1 /* Libart_LGPL - library of basic graphic primitives
2 * Copyright (C) 1998 Raph Levien
3 *
4 * This library is free software; you can redistribute it and/or
5 * modify it under the terms of the GNU Library General Public
6 * License as published by the Free Software Foundation; either
7 * version 2 of the License, or (at your option) any later version.
8 *
9 * This library is distributed in the hope that it will be useful,
10 * but WITHOUT ANY WARRANTY; without even the implied warranty of
11 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
12 * Library General Public License for more details.
13 *
14 * You should have received a copy of the GNU Library General Public
15 * License along with this library; if not, write to the
16 * Free Software Foundation, Inc., 59 Temple Place - Suite 330,
17 * Boston, MA 02111-1307, USA.
18 */
19
20 /* Simple manipulations with affine transformations */
21
22 #include "config.h"
23 #include "art_affine.h"
24 #include "art_misc.h" /* for M_PI */
25
26 #include <math.h>
27 #include <stdio.h> /* for sprintf */
28 #include <string.h> /* for strcpy */
29
30
31 /* According to a strict interpretation of the libart structure, this
32 routine should go into its own module, art_point_affine. However,
33 it's only two lines of code, and it can be argued that it is one of
34 the natural basic functions of an affine transformation.
35 */
36
37 /**
38 * art_affine_point: Do an affine transformation of a point.
39 * @dst: Where the result point is stored.
40 * @src: The original point.
41 @ @affine: The affine transformation.
42 **/
43 void
art_affine_point(ArtPoint * dst,const ArtPoint * src,const double affine[6])44 art_affine_point (ArtPoint *dst, const ArtPoint *src,
45 const double affine[6])
46 {
47 double x, y;
48
49 x = src->x;
50 y = src->y;
51 dst->x = x * affine[0] + y * affine[2] + affine[4];
52 dst->y = x * affine[1] + y * affine[3] + affine[5];
53 }
54
55 /**
56 * art_affine_invert: Find the inverse of an affine transformation.
57 * @dst: Where the resulting affine is stored.
58 * @src: The original affine transformation.
59 *
60 * All non-degenerate affine transforms are invertible. If the original
61 * affine is degenerate or nearly so, expect numerical instability and
62 * very likely core dumps on Alpha and other fp-picky architectures.
63 * Otherwise, @dst multiplied with @src, or @src multiplied with @dst
64 * will be (to within roundoff error) the identity affine.
65 **/
66 void
art_affine_invert(double dst[6],const double src[6])67 art_affine_invert (double dst[6], const double src[6])
68 {
69 double r_det;
70
71 r_det = 1.0 / (src[0] * src[3] - src[1] * src[2]);
72 dst[0] = src[3] * r_det;
73 dst[1] = -src[1] * r_det;
74 dst[2] = -src[2] * r_det;
75 dst[3] = src[0] * r_det;
76 dst[4] = -src[4] * dst[0] - src[5] * dst[2];
77 dst[5] = -src[4] * dst[1] - src[5] * dst[3];
78 }
79
80 /**
81 * art_affine_flip: Flip an affine transformation horizontally and/or vertically.
82 * @dst_affine: Where the resulting affine is stored.
83 * @src_affine: The original affine transformation.
84 * @horiz: Whether or not to flip horizontally.
85 * @vert: Whether or not to flip horizontally.
86 *
87 * Flips the affine transform. FALSE for both @horiz and @vert implements
88 * a simple copy operation. TRUE for both @horiz and @vert is a
89 * 180 degree rotation. It is ok for @src_affine and @dst_affine to
90 * be equal pointers.
91 **/
92 void
art_affine_flip(double dst_affine[6],const double src_affine[6],int horz,int vert)93 art_affine_flip (double dst_affine[6], const double src_affine[6], int horz, int vert)
94 {
95 dst_affine[0] = horz ? - src_affine[0] : src_affine[0];
96 dst_affine[1] = horz ? - src_affine[1] : src_affine[1];
97 dst_affine[2] = vert ? - src_affine[2] : src_affine[2];
98 dst_affine[3] = vert ? - src_affine[3] : src_affine[3];
99 dst_affine[4] = horz ? - src_affine[4] : src_affine[4];
100 dst_affine[5] = vert ? - src_affine[5] : src_affine[5];
101 }
102
103 #define EPSILON 1e-6
104
105 /* It's ridiculous I have to write this myself. This is hardcoded to
106 six digits of precision, which is good enough for PostScript.
107
108 The return value is the number of characters (i.e. strlen (str)).
109 It is no more than 12. */
110 static int
art_ftoa(char str[80],double x)111 art_ftoa (char str[80], double x)
112 {
113 char *p = str;
114 int i, j;
115
116 p = str;
117 if (fabs (x) < EPSILON / 2)
118 {
119 strcpy (str, "0");
120 return 1;
121 }
122 if (x < 0)
123 {
124 *p++ = '-';
125 x = -x;
126 }
127 if ((int)floor ((x + EPSILON / 2) < 1))
128 {
129 *p++ = '0';
130 *p++ = '.';
131 i = sprintf (p, "%06d", (int)floor ((x + EPSILON / 2) * 1e6));
132 while (i && p[i - 1] == '0')
133 i--;
134 if (i == 0)
135 i--;
136 p += i;
137 }
138 else if (x < 1e6)
139 {
140 i = sprintf (p, "%d", (int)floor (x + EPSILON / 2));
141 p += i;
142 if (i < 6)
143 {
144 int ix;
145
146 *p++ = '.';
147 x -= floor (x + EPSILON / 2);
148 for (j = i; j < 6; j++)
149 x *= 10;
150 ix = floor (x + 0.5);
151
152 for (j = 0; j < i; j++)
153 ix *= 10;
154
155 /* A cheap hack, this routine can round wrong for fractions
156 near one. */
157 if (ix == 1000000)
158 ix = 999999;
159
160 sprintf (p, "%06d", ix);
161 i = 6 - i;
162 while (i && p[i - 1] == '0')
163 i--;
164 if (i == 0)
165 i--;
166 p += i;
167 }
168 }
169 else
170 p += sprintf (p, "%g", x);
171
172 *p = '\0';
173 return p - str;
174 }
175
176
177
178 #include <stdlib.h>
179 /**
180 * art_affine_to_string: Convert affine transformation to concise PostScript string representation.
181 * @str: Where to store the resulting string.
182 * @src: The affine transform.
183 *
184 * Converts an affine transform into a bit of PostScript code that
185 * implements the transform. Special cases of scaling, rotation, and
186 * translation are detected, and the corresponding PostScript
187 * operators used (this greatly aids understanding the output
188 * generated). The identity transform is mapped to the null string.
189 **/
190 void
art_affine_to_string(char str[128],const double src[6])191 art_affine_to_string (char str[128], const double src[6])
192 {
193 char tmp[80];
194 int i, ix;
195
196 #if 0
197 for (i = 0; i < 1000; i++)
198 {
199 double d = rand () * .1 / RAND_MAX;
200 art_ftoa (tmp, d);
201 printf ("%g %f %s\n", d, d, tmp);
202 }
203 #endif
204 if (fabs (src[4]) < EPSILON && fabs (src[5]) < EPSILON)
205 {
206 /* could be scale or rotate */
207 if (fabs (src[1]) < EPSILON && fabs (src[2]) < EPSILON)
208 {
209 /* scale */
210 if (fabs (src[0] - 1) < EPSILON && fabs (src[3] - 1) < EPSILON)
211 {
212 /* identity transform */
213 str[0] = '\0';
214 return;
215 }
216 else
217 {
218 ix = 0;
219 ix += art_ftoa (str + ix, src[0]);
220 str[ix++] = ' ';
221 ix += art_ftoa (str + ix, src[3]);
222 strcpy (str + ix, " scale");
223 return;
224 }
225 }
226 else
227 {
228 /* could be rotate */
229 if (fabs (src[0] - src[3]) < EPSILON &&
230 fabs (src[1] + src[2]) < EPSILON &&
231 fabs (src[0] * src[0] + src[1] * src[1] - 1) < 2 * EPSILON)
232 {
233 double theta;
234
235 theta = (180 / M_PI) * atan2 (src[1], src[0]);
236 art_ftoa (tmp, theta);
237 sprintf (str, "%s rotate", tmp);
238 return;
239 }
240 }
241 }
242 else
243 {
244 /* could be translate */
245 if (fabs (src[0] - 1) < EPSILON && fabs (src[1]) < EPSILON &&
246 fabs (src[2]) < EPSILON && fabs (src[3] - 1) < EPSILON)
247 {
248 ix = 0;
249 ix += art_ftoa (str + ix, src[4]);
250 str[ix++] = ' ';
251 ix += art_ftoa (str + ix, src[5]);
252 strcpy (str + ix, " translate");
253 return;
254 }
255 }
256
257 ix = 0;
258 str[ix++] = '[';
259 str[ix++] = ' ';
260 for (i = 0; i < 6; i++)
261 {
262 ix += art_ftoa (str + ix, src[i]);
263 str[ix++] = ' ';
264 }
265 strcpy (str + ix, "] concat");
266 }
267
268 /**
269 * art_affine_multiply: Multiply two affine transformation matrices.
270 * @dst: Where to store the result.
271 * @src1: The first affine transform to multiply.
272 * @src2: The second affine transform to multiply.
273 *
274 * Multiplies two affine transforms together, i.e. the resulting @dst
275 * is equivalent to doing first @src1 then @src2. Note that the
276 * PostScript concat operator multiplies on the left, i.e. "M concat"
277 * is equivalent to "CTM = multiply (M, CTM)";
278 *
279 * It is safe to call this function with @dst equal to @src1 or @src2.
280 **/
281 void
art_affine_multiply(double dst[6],const double src1[6],const double src2[6])282 art_affine_multiply (double dst[6], const double src1[6], const double src2[6])
283 {
284 double d0, d1, d2, d3, d4, d5;
285
286 d0 = src1[0] * src2[0] + src1[1] * src2[2];
287 d1 = src1[0] * src2[1] + src1[1] * src2[3];
288 d2 = src1[2] * src2[0] + src1[3] * src2[2];
289 d3 = src1[2] * src2[1] + src1[3] * src2[3];
290 d4 = src1[4] * src2[0] + src1[5] * src2[2] + src2[4];
291 d5 = src1[4] * src2[1] + src1[5] * src2[3] + src2[5];
292 dst[0] = d0;
293 dst[1] = d1;
294 dst[2] = d2;
295 dst[3] = d3;
296 dst[4] = d4;
297 dst[5] = d5;
298 }
299
300 /**
301 * art_affine_identity: Set up the identity matrix.
302 * @dst: Where to store the resulting affine transform.
303 *
304 * Sets up an identity matrix.
305 **/
306 void
art_affine_identity(double dst[6])307 art_affine_identity (double dst[6])
308 {
309 dst[0] = 1;
310 dst[1] = 0;
311 dst[2] = 0;
312 dst[3] = 1;
313 dst[4] = 0;
314 dst[5] = 0;
315 }
316
317
318 /**
319 * art_affine_scale: Set up a scaling matrix.
320 * @dst: Where to store the resulting affine transform.
321 * @sx: X scale factor.
322 * @sy: Y scale factor.
323 *
324 * Sets up a scaling matrix.
325 **/
326 void
art_affine_scale(double dst[6],double sx,double sy)327 art_affine_scale (double dst[6], double sx, double sy)
328 {
329 dst[0] = sx;
330 dst[1] = 0;
331 dst[2] = 0;
332 dst[3] = sy;
333 dst[4] = 0;
334 dst[5] = 0;
335 }
336
337 /**
338 * art_affine_rotate: Set up a rotation affine transform.
339 * @dst: Where to store the resulting affine transform.
340 * @theta: Rotation angle in degrees.
341 *
342 * Sets up a rotation matrix. In the standard libart coordinate
343 * system, in which increasing y moves downward, this is a
344 * counterclockwise rotation. In the standard PostScript coordinate
345 * system, which is reversed in the y direction, it is a clockwise
346 * rotation.
347 **/
348 void
art_affine_rotate(double dst[6],double theta)349 art_affine_rotate (double dst[6], double theta)
350 {
351 double s, c;
352
353 s = sin (theta * M_PI / 180.0);
354 c = cos (theta * M_PI / 180.0);
355 dst[0] = c;
356 dst[1] = s;
357 dst[2] = -s;
358 dst[3] = c;
359 dst[4] = 0;
360 dst[5] = 0;
361 }
362
363 /**
364 * art_affine_shear: Set up a shearing matrix.
365 * @dst: Where to store the resulting affine transform.
366 * @theta: Shear angle in degrees.
367 *
368 * Sets up a shearing matrix. In the standard libart coordinate system
369 * and a small value for theta, || becomes \\. Horizontal lines remain
370 * unchanged.
371 **/
372 void
art_affine_shear(double dst[6],double theta)373 art_affine_shear (double dst[6], double theta)
374 {
375 double t;
376
377 t = tan (theta * M_PI / 180.0);
378 dst[0] = 1;
379 dst[1] = 0;
380 dst[2] = t;
381 dst[3] = 1;
382 dst[4] = 0;
383 dst[5] = 0;
384 }
385
386 /**
387 * art_affine_translate: Set up a translation matrix.
388 * @dst: Where to store the resulting affine transform.
389 * @tx: X translation amount.
390 * @tx: Y translation amount.
391 *
392 * Sets up a translation matrix.
393 **/
394 void
art_affine_translate(double dst[6],double tx,double ty)395 art_affine_translate (double dst[6], double tx, double ty)
396 {
397 dst[0] = 1;
398 dst[1] = 0;
399 dst[2] = 0;
400 dst[3] = 1;
401 dst[4] = tx;
402 dst[5] = ty;
403 }
404
405 /**
406 * art_affine_expansion: Find the affine's expansion factor.
407 * @src: The affine transformation.
408 *
409 * Finds the expansion factor, i.e. the square root of the factor
410 * by which the affine transform affects area. In an affine transform
411 * composed of scaling, rotation, shearing, and translation, returns
412 * the amount of scaling.
413 *
414 * Return value: the expansion factor.
415 **/
416 double
art_affine_expansion(const double src[6])417 art_affine_expansion (const double src[6])
418 {
419 return sqrt (fabs (src[0] * src[3] - src[1] * src[2]));
420 }
421
422 /**
423 * art_affine_rectilinear: Determine whether the affine transformation is rectilinear.
424 * @src: The original affine transformation.
425 *
426 * Determines whether @src is rectilinear, i.e. grid-aligned
427 * rectangles are transformed to other grid-aligned rectangles. The
428 * implementation has epsilon-tolerance for roundoff errors.
429 *
430 * Return value: TRUE if @src is rectilinear.
431 **/
432 int
art_affine_rectilinear(const double src[6])433 art_affine_rectilinear (const double src[6])
434 {
435 return ((fabs (src[1]) < EPSILON && fabs (src[2]) < EPSILON) ||
436 (fabs (src[0]) < EPSILON && fabs (src[3]) < EPSILON));
437 }
438
439 /**
440 * art_affine_equal: Determine whether two affine transformations are equal.
441 * @matrix1: An affine transformation.
442 * @matrix2: Another affine transformation.
443 *
444 * Determines whether @matrix1 and @matrix2 are equal, with
445 * epsilon-tolerance for roundoff errors.
446 *
447 * Return value: TRUE if @matrix1 and @matrix2 are equal.
448 **/
449 int
art_affine_equal(double matrix1[6],double matrix2[6])450 art_affine_equal (double matrix1[6], double matrix2[6])
451 {
452 return (fabs (matrix1[0] - matrix2[0]) < EPSILON &&
453 fabs (matrix1[1] - matrix2[1]) < EPSILON &&
454 fabs (matrix1[2] - matrix2[2]) < EPSILON &&
455 fabs (matrix1[3] - matrix2[3]) < EPSILON &&
456 fabs (matrix1[4] - matrix2[4]) < EPSILON &&
457 fabs (matrix1[5] - matrix2[5]) < EPSILON);
458 }
459