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1 /***************************************************************************/
2 /* */
3 /* fttrigon.c */
4 /* */
5 /* FreeType trigonometric functions (body). */
6 /* */
7 /* Copyright 2001, 2002, 2003, 2004, 2005 by */
8 /* David Turner, Robert Wilhelm, and Werner Lemberg. */
9 /* */
10 /* This file is part of the FreeType project, and may only be used, */
11 /* modified, and distributed under the terms of the FreeType project */
12 /* license, LICENSE.TXT. By continuing to use, modify, or distribute */
13 /* this file you indicate that you have read the license and */
14 /* understand and accept it fully. */
15 /* */
16 /***************************************************************************/
17
18
19 #include <ft2build.h>
20 #include FT_INTERNAL_OBJECTS_H
21 #include FT_TRIGONOMETRY_H
22
23
24 /* the following is 0.2715717684432231 * 2^30 */
25 #define FT_TRIG_COSCALE 0x11616E8EUL
26
27 /* this table was generated for FT_PI = 180L << 16, i.e. degrees */
28 #define FT_TRIG_MAX_ITERS 23
29
30 static const FT_Fixed
31 ft_trig_arctan_table[24] =
32 {
33 4157273L, 2949120L, 1740967L, 919879L, 466945L, 234379L, 117304L,
34 58666L, 29335L, 14668L, 7334L, 3667L, 1833L, 917L, 458L, 229L, 115L,
35 57L, 29L, 14L, 7L, 4L, 2L, 1L
36 };
37
38 /* the Cordic shrink factor, multiplied by 2^32 */
39 #define FT_TRIG_SCALE 1166391785UL /* 0x4585BA38UL */
40
41
42 #ifdef FT_CONFIG_HAS_INT64
43
44 /* multiply a given value by the CORDIC shrink factor */
45 static FT_Fixed
46 ft_trig_downscale( FT_Fixed val )
47 {
48 FT_Fixed s;
49 FT_Int64 v;
50
51
52 s = val;
53 val = ( val >= 0 ) ? val : -val;
54
55 v = ( val * (FT_Int64)FT_TRIG_SCALE ) + 0x100000000UL;
56 val = (FT_Fixed)( v >> 32 );
57
58 return ( s >= 0 ) ? val : -val;
59 }
60
61 #else /* !FT_CONFIG_HAS_INT64 */
62
63 /* multiply a given value by the CORDIC shrink factor */
64 static FT_Fixed
65 ft_trig_downscale( FT_Fixed val )
66 {
67 FT_Fixed s;
68 FT_UInt32 v1, v2, k1, k2, hi, lo1, lo2, lo3;
69
70
71 s = val;
72 val = ( val >= 0 ) ? val : -val;
73
74 v1 = (FT_UInt32)val >> 16;
75 v2 = (FT_UInt32)val & 0xFFFFL;
76
77 k1 = FT_TRIG_SCALE >> 16; /* constant */
78 k2 = FT_TRIG_SCALE & 0xFFFFL; /* constant */
79
80 hi = k1 * v1;
81 lo1 = k1 * v2 + k2 * v1; /* can't overflow */
82
83 lo2 = ( k2 * v2 ) >> 16;
84 lo3 = ( lo1 >= lo2 ) ? lo1 : lo2;
85 lo1 += lo2;
86
87 hi += lo1 >> 16;
88 if ( lo1 < lo3 )
89 hi += 0x10000UL;
90
91 val = (FT_Fixed)hi;
92
93 return ( s >= 0 ) ? val : -val;
94 }
95
96 #endif /* !FT_CONFIG_HAS_INT64 */
97
98
99 static FT_Int
100 ft_trig_prenorm( FT_Vector* vec )
101 {
102 FT_Fixed x, y, z;
103 FT_Int shift;
104
105
106 x = vec->x;
107 y = vec->y;
108
109 z = ( ( x >= 0 ) ? x : - x ) | ( (y >= 0) ? y : -y );
110 shift = 0;
111
112 #if 1
113 /* determine msb bit index in `shift' */
114 if ( z >= ( 1L << 16 ) )
115 {
116 z >>= 16;
117 shift += 16;
118 }
119 if ( z >= ( 1L << 8 ) )
120 {
121 z >>= 8;
122 shift += 8;
123 }
124 if ( z >= ( 1L << 4 ) )
125 {
126 z >>= 4;
127 shift += 4;
128 }
129 if ( z >= ( 1L << 2 ) )
130 {
131 z >>= 2;
132 shift += 2;
133 }
134 if ( z >= ( 1L << 1 ) )
135 {
136 z >>= 1;
137 shift += 1;
138 }
139
140 if ( shift <= 27 )
141 {
142 shift = 27 - shift;
143 vec->x = x << shift;
144 vec->y = y << shift;
145 }
146 else
147 {
148 shift -= 27;
149 vec->x = x >> shift;
150 vec->y = y >> shift;
151 shift = -shift;
152 }
153
154 #else /* 0 */
155
156 if ( z < ( 1L << 27 ) )
157 {
158 do
159 {
160 shift++;
161 z <<= 1;
162 } while ( z < ( 1L << 27 ) );
163 vec->x = x << shift;
164 vec->y = y << shift;
165 }
166 else if ( z > ( 1L << 28 ) )
167 {
168 do
169 {
170 shift++;
171 z >>= 1;
172 } while ( z > ( 1L << 28 ) );
173
174 vec->x = x >> shift;
175 vec->y = y >> shift;
176 shift = -shift;
177 }
178
179 #endif /* 0 */
180
181 return shift;
182 }
183
184
185 static void
186 ft_trig_pseudo_rotate( FT_Vector* vec,
187 FT_Angle theta )
188 {
189 FT_Int i;
190 FT_Fixed x, y, xtemp;
191 const FT_Fixed *arctanptr;
192
193
194 x = vec->x;
195 y = vec->y;
196
197 /* Get angle between -90 and 90 degrees */
198 while ( theta <= -FT_ANGLE_PI2 )
199 {
200 x = -x;
201 y = -y;
202 theta += FT_ANGLE_PI;
203 }
204
205 while ( theta > FT_ANGLE_PI2 )
206 {
207 x = -x;
208 y = -y;
209 theta -= FT_ANGLE_PI;
210 }
211
212 /* Initial pseudorotation, with left shift */
213 arctanptr = ft_trig_arctan_table;
214
215 if ( theta < 0 )
216 {
217 xtemp = x + ( y << 1 );
218 y = y - ( x << 1 );
219 x = xtemp;
220 theta += *arctanptr++;
221 }
222 else
223 {
224 xtemp = x - ( y << 1 );
225 y = y + ( x << 1 );
226 x = xtemp;
227 theta -= *arctanptr++;
228 }
229
230 /* Subsequent pseudorotations, with right shifts */
231 i = 0;
232 do
233 {
234 if ( theta < 0 )
235 {
236 xtemp = x + ( y >> i );
237 y = y - ( x >> i );
238 x = xtemp;
239 theta += *arctanptr++;
240 }
241 else
242 {
243 xtemp = x - ( y >> i );
244 y = y + ( x >> i );
245 x = xtemp;
246 theta -= *arctanptr++;
247 }
248 } while ( ++i < FT_TRIG_MAX_ITERS );
249
250 vec->x = x;
251 vec->y = y;
252 }
253
254
255 static void
256 ft_trig_pseudo_polarize( FT_Vector* vec )
257 {
258 FT_Fixed theta;
259 FT_Fixed yi, i;
260 FT_Fixed x, y;
261 const FT_Fixed *arctanptr;
262
263
264 x = vec->x;
265 y = vec->y;
266
267 /* Get the vector into the right half plane */
268 theta = 0;
269 if ( x < 0 )
270 {
271 x = -x;
272 y = -y;
273 theta = 2 * FT_ANGLE_PI2;
274 }
275
276 if ( y > 0 )
277 theta = - theta;
278
279 arctanptr = ft_trig_arctan_table;
280
281 if ( y < 0 )
282 {
283 /* Rotate positive */
284 yi = y + ( x << 1 );
285 x = x - ( y << 1 );
286 y = yi;
287 theta -= *arctanptr++; /* Subtract angle */
288 }
289 else
290 {
291 /* Rotate negative */
292 yi = y - ( x << 1 );
293 x = x + ( y << 1 );
294 y = yi;
295 theta += *arctanptr++; /* Add angle */
296 }
297
298 i = 0;
299 do
300 {
301 if ( y < 0 )
302 {
303 /* Rotate positive */
304 yi = y + ( x >> i );
305 x = x - ( y >> i );
306 y = yi;
307 theta -= *arctanptr++;
308 }
309 else
310 {
311 /* Rotate negative */
312 yi = y - ( x >> i );
313 x = x + ( y >> i );
314 y = yi;
315 theta += *arctanptr++;
316 }
317 } while ( ++i < FT_TRIG_MAX_ITERS );
318
319 /* round theta */
320 if ( theta >= 0 )
321 theta = FT_PAD_ROUND( theta, 32 );
322 else
323 theta = -FT_PAD_ROUND( -theta, 32 );
324
325 vec->x = x;
326 vec->y = theta;
327 }
328
329
330 /* documentation is in fttrigon.h */
331
332 FT_EXPORT_DEF( FT_Fixed )
333 FT_Cos( FT_Angle angle )
334 {
335 FT_Vector v;
336
337
338 v.x = FT_TRIG_COSCALE >> 2;
339 v.y = 0;
340 ft_trig_pseudo_rotate( &v, angle );
341
342 return v.x / ( 1 << 12 );
343 }
344
345
346 /* documentation is in fttrigon.h */
347
348 FT_EXPORT_DEF( FT_Fixed )
349 FT_Sin( FT_Angle angle )
350 {
351 return FT_Cos( FT_ANGLE_PI2 - angle );
352 }
353
354
355 /* documentation is in fttrigon.h */
356
357 FT_EXPORT_DEF( FT_Fixed )
358 FT_Tan( FT_Angle angle )
359 {
360 FT_Vector v;
361
362
363 v.x = FT_TRIG_COSCALE >> 2;
364 v.y = 0;
365 ft_trig_pseudo_rotate( &v, angle );
366
367 return FT_DivFix( v.y, v.x );
368 }
369
370
371 /* documentation is in fttrigon.h */
372
373 FT_EXPORT_DEF( FT_Angle )
374 FT_Atan2( FT_Fixed dx,
375 FT_Fixed dy )
376 {
377 FT_Vector v;
378
379
380 if ( dx == 0 && dy == 0 )
381 return 0;
382
383 v.x = dx;
384 v.y = dy;
385 ft_trig_prenorm( &v );
386 ft_trig_pseudo_polarize( &v );
387
388 return v.y;
389 }
390
391
392 /* documentation is in fttrigon.h */
393
394 FT_EXPORT_DEF( void )
395 FT_Vector_Unit( FT_Vector* vec,
396 FT_Angle angle )
397 {
398 vec->x = FT_TRIG_COSCALE >> 2;
399 vec->y = 0;
400 ft_trig_pseudo_rotate( vec, angle );
401 vec->x >>= 12;
402 vec->y >>= 12;
403 }
404
405
406 /* these macros return 0 for positive numbers,
407 and -1 for negative ones */
408 #define FT_SIGN_LONG( x ) ( (x) >> ( FT_SIZEOF_LONG * 8 - 1 ) )
409 #define FT_SIGN_INT( x ) ( (x) >> ( FT_SIZEOF_INT * 8 - 1 ) )
410 #define FT_SIGN_INT32( x ) ( (x) >> 31 )
411 #define FT_SIGN_INT16( x ) ( (x) >> 15 )
412
413
414 /* documentation is in fttrigon.h */
415
416 FT_EXPORT_DEF( void )
417 FT_Vector_Rotate( FT_Vector* vec,
418 FT_Angle angle )
419 {
420 FT_Int shift;
421 FT_Vector v;
422
423
424 v.x = vec->x;
425 v.y = vec->y;
426
427 if ( angle && ( v.x != 0 || v.y != 0 ) )
428 {
429 shift = ft_trig_prenorm( &v );
430 ft_trig_pseudo_rotate( &v, angle );
431 v.x = ft_trig_downscale( v.x );
432 v.y = ft_trig_downscale( v.y );
433
434 if ( shift > 0 )
435 {
436 FT_Int32 half = 1L << ( shift - 1 );
437
438
439 vec->x = ( v.x + half + FT_SIGN_LONG( v.x ) ) >> shift;
440 vec->y = ( v.y + half + FT_SIGN_LONG( v.y ) ) >> shift;
441 }
442 else
443 {
444 shift = -shift;
445 vec->x = v.x << shift;
446 vec->y = v.y << shift;
447 }
448 }
449 }
450
451
452 /* documentation is in fttrigon.h */
453
454 FT_EXPORT_DEF( FT_Fixed )
455 FT_Vector_Length( FT_Vector* vec )
456 {
457 FT_Int shift;
458 FT_Vector v;
459
460
461 v = *vec;
462
463 /* handle trivial cases */
464 if ( v.x == 0 )
465 {
466 return ( v.y >= 0 ) ? v.y : -v.y;
467 }
468 else if ( v.y == 0 )
469 {
470 return ( v.x >= 0 ) ? v.x : -v.x;
471 }
472
473 /* general case */
474 shift = ft_trig_prenorm( &v );
475 ft_trig_pseudo_polarize( &v );
476
477 v.x = ft_trig_downscale( v.x );
478
479 if ( shift > 0 )
480 return ( v.x + ( 1 << ( shift - 1 ) ) ) >> shift;
481
482 return v.x << -shift;
483 }
484
485
486 /* documentation is in fttrigon.h */
487
488 FT_EXPORT_DEF( void )
489 FT_Vector_Polarize( FT_Vector* vec,
490 FT_Fixed *length,
491 FT_Angle *angle )
492 {
493 FT_Int shift;
494 FT_Vector v;
495
496
497 v = *vec;
498
499 if ( v.x == 0 && v.y == 0 )
500 return;
501
502 shift = ft_trig_prenorm( &v );
503 ft_trig_pseudo_polarize( &v );
504
505 v.x = ft_trig_downscale( v.x );
506
507 *length = ( shift >= 0 ) ? ( v.x >> shift ) : ( v.x << -shift );
508 *angle = v.y;
509 }
510
511
512 /* documentation is in fttrigon.h */
513
514 FT_EXPORT_DEF( void )
515 FT_Vector_From_Polar( FT_Vector* vec,
516 FT_Fixed length,
517 FT_Angle angle )
518 {
519 vec->x = length;
520 vec->y = 0;
521
522 FT_Vector_Rotate( vec, angle );
523 }
524
525
526 /* documentation is in fttrigon.h */
527
528 FT_EXPORT_DEF( FT_Angle )
529 FT_Angle_Diff( FT_Angle angle1,
530 FT_Angle angle2 )
531 {
532 FT_Angle delta = angle2 - angle1;
533
534
535 delta %= FT_ANGLE_2PI;
536 if ( delta < 0 )
537 delta += FT_ANGLE_2PI;
538
539 if ( delta > FT_ANGLE_PI )
540 delta -= FT_ANGLE_2PI;
541
542 return delta;
543 }
544
545
546 /* END */