis-aligned bounding box (AABB). */
typedef struct dmnsn_bounding_box {
dmnsn_vector min; /**< The coordinate-wise minimum extent of the box */
dmnsn_vector max; /**< The coordinate-wise maximum extent of the box */
} dmnsn_bounding_box;
/** A standard format string for bounding boxes. */
#define DMNSN_BOUNDING_BOX_FORMAT "(<%g, %g, %g> ==> <%g, %g, %g>)"
/** The appropriate arguements to printf() a bounding box. */
#define DMNSN_BOUNDING_BOX_PRINTF(box) \
DMNSN_VECTOR_PRINTF((box).min), DMNSN_VECTOR_PRINTF((box).max)
/* Constants */
/** The smallest value considered non-zero by some numerical algorithms */
#define dmnsn_epsilon 1.0e-10
/** The zero vector */
static const dmnsn_vector dmnsn_zero = { 0.0, 0.0, 0.0 };
/** The x vector. */
static const dmnsn_vector dmnsn_x = { 1.0, 0.0, 0.0 };
/** The y vector. */
static const dmnsn_vector dmnsn_y = { 0.0, 1.0, 0.0 };
/** The z vector. */
static const dmnsn_vector dmnsn_z = { 0.0, 0.0, 1.0 };
/* Scalar functions */
/** Find the minimum of two scalars. */
DMNSN_INLINE double
dmnsn_min(double a, double b)
{
return a < b ? a : b;
}
/** Find the maximum of two scalars. */
DMNSN_INLINE double
dmnsn_max(double a, double b)
{
return a > b ? a : b;
}
/** Convert degrees to radians */
DMNSN_INLINE double
dmnsn_radians(double degrees)
{
return degrees*atan(1.0)/45.0;
}
/** Convert radians to degrees */
DMNSN_INLINE double
dmnsn_degrees(double radians)
{
return radians*45.0/atan(1.0);
}
/** Return the sign bit of a scalar. */
DMNSN_INLINE int
dmnsn_signbit(double n)
{
/* Guarantee a 1 or 0 return, to allow testing two signs for equality */
return signbit(n) ? 1 : 0;
}
/* Shorthand for vector/matrix construction */
/** Construct a new vector */
DMNSN_INLINE dmnsn_vector
dmnsn_new_vector(double x, double y, double z)
{
dmnsn_vector v = { x, y, z };
return v;
}
/** Construct a new matrix */
DMNSN_INLINE dmnsn_matrix
dmnsn_new_matrix(double a0, double a1, double a2, double a3,
double b0, double b1, double b2, double b3,
double c0, double c1, double c2, double c3,
double d0, double d1, double d2, double d3)
{
dmnsn_matrix m = { { { a0, a1, a2, a3 },
{ b0, b1, b2, b3 },
{ c0, c1, c2, c3 },
{ d0, d1, d2, d3 } } };
return m;
}
/** Return the identity matrix */
dmnsn_matrix dmnsn_identity_matrix(void);
/**
* A scale transformation.
* @param[in] s A vector with components representing the scaling factor in
* each axis.
* @return The transformation matrix.
*/
dmnsn_matrix dmnsn_scale_matrix(dmnsn_vector s);
/**
* A translation.
* @param[in] d The vector to translate by.
* @return The transformation matrix.
*/
dmnsn_matrix dmnsn_translation_matrix(dmnsn_vector d);
/**
* A left-handed rotation.
* @param[in] theta A vector representing an axis and angle.
* @f$ axis = \vec{\theta}/|\vec{\theta}| @f$,
* @f$ angle = |\vec{\theta}| @f$
* @return The transformation matrix.
*/
dmnsn_matrix dmnsn_rotation_matrix(dmnsn_vector theta);
/**
* Construct a new line.
* @param[in] x0 A point on the line.
* @param[in] n The direction of the line.
* @return The new line.
*/
DMNSN_INLINE dmnsn_line
dmnsn_new_line(dmnsn_vector x0, dmnsn_vector n)
{
dmnsn_line l = { x0, n };
return l;
}
/** Return the bounding box which contains nothing. */
DMNSN_INLINE dmnsn_bounding_box
dmnsn_zero_bounding_box(void)
{
dmnsn_bounding_box box = {
{ INFINITY, INFINITY, INFINITY },
{ -INFINITY, -INFINITY, -INFINITY }
};
return box;
}
/** Return the bounding box which contains everything. */
DMNSN_INLINE dmnsn_bounding_box
dmnsn_infinite_bounding_box(void)
{
dmnsn_bounding_box box = {
{ -INFINITY, -INFINITY, -INFINITY },
{ INFINITY, INFINITY, INFINITY }
};
return box;
}
/* Vector element access */
/** Constants for indexing a vector like an array. */
enum {
DMNSN_X, /**< The x component. */
DMNSN_Y, /**< The y component. */
DMNSN_Z /**< The z component. */
};
/**
* Index a vector like an array.
* @param[in] n The vector to index.
* @param[in] elem Which element to access; one of \ref DMNSN_X, \ref DMNSN_Y,
* or \ref DMNSN_Z.
* @return The requested element.
*/
DMNSN_INLINE double
dmnsn_vector_element(dmnsn_vector n, int elem)
{
switch (elem) {
case DMNSN_X:
return n.x;
case DMNSN_Y:
return n.y;
case DMNSN_Z:
return n.z;
default:
dmnsn_assert(false, "Wrong vector element requested.");
return 0.0;
}
}
/* Vector and matrix arithmetic */
/** Negate a vector */
DMNSN_INLINE dmnsn_vector
dmnsn_vector_negate(dmnsn_vector rhs)
{
/* 3 negations */
dmnsn_vector v = { -rhs.x, -rhs.y, -rhs.z };
return v;
}
/** Add two vectors */
DMNSN_INLINE dmnsn_vector
dmnsn_vector_add(dmnsn_vector lhs, dmnsn_vector rhs)
{
/* 3 additions */
dmnsn_vector v = { lhs.x + rhs.x, lhs.y + rhs.y, lhs.z + rhs.z };
return v;
}
/** Subtract two vectors */
DMNSN_INLINE dmnsn_vector
dmnsn_vector_sub(dmnsn_vector lhs, dmnsn_vector rhs)
{
/* 3 additions */
dmnsn_vector v = { lhs.x - rhs.x, lhs.y - rhs.y, lhs.z - rhs.z };
return v;
}
/** Multiply a vector by a scalar. */
DMNSN_INLINE dmnsn_vector
dmnsn_vector_mul(double lhs, dmnsn_vector rhs)
{
/* 3 multiplications */
dmnsn_vector v = { lhs*rhs.x, lhs*rhs.y, lhs*rhs.z };
return v;
}
/** Divide a vector by a scalar. */
DMNSN_INLINE dmnsn_vector
dmnsn_vector_div(dmnsn_vector lhs, double rhs)
{
/* 3 divisions */
dmnsn_vector v = { lhs.x/rhs, lhs.y/rhs, lhs.z/rhs };
return v;
}
/** Return the dot product of two vectors. */
DMNSN_INLINE double
dmnsn_vector_dot(dmnsn_vector lhs, dmnsn_vector rhs)
{
/* 3 multiplications, 2 additions */
return lhs.x*rhs.x + lhs.y*rhs.y + lhs.z*rhs.z;
}
/** Return the cross product of two vectors. */
DMNSN_INLINE dmnsn_vector
dmnsn_vector_cross(dmnsn_vector lhs, dmnsn_vector rhs)
{
/* 6 multiplications, 3 additions */
dmnsn_vector v = { lhs.y*rhs.z - lhs.z*rhs.y,
lhs.z*rhs.x - lhs.x*rhs.z,
lhs.x*rhs.y - lhs.y*rhs.x };
return v;
}
/** Return the projection of \p u onto \p d. */
DMNSN_INLINE dmnsn_vector
dmnsn_vector_proj(dmnsn_vector u, dmnsn_vector d)
{
/* 1 division, 9 multiplications, 4 additions */
return dmnsn_vector_mul(dmnsn_vector_dot(u, d)/dmnsn_vector_dot(d, d), d);
}
/** Return the magnitude of a vector. */
DMNSN_INLINE double
dmnsn_vector_norm(dmnsn_vector n)
{
/* 1 sqrt, 3 multiplications, 2 additions */
return sqrt(dmnsn_vector_dot(n, n));
}
/** Return the direction of a vector. */
DMNSN_INLINE dmnsn_vector
dmnsn_vector_normalize(dmnsn_vector n)
{
/* 1 sqrt, 3 divisions, 3 multiplications, 2 additions */
return dmnsn_vector_div(n, dmnsn_vector_norm(n));
}
/** Return the component-wise minimum of two vectors. */
DMNSN_INLINE dmnsn_vector
dmnsn_vector_min(dmnsn_vector a, dmnsn_vector b)
{
return dmnsn_new_vector(
dmnsn_min(a.x, b.x),
dmnsn_min(a.y, b.y),
dmnsn_min(a.z, b.z)
);
}
/** Return the component-wise maximum of two vectors. */
DMNSN_INLINE dmnsn_vector
dmnsn_vector_max(dmnsn_vector a, dmnsn_vector b)
{
return dmnsn_new_vector(
dmnsn_max(a.x, b.x),
dmnsn_max(a.y, b.y),
dmnsn_max(a.z, b.z)
);
}
/** Return the angle between two vectors with respect to an axis. */
double dmnsn_vector_axis_angle(dmnsn_vector v1, dmnsn_vector v2,
dmnsn_vector axis);
/** Invert a matrix. */
dmnsn_matrix dmnsn_matrix_inverse(dmnsn_matrix A);
/** Multiply two matricies. */
dmnsn_matrix dmnsn_matrix_mul(dmnsn_matrix lhs, dmnsn_matrix rhs);
/** Transform a vector by a matrix. */
DMNSN_INLINE dmnsn_vector
dmnsn_transform_vector(dmnsn_matrix T, dmnsn_vector v)
{
/* 12 multiplications, 3 divisions, 12 additions */
dmnsn_vector r;
double w;
r.x = T.n[0][0]*v.x + T.n[0][1]*v.y + T.n[0][2]*v.z + T.n[0][3];
r.y = T.n[1][0]*v.x + T.n[1][1]*v.y + T.n[1][2]*v.z + T.n[1][3];
r.z = T.n[2][0]*v.x + T.n[2][1]*v.y + T.n[2][2]*v.z + T.n[2][3];
w = T.n[3][0]*v.x + T.n[3][1]*v.y + T.n[3][2]*v.z + T.n[3][3];
return dmnsn_vector_div(r, w);
}
/** Transform a bounding box by a matrix. */
dmnsn_bounding_box dmnsn_transform_bounding_box(dmnsn_matrix T,
dmnsn_bounding_box box);
/**
* Transform a line by a matrix.
* \f$ n' = T(l.\vec{x_0} + l.\vec{n}) - T(l.\vec{x_0}) \f$,
* \f$ \vec{x_0}' = T(l.\vec{x_0}) \f$
*/
DMNSN_INLINE dmnsn_line
dmnsn_transform_line(dmnsn_matrix T, dmnsn_line l)
{
/* 24 multiplications, 6 divisions, 30 additions */
dmnsn_line ret;
ret.x0 = dmnsn_transform_vector(T, l.x0);
ret.n = dmnsn_vector_sub(
dmnsn_transform_vector(T, dmnsn_vector_add(l.x0, l.n)),
ret.x0
);
return ret;
}
/**
* Return the point at \p t on a line.
* The point is defined by \f$ l.\vec{x_0} + t \cdot l.\vec{n} \f$
*/
DMNSN_INLINE dmnsn_vector
dmnsn_line_point(dmnsn_line l, double t)
{
return dmnsn_vector_add(l.x0, dmnsn_vector_mul(t, l.n));
}
/** Add epsilon*l.n to l.x0, to avoid self-intersections */
DMNSN_INLINE dmnsn_line
dmnsn_line_add_epsilon(dmnsn_line l)
{
return dmnsn_new_line(
dmnsn_vector_add(
l.x0,
dmnsn_vector_mul(1.0e3*dmnsn_epsilon, l.n)
),
l.n
);
}
/** Return whether \p p is within the axis-aligned bounding box */
DMNSN_INLINE bool
dmnsn_bounding_box_contains(dmnsn_bounding_box box, dmnsn_vector p)
{
return (p.x >= box.min.x && p.y >= box.min.y && p.z >= box.min.z)
&& (p.x <= box.max.x && p.y <= box.max.y && p.z <= box.max.z);
}
/** Return whether a bounding box is infinite */
DMNSN_INLINE bool
dmnsn_bounding_box_is_infinite(dmnsn_bounding_box box)
{
return box.min.x == -INFINITY;
}
#endif /* DIMENSION_GEOMETRY_H */
|