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
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
|
/***************************************************************************/
/* */
/* fttrigon.c */
/* */
/* FreeType trigonometric functions (body). */
/* */
/* Copyright 2001-2005, 2012-2013 by */
/* David Turner, Robert Wilhelm, and Werner Lemberg. */
/* */
/* This file is part of the FreeType project, and may only be used, */
/* modified, and distributed under the terms of the FreeType project */
/* license, LICENSE.TXT. By continuing to use, modify, or distribute */
/* this file you indicate that you have read the license and */
/* understand and accept it fully. */
/* */
/***************************************************************************/
/*************************************************************************/
/* */
/* This is a fixed-point CORDIC implementation of trigonometric */
/* functions as well as transformations between Cartesian and polar */
/* coordinates. The angles are represented as 16.16 fixed-point values */
/* in degrees, i.e., the angular resolution is 2^-16 degrees. Note that */
/* only vectors longer than 2^16*180/pi (or at least 22 bits) on a */
/* discrete Cartesian grid can have the same or better angular */
/* resolution. Therefore, to maintain this precision, some functions */
/* require an interim upscaling of the vectors, whereas others operate */
/* with 24-bit long vectors directly. */
/* */
/*************************************************************************/
#include <ft2build.h>
#include FT_INTERNAL_OBJECTS_H
#include FT_INTERNAL_CALC_H
#include FT_TRIGONOMETRY_H
/* the Cordic shrink factor 0.858785336480436 * 2^32 */
#define FT_TRIG_SCALE 0xDBD95B16UL
/* the highest bit in overflow-safe vector components, */
/* MSB of 0.858785336480436 * sqrt(0.5) * 2^30 */
#define FT_TRIG_SAFE_MSB 29
/* this table was generated for FT_PI = 180L << 16, i.e. degrees */
#define FT_TRIG_MAX_ITERS 23
static const FT_Fixed
ft_trig_arctan_table[] =
{
1740967L, 919879L, 466945L, 234379L, 117304L, 58666L, 29335L,
14668L, 7334L, 3667L, 1833L, 917L, 458L, 229L, 115L,
57L, 29L, 14L, 7L, 4L, 2L, 1L
};
#ifdef FT_LONG64
/* multiply a given value by the CORDIC shrink factor */
static FT_Fixed
ft_trig_downscale( FT_Fixed val )
{
FT_Fixed s;
FT_Int64 v;
s = val;
val = FT_ABS( val );
v = ( val * (FT_Int64)FT_TRIG_SCALE ) + 0x100000000UL;
val = (FT_Fixed)( v >> 32 );
return ( s >= 0 ) ? val : -val;
}
#else /* !FT_LONG64 */
/* multiply a given value by the CORDIC shrink factor */
static FT_Fixed
ft_trig_downscale( FT_Fixed val )
{
FT_Fixed s;
FT_UInt32 v1, v2, k1, k2, hi, lo1, lo2, lo3;
s = val;
val = FT_ABS( val );
v1 = (FT_UInt32)val >> 16;
v2 = (FT_UInt32)( val & 0xFFFFL );
k1 = (FT_UInt32)FT_TRIG_SCALE >> 16; /* constant */
k2 = (FT_UInt32)( FT_TRIG_SCALE & 0xFFFFL ); /* constant */
hi = k1 * v1;
lo1 = k1 * v2 + k2 * v1; /* can't overflow */
lo2 = ( k2 * v2 ) >> 16;
lo3 = FT_MAX( lo1, lo2 );
lo1 += lo2;
hi += lo1 >> 16;
if ( lo1 < lo3 )
hi += (FT_UInt32)0x10000UL;
val = (FT_Fixed)hi;
return ( s >= 0 ) ? val : -val;
}
#endif /* !FT_LONG64 */
static FT_Int
ft_trig_prenorm( FT_Vector* vec )
{
FT_Pos x, y;
FT_Int shift;
x = vec->x;
y = vec->y;
shift = FT_MSB( FT_ABS( x ) | FT_ABS( y ) );
if ( shift <= FT_TRIG_SAFE_MSB )
{
shift = FT_TRIG_SAFE_MSB - shift;
vec->x = (FT_Pos)( (FT_ULong)x << shift );
vec->y = (FT_Pos)( (FT_ULong)y << shift );
}
else
{
shift -= FT_TRIG_SAFE_MSB;
vec->x = x >> shift;
vec->y = y >> shift;
shift = -shift;
}
return shift;
}
static void
ft_trig_pseudo_rotate( FT_Vector* vec,
FT_Angle theta )
{
FT_Int i;
FT_Fixed x, y, xtemp, b;
const FT_Fixed *arctanptr;
x = vec->x;
y = vec->y;
/* Rotate inside [-PI/4,PI/4] sector */
while ( theta < -FT_ANGLE_PI4 )
{
xtemp = y;
y = -x;
x = xtemp;
theta += FT_ANGLE_PI2;
}
while ( theta > FT_ANGLE_PI4 )
{
xtemp = -y;
y = x;
x = xtemp;
theta -= FT_ANGLE_PI2;
}
arctanptr = ft_trig_arctan_table;
/* Pseudorotations, with right shifts */
for ( i = 1, b = 1; i < FT_TRIG_MAX_ITERS; b <<= 1, i++ )
{
if ( theta < 0 )
{
xtemp = x + ( ( y + b ) >> i );
y = y - ( ( x + b ) >> i );
x = xtemp;
theta += *arctanptr++;
}
else
{
xtemp = x - ( ( y + b ) >> i );
y = y + ( ( x + b ) >> i );
x = xtemp;
theta -= *arctanptr++;
}
}
vec->x = x;
vec->y = y;
}
static void
ft_trig_pseudo_polarize( FT_Vector* vec )
{
FT_Angle theta;
FT_Int i;
FT_Fixed x, y, xtemp, b;
const FT_Fixed *arctanptr;
x = vec->x;
y = vec->y;
/* Get the vector into [-PI/4,PI/4] sector */
if ( y > x )
{
if ( y > -x )
{
theta = FT_ANGLE_PI2;
xtemp = y;
y = -x;
x = xtemp;
}
else
{
theta = y > 0 ? FT_ANGLE_PI : -FT_ANGLE_PI;
x = -x;
y = -y;
}
}
else
{
if ( y < -x )
{
theta = -FT_ANGLE_PI2;
xtemp = -y;
y = x;
x = xtemp;
}
else
{
theta = 0;
}
}
arctanptr = ft_trig_arctan_table;
/* Pseudorotations, with right shifts */
for ( i = 1, b = 1; i < FT_TRIG_MAX_ITERS; b <<= 1, i++ )
{
if ( y > 0 )
{
xtemp = x + ( ( y + b ) >> i );
y = y - ( ( x + b ) >> i );
x = xtemp;
theta += *arctanptr++;
}
else
{
xtemp = x - ( ( y + b ) >> i );
y = y + ( ( x + b ) >> i );
x = xtemp;
theta -= *arctanptr++;
}
}
/* round theta */
if ( theta >= 0 )
theta = FT_PAD_ROUND( theta, 32 );
else
theta = -FT_PAD_ROUND( -theta, 32 );
vec->x = x;
vec->y = theta;
}
/* documentation is in fttrigon.h */
FT_EXPORT_DEF( FT_Fixed )
FT_Cos( FT_Angle angle )
{
FT_Vector v;
v.x = FT_TRIG_SCALE >> 8;
v.y = 0;
ft_trig_pseudo_rotate( &v, angle );
return ( v.x + 0x80L ) >> 8;
}
/* documentation is in fttrigon.h */
FT_EXPORT_DEF( FT_Fixed )
FT_Sin( FT_Angle angle )
{
return FT_Cos( FT_ANGLE_PI2 - angle );
}
/* documentation is in fttrigon.h */
FT_EXPORT_DEF( FT_Fixed )
FT_Tan( FT_Angle angle )
{
FT_Vector v;
v.x = FT_TRIG_SCALE >> 8;
v.y = 0;
ft_trig_pseudo_rotate( &v, angle );
return FT_DivFix( v.y, v.x );
}
/* documentation is in fttrigon.h */
FT_EXPORT_DEF( FT_Angle )
FT_Atan2( FT_Fixed dx,
FT_Fixed dy )
{
FT_Vector v;
if ( dx == 0 && dy == 0 )
return 0;
v.x = dx;
v.y = dy;
ft_trig_prenorm( &v );
ft_trig_pseudo_polarize( &v );
return v.y;
}
/* documentation is in fttrigon.h */
FT_EXPORT_DEF( void )
FT_Vector_Unit( FT_Vector* vec,
FT_Angle angle )
{
vec->x = FT_TRIG_SCALE >> 8;
vec->y = 0;
ft_trig_pseudo_rotate( vec, angle );
vec->x = ( vec->x + 0x80L ) >> 8;
vec->y = ( vec->y + 0x80L ) >> 8;
}
/* these macros return 0 for positive numbers,
and -1 for negative ones */
#define FT_SIGN_LONG( x ) ( (x) >> ( FT_SIZEOF_LONG * 8 - 1 ) )
#define FT_SIGN_INT( x ) ( (x) >> ( FT_SIZEOF_INT * 8 - 1 ) )
#define FT_SIGN_INT32( x ) ( (x) >> 31 )
#define FT_SIGN_INT16( x ) ( (x) >> 15 )
/* documentation is in fttrigon.h */
FT_EXPORT_DEF( void )
FT_Vector_Rotate( FT_Vector* vec,
FT_Angle angle )
{
FT_Int shift;
FT_Vector v;
v.x = vec->x;
v.y = vec->y;
if ( angle && ( v.x != 0 || v.y != 0 ) )
{
shift = ft_trig_prenorm( &v );
ft_trig_pseudo_rotate( &v, angle );
v.x = ft_trig_downscale( v.x );
v.y = ft_trig_downscale( v.y );
if ( shift > 0 )
{
FT_Int32 half = (FT_Int32)1L << ( shift - 1 );
vec->x = ( v.x + half + FT_SIGN_LONG( v.x ) ) >> shift;
vec->y = ( v.y + half + FT_SIGN_LONG( v.y ) ) >> shift;
}
else
{
shift = -shift;
vec->x = (FT_Pos)( (FT_ULong)v.x << shift );
vec->y = (FT_Pos)( (FT_ULong)v.y << shift );
}
}
}
/* documentation is in fttrigon.h */
FT_EXPORT_DEF( FT_Fixed )
FT_Vector_Length( FT_Vector* vec )
{
FT_Int shift;
FT_Vector v;
v = *vec;
/* handle trivial cases */
if ( v.x == 0 )
{
return FT_ABS( v.y );
}
else if ( v.y == 0 )
{
return FT_ABS( v.x );
}
/* general case */
shift = ft_trig_prenorm( &v );
ft_trig_pseudo_polarize( &v );
v.x = ft_trig_downscale( v.x );
if ( shift > 0 )
return ( v.x + ( 1 << ( shift - 1 ) ) ) >> shift;
return (FT_Fixed)( (FT_UInt32)v.x << -shift );
}
/* documentation is in fttrigon.h */
FT_EXPORT_DEF( void )
FT_Vector_Polarize( FT_Vector* vec,
FT_Fixed *length,
FT_Angle *angle )
{
FT_Int shift;
FT_Vector v;
v = *vec;
if ( v.x == 0 && v.y == 0 )
return;
shift = ft_trig_prenorm( &v );
ft_trig_pseudo_polarize( &v );
v.x = ft_trig_downscale( v.x );
*length = ( shift >= 0 ) ? ( v.x >> shift )
: (FT_Fixed)( (FT_UInt32)v.x << -shift );
*angle = v.y;
}
/* documentation is in fttrigon.h */
FT_EXPORT_DEF( void )
FT_Vector_From_Polar( FT_Vector* vec,
FT_Fixed length,
FT_Angle angle )
{
vec->x = length;
vec->y = 0;
FT_Vector_Rotate( vec, angle );
}
/* documentation is in fttrigon.h */
FT_EXPORT_DEF( FT_Angle )
FT_Angle_Diff( FT_Angle angle1,
FT_Angle angle2 )
{
FT_Angle delta = angle2 - angle1;
delta %= FT_ANGLE_2PI;
if ( delta < 0 )
delta += FT_ANGLE_2PI;
if ( delta > FT_ANGLE_PI )
delta -= FT_ANGLE_2PI;
return delta;
}
/* END */
|
