summaryrefslogtreecommitdiff
path: root/src/Type1/spaces.c
blob: 9c79954b8ce4f99a2c43fa69171a083ddce9eea1 (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
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
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
/* $Xorg: spaces.c,v 1.4 2000/08/17 19:46:32 cpqbld Exp $ */
/* Copyright International Business Machines, Corp. 1991
 * All Rights Reserved
 * Copyright Lexmark International, Inc. 1991
 * All Rights Reserved
 *
 * License to use, copy, modify, and distribute this software and its
 * documentation for any purpose and without fee is hereby granted,
 * provided that the above copyright notice appear in all copies and that
 * both that copyright notice and this permission notice appear in
 * supporting documentation, and that the name of IBM or Lexmark not be
 * used in advertising or publicity pertaining to distribution of the
 * software without specific, written prior permission.
 *
 * IBM AND LEXMARK PROVIDE THIS SOFTWARE "AS IS", WITHOUT ANY WARRANTIES OF
 * ANY KIND, EITHER EXPRESS OR IMPLIED, INCLUDING, BUT NOT LIMITED TO ANY
 * IMPLIED WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE,
 * AND NONINFRINGEMENT OF THIRD PARTY RIGHTS.  THE ENTIRE RISK AS TO THE
 * QUALITY AND PERFORMANCE OF THE SOFTWARE, INCLUDING ANY DUTY TO SUPPORT
 * OR MAINTAIN, BELONGS TO THE LICENSEE.  SHOULD ANY PORTION OF THE
 * SOFTWARE PROVE DEFECTIVE, THE LICENSEE (NOT IBM OR LEXMARK) ASSUMES THE
 * ENTIRE COST OF ALL SERVICING, REPAIR AND CORRECTION.  IN NO EVENT SHALL
 * IBM OR LEXMARK BE LIABLE FOR ANY SPECIAL, INDIRECT OR CONSEQUENTIAL
 * DAMAGES OR ANY DAMAGES WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR
 * PROFITS, WHETHER IN AN ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS
 * ACTION, ARISING OUT OF OR IN CONNECTION WITH THE USE OR PERFORMANCE OF
 * THIS SOFTWARE.
 */
/* $XFree86: xc/lib/font/Type1/spaces.c,v 3.11 2003/05/27 22:26:47 tsi Exp $ */
 /* SPACES   CWEB         V0021 ********                             */
/*
:h1 id=spaces.SPACES Module - Handles Coordinate Spaces
 
This module is responsible for handling the TYPE1IMAGER "XYspace" object.
 
&author. Jeffrey B. Lotspiech (lotspiech@almaden.ibm.com)
 
 
:h3.Include Files
*/

#ifdef FONTMODULE
#include "Xdefs.h"	/* Bool declaration ??? */
#include "Xmd.h"	/* INT32 declaration ??? */
#include "os.h"
#include "xf86_ansic.h"
#else
#include "X11/Xos.h"
#include "os.h"
#endif
#include "objects.h"
#include "spaces.h"
#include "paths.h"
#include "pictures.h"
#include "fonts.h"
#include "arith.h"
#include "trig.h"

static void FindFfcn ( double cx, double cy, 
			      convertFunc *fcnP );
static void FindIfcn ( double cx, double cy, 
			      fractpel *icxP, fractpel *icyP, 
			      iconvertFunc *fcnP );

/*
:h3.Entry Points Provided to the TYPE1IMAGER User
*/
 
/*SHARED LINE(S) ORIGINATED HERE*/
 
/*
:h3.Entry Points Provided to Other Modules
*/
 
/*
In addition, other modules call the SPACES module through function
vectors in the "XYspace" structure.  The entry points accessed that
way are "FConvert()", "IConvert()", and "ForceFloat()".
*/
 
/*SHARED LINE(S) ORIGINATED HERE*/
/*
:h3.Macros and Typedefs Provided to Other Modules
 
:h4.Duplicating and Killing Spaces
 
Destroying XYspaces is so simple we can do it with a
macro:
*/
 
/*SHARED LINE(S) ORIGINATED HERE*/
/*
On the other hand, duplicating XYspaces is slightly more difficult
because of the need to keep a unique ID in the space, see
:hdref refid=dupspace..
 
:h4.Fixed Point Pel Representation
 
We represent pel positions with fixed point numbers.  This does NOT
mean integer, but truly means fixed point, with a certain number
of binary digits (FRACTBITS) representing the fractional part of the
pel.
*/
 
/*SHARED LINE(S) ORIGINATED HERE*/
/*
:h2.Data Structures for Coordinate Spaces and Points
*/
/*
:h3 id=matrix.Matrices
 
TYPE1IMAGER uses 2x2 transformation matrices.  We'll use C notation for
such a matrix (M[2][2]), the first index being rows, the second columns.
*/
 
/*
:h3.The "doublematrix" Structure
 
We frequently find it desirable to store both a matrix and its
inverse.  We store these in a "doublematrix" structure.
*/
 
/*SHARED LINE(S) ORIGINATED HERE*/
 
/*
:h3.The "XYspace" Structure
 
The XYspace structure represents the XYspace object.
*/
 
/*SHARED LINE(S) ORIGINATED HERE*/
#define    RESERVED  10      /* 'n' IDs are reserved for invalid & immortal spaces */
/*
*/
#define    NEXTID    ((SpaceID < RESERVED) ? (SpaceID = RESERVED) : ++SpaceID)
 
static unsigned int SpaceID = 1;
 
struct XYspace *
CopySpace(struct XYspace *S)
{
       S = (struct XYspace *)Allocate(sizeof(struct XYspace), S, 0);
       S->ID = NEXTID;
       return(S);
}
/*
:h3.The "fractpoint" Structure
 
A fractional point is just a "fractpel" x and y:
*/
 
/*SHARED LINE(S) ORIGINATED HERE*/
 
/*
:h3.Lazy Evaluation of Matrix Inverses
 
Calculating the inverse of a matrix is somewhat involved, and we usually
do not need them.  So, we flag whether or not the space has the inverse
already calculated:
*/
 
#define    HASINVERSE(flag)   ((flag)&0x80)
 
/*
The following macro forces a space to have an inverse:
*/
 
#define    CoerceInverse(S)   if (!HASINVERSE((S)->flag)) { \
    MatrixInvert((S)->tofract.normal, (S)->tofract.inverse); (S)->flag |= HASINVERSE(ON); }
/*
:h3.IDENTITY Space
 
IDENTITY space is (logically) the space corresponding to the identity
transformation matrix.  However, since all our transformation matrices
have a common FRACTFLOAT scale factor to convert to 'fractpel's, that
is actually what we store in 'tofract' matrix of IDENTITY:
*/
 
static struct XYspace identity = { SPACETYPE, ISPERMANENT(ON) + ISIMMORTAL(ON)
                        + HASINVERSE(ON), 2, /* added 3-26-91 PNM */
                        NULL, NULL,
                        NULL, NULL, NULL, NULL,
                        INVALIDID + 1, 0,
                        {{{FRACTFLOAT, 0.0}, {0.0, FRACTFLOAT}},
                        {{1.0/FRACTFLOAT, 0.0}, {0.0, 1.0/FRACTFLOAT}}},
                        {{0, 0}, {0, 0}}};
struct XYspace *IDENTITY = &identity;
 
/*
*/
#define  MAXCONTEXTS   16
 
static struct doublematrix contexts[MAXCONTEXTS];
 
#ifdef notdef

static int nextcontext = 1;
 
/*SHARED LINE(S) ORIGINATED HERE*/
 
/*
:h3.FindDeviceContext() - Find the Context Given a Device
 
This routine, given a device, returns the index of the device's
transformation matrix in the context array.  If it cannot find it,
it will allocate a new array entry and fill it out.
*/
 
static int 
FindDeviceContext(pointer device) /* device token                            */
{
       double M[2][2];       /* temporary matrix                             */
       float Xres,Yres;      /* device  resolution                           */
       int orient = -1;      /* device orientation                           */
       int rc = -1;          /* return code for QueryDeviceState             */
 
       if (rc != 0)          /* we only bother with this check once          */
               Abort("Context:  QueryDeviceState didn't work");
 
       M[0][0] = M[1][0] = M[0][1] = M[1][1] = 0.0;
 
       switch (orient) {
           case 0:
               M[0][0] = Xres;  M[1][1] = -Yres;
               break;
           case 1:
               M[1][0] = Yres;  M[0][1] = Xres;
               break;
           case 2:
               M[0][0] = -Xres;  M[1][1] = Yres;
               break;
           case 3:
               M[1][0] = -Yres;  M[0][1] = -Xres;
               break;
           default:
               Abort("QueryDeviceState returned invalid orientation");
       }
       return(FindContext(M));
}
 
/*
:h3.FindContext() - Find the Context Given a Matrix
 
This routine, given a matrix, returns the index of that matrix matrix in
the context array.  If it cannot find it, it will allocate a new array
entry and fill it out.
*/
 
int 
FindContext(double M[2][2])  /* array to search for                          */
{
       register int i;       /* loop variable for search                     */
       for (i=0; i < nextcontext; i++)
               if (M[0][0] == contexts[i].normal[0][0] && M[1][0] == contexts[i].normal[1][0]
                   && M[0][1] == contexts[i].normal[0][1] && M[1][1] == contexts[i].normal[1][1])
                       break;
 
       if (i >= nextcontext) {
               if (i >= MAXCONTEXTS)
                       Abort("Context:  out of them");
               LONGCOPY(contexts[i].normal, M, sizeof(contexts[i].normal));
               MatrixInvert(M, contexts[i].inverse);
               nextcontext++;
       }
 
       return(i);
}
 
/*
:h3.Context() - Create a Coordinate Space for a Device
 
This user operator is implemented by first finding the device context
array index, then transforming IDENTITY space to create an appropriate
cooridnate space.
*/
 
struct XYspace *
Context(pointer device,      /* device token                                 */
	double units)        /* multiples of one inch                        */
{
       double M[2][2];       /* device transformation matrix                 */
       register int n;       /* will hold device context number              */
       register struct XYspace *S;  /* XYspace constructed                   */
 
       ARGCHECK((device == NULL), "Context of NULLDEVICE not allowed",
                    NULL, IDENTITY, (0), struct XYspace *);
       ARGCHECK((units == 0.0), "Context: bad units", NULL, IDENTITY, (0), struct XYspace *);
 
       n = FindDeviceContext(device);
 
       LONGCOPY(M, contexts[n].normal, sizeof(M));
 
       M[0][0] *= units;
       M[0][1] *= units;
       M[1][0] *= units;
       M[1][1] *= units;
 
       S = (struct XYspace *)Xform(IDENTITY, M);
 
       S->context = n;
       return(S);
}
#endif
 
/*
:h3.ConsiderContext() - Adjust a Matrix to Take Out Device Transform
 
Remember, we have :f/x times U times D/ and :f/M/ and and we want :f/x
times U times M times D/.  An easy way to do this is to calculate
:f/D sup <-1> times M times D/, because:
:formula.
x times U times D times D sup <-1> times M times D = x times U times M times D
:formula.
So this subroutine, given an :f/M/and an object, finds the :f/D/ for that
object and modifies :f/M/ so it is :f/D sup <-1> times M times D/.
*/
 
static void 
ConsiderContext(struct xobject *obj,  /* object to be transformed            */
		double M[2][2])    /* matrix (may be changed)                */
{
       register int context = 0; /* index in contexts array                  */
 
       if (obj == NULL) return;
 
       if (ISPATHTYPE(obj->type)) {
               struct segment *path = (struct segment *) obj;
 
               context = path->context;
       }
       else if (obj->type == SPACETYPE) {
               struct XYspace *S = (struct XYspace *) obj;
 
               context = S->context;
       }
       else if (obj->type == PICTURETYPE) {

       }
       else
               context = NULLCONTEXT;
 
       if (context != NULLCONTEXT) {
               MatrixMultiply(contexts[context].inverse, M, M);
               MatrixMultiply(M, contexts[context].normal, M);
       }
}
 
/*
:h2.Conversion from User's X,Y to "fractpel" X,Y
 
When the user is building paths (lines, moves, curves, etc.) he passes
the control points (x,y) for the paths together with an XYspace.  We
must convert from the user's (x,y) to our internal representation
which is in pels (fractpels, actually).  This involves transforming
the user's (x,y) under the coordinate space transformation.  It is
important that we do this quickly.  So, we store pointers to different
conversion functions right in the XYspace structure.  This allows us
to have simpler special case functions for the more commonly
encountered types of transformations.
 
:h3.Convert(), IConvert(), and ForceFloat() - Called Through "XYspace" Structure
 
These are functions that fit in the "convert" and "iconvert" function
pointers in the XYspace structure.  They call the "xconvert", "yconvert",
"ixconvert", and "iyconvert" as appropriate to actually do the work.
These secondary routines come in many flavors to handle different
special cases as quickly as possible.
*/
 
static void 
FXYConvert(struct fractpoint *pt, /* point to set                            */
	   struct XYspace *S,     /* relevant coordinate space               */
	   double x, double y)    /* user's coordinates of point             */
{
       pt->x = (*S->xconvert)(S->tofract.normal[0][0], S->tofract.normal[1][0], x, y);
       pt->y = (*S->yconvert)(S->tofract.normal[0][1], S->tofract.normal[1][1], x, y);
}
 
static void 
IXYConvert(struct fractpoint *pt,   /* point to set                          */
	   struct XYspace *S,       /* relevant coordinate space             */
	   long x, long y)          /* user's coordinates of point           */
{
       pt->x = (*S->ixconvert)(S->itofract[0][0], S->itofract[1][0], x, y);
       pt->y = (*S->iyconvert)(S->itofract[0][1], S->itofract[1][1], x, y);
}
 
/*
ForceFloat is a substitute for IConvert(), when we just do not have
enough significant digits in the coefficients to get high enough
precision in the answer with fixed point arithmetic.  So, we force the
integers to floats, and do the arithmetic all with floats:
*/
 
static void 
ForceFloat(struct fractpoint *pt,  /* point to set                           */
	   struct XYspace *S,      /* relevant coordinate space              */
	   long x, long y)         /* user's coordinates of point            */
{
       (*S->convert)(pt, S, (double) x, (double) y);
}
 
/*
:h3.FXYboth(), FXonly(), FYonly() - Floating Point Conversion
 
These are the routines we use when the user has given us floating
point numbers for x and y. FXYboth() is the general purpose routine;
FXonly() and FYonly() are special cases when one of the coefficients
is 0.0.
*/
 
static fractpel 
FXYboth(double cx, double cy,  /* x and y coefficients                       */
	double x, double y)    /* user x,y                                   */
{
       register double r;    /* temporary float                              */
 
       r = x * cx + y * cy;
       return((fractpel) r);
}
 
/*ARGSUSED*/
static fractpel 
FXonly(double cx, double cy,  /* x and y coefficients                        */
       double x, double y)    /* user x,y                                    */
{
       register double r;    /* temporary float                              */
 
       r = x * cx;
       return((fractpel) r);
}
 
/*ARGSUSED*/
static fractpel 
FYonly(double cx, double cy,   /* x and y coefficients                       */
       double x, double y)     /* user x,y                                   */
{
       register double r;    /* temporary float                              */
 
       r = y * cy;
       return((fractpel) r);
}
 
/*
:h3.IXYboth(), IXonly(), IYonly() - Simple Integer Conversion
 
These are the routines we use when the user has given us integers for
x and y, and the coefficients have enough significant digits to
provide precise answers with only "long" (32 bit?) multiplication.
IXYboth() is the general purpose routine; IXonly() and IYonly() are
special cases when one of the coefficients is 0.
*/
 
static fractpel 
IXYboth(fractpel cx, fractpel cy, /* x and y coefficients                    */
	long x, long y)           /* user x,y                                */
{
       return(x * cx + y * cy);
}
 
/*ARGSUSED*/
static fractpel
IXonly(fractpel cx, fractpel cy, /* x and y coefficients                     */
       long x, long y)           /* user x,y                                 */
{
       return(x * cx);
}
 
/*ARGSUSED*/
static fractpel 
IYonly(fractpel cx, fractpel cy, /* x and y coefficients                     */
       long x, long y)           /* user x,y                                 */
{
       return(y * cy);
}
 
 
/*
:h3.FPXYboth(), FPXonly(), FPYonly() - More Involved Integer Conversion
 
These are the routines we use when the user has given us integers for
x and y, but the coefficients do not have enough significant digits to
provide precise answers with only "long" (32 bit?)  multiplication.
We have increased the number of significant bits in the coefficients
by FRACTBITS; therefore we must use "double long" (64 bit?)
multiplication by calling FPmult().  FPXYboth() is the general purpose
routine; FPXonly() and FPYonly() are special cases when one of the
coefficients is 0.
 
Note that it is perfectly possible for us to calculate X with the
"FP" method and Y with the "I" method, or vice versa.  It all depends
on how the functions in the XYspace structure are filled out.
*/
 
static fractpel 
FPXYboth(fractpel cx, fractpel cy, /* x and y coefficients                   */
	 long x, long y)           /* user x,y                               */
{
       return( FPmult(x, cx) + FPmult(y, cy) );
}
 
/*ARGSUSED*/
static fractpel 
FPXonly(fractpel cx, fractpel cy, /* x and y coefficients                    */
	long x, long y)           /* user x,y                                */
{
       return( FPmult(x, cx) );
}
 
/*ARGSUSED*/
static fractpel 
FPYonly(fractpel cx, fractpel cy, /* x and y coefficients                    */
	long x, long y)           /* user x,y                                */
{
       return( FPmult(y, cy) );
}
 
 
 
/*
:h3.FillOutFcns() - Determine the Appropriate Functions to Use for Conversion
 
This function fills out the "convert" and "iconvert" function pointers
in an XYspace structure, and also fills the "helper"
functions that actually do the work.
*/
 
static void 
FillOutFcns(struct XYspace *S)    /* functions will be set in this structure */
{
       S->convert = FXYConvert;
       S->iconvert = IXYConvert;
 
       FindFfcn(S->tofract.normal[0][0], S->tofract.normal[1][0], &S->xconvert);
       FindFfcn(S->tofract.normal[0][1], S->tofract.normal[1][1], &S->yconvert);
       FindIfcn(S->tofract.normal[0][0], S->tofract.normal[1][0],
                &S->itofract[0][0], &S->itofract[1][0], &S->ixconvert);
       FindIfcn(S->tofract.normal[0][1], S->tofract.normal[1][1],
                &S->itofract[0][1], &S->itofract[1][1], &S->iyconvert);
 
       if (S->ixconvert == NULL || S->iyconvert == NULL)
                S->iconvert = ForceFloat;
}
 
/*
:h4.FindFfcn() - Subroutine of FillOutFcns() to Fill Out Floating Functions
 
This function tests for the special case of one of the coefficients
being zero:
*/
 
static void 
FindFfcn(double cx, double cy, /* x and y coefficients                       */
	 convertFunc *fcnP)    /* pointer to function to set                 */
{
       if (cx == 0.0)
               *fcnP = FYonly;
       else if (cy == 0.0)
               *fcnP = FXonly;
       else
               *fcnP = FXYboth;
}
 
/*
:h4.FindIfcn() - Subroutine of FillOutFcns() to Fill Out Integer Functions
 
There are two types of integer functions, the 'I' type and the 'FP' type.
We use the I type functions when we are satisfied with simple integer
arithmetic.  We used the FP functions when we feel we need higher
precision (but still fixed point) arithmetic.  If all else fails,
we store a NULL indicating that this we should do the conversion in
floating point.
*/
 
static void 
FindIfcn(double cx, double cy, /* x and y coefficients                       */
	 fractpel *icxP, fractpel *icyP, /* fixed point coefficients to set  */
	 iconvertFunc *fcnP)          /* pointer to function to set          */
{
       register fractpel imax;  /* maximum of cx and cy                      */
 
       *icxP = cx;
       *icyP = cy;
 
       if (cx != (float) (*icxP) || cy != (float) (*icyP)) {
/*
At this point we know our integer approximations of the coefficients
are not exact.  However, we will still use them if the maximum
coefficient will not fit in a 'fractpel'.   Of course, we have little
choice at that point, but we haven't lost that much precision by
staying with integer arithmetic.  We have enough significant digits
so that
any error we introduce is less than one part in 2:sup/16/.
*/
 
               imax = MAX(ABS(*icxP), ABS(*icyP));
               if (imax < (fractpel) (1<<(FRACTBITS-1)) ) {
/*
At this point we know our integer approximations just do not have
enough significant digits for accuracy.  We will add FRACTBITS
significant digits to the coefficients (by multiplying them by
1<<FRACTBITS) and go to the "FP" form of the functions.  First, we
check to see if we have ANY significant digits at all (that is, if
imax == 0).  If we don't, we suspect that adding FRACTBITS digits
won't help, so we punt the whole thing.
*/
                       if (imax == 0) {
                               *fcnP = NULL;
                               return;
                       }
                       cx *= FRACTFLOAT;
                       cy *= FRACTFLOAT;
                       *icxP = cx;
                       *icyP = cy;
                       *fcnP = FPXYboth;
               }
               else
                       *fcnP = IXYboth;
       }
       else
               *fcnP = IXYboth;
/*
Now we check for special cases where one coefficient is zero (after
integer conversion):
*/
       if (*icxP == 0)
               *fcnP = (*fcnP == FPXYboth) ? FPYonly : IYonly;
       else if (*icyP == 0)
               *fcnP = (*fcnP == FPXYboth) ? FPXonly : IXonly;
}
/*
:h3.UnConvert() - Find User Coordinates From FractPoints
 
The interesting thing with this routine is that we avoid calculating
the matrix inverse of the device transformation until we really need
it, which is to say, until this routine is called for the first time
with a given coordinate space.
 
We also only calculate it only once.  If the inverted matrix is valid,
we don't calculate it; if not, we do.  We never expect matrices with
zero determinants, so by convention, we mark the matrix is invalid by
marking both X terms zero.
*/
 
void 
UnConvert(struct XYspace *S,      /* relevant coordinate space               */
	  struct fractpoint *pt,  /* device coordinates                      */
	  double *xp, double *yp) /* where to store resulting x,y            */
{
       double x,y;
 
       CoerceInverse(S);
       x = pt->x;
       y = pt->y;
       *xp = S->tofract.inverse[0][0] * x + S->tofract.inverse[1][0] * y;
       *yp = S->tofract.inverse[0][1] * x + S->tofract.inverse[1][1] * y;
}
 
/*
:h2.Transformations
*/
/*
:h3 id=xform.Xform() - Transform Object in X and Y
 
TYPE1IMAGER wants transformations of objects like paths to be identical
to transformations of spaces.  For example, if you scale a line(1,1)
by 10 it should yield the same result as generating the line(1,1) in
a coordinate space that has been scaled by 10.
 
We handle fonts by storing the accumulated transform, for example, SR
(accumulating on the right).  Then when we map the font through space TD,
for example, we multiply the accumulated font transform on the left by
the space transform on the right, yielding SRTD in this case.  We will
get the same result if we did S, then R, then T on the space and mapping
an unmodified font through that space.
*/
 
struct xobject *
t1_Xform(struct xobject *obj,   /* object to transform                       */
	 double M[2][2])        /* transformation matrix                     */
{
       if (obj == NULL)
               return(NULL);
 
       if (obj->type == FONTTYPE) {
               register struct font *F = (struct font *) obj;
 
               F = UniqueFont(F);
               return((struct xobject*)F);
       }
       if (obj->type == PICTURETYPE) {
/*
In the case of a picture, we choose both to update the picture's
transformation matrix and keep the handles up to date.
*/
               register struct picture *P = (struct picture *) obj;
               register struct segment *handles;  /* temporary path to transform handles */
 
               P = UniquePicture(P);
               handles = PathSegment(LINETYPE, P->origin.x, P->origin.y);
               handles = Join(handles,
                              PathSegment(LINETYPE, P->ending.x, P->ending.y) );
               handles = (struct segment *)Xform((struct xobject *) handles, M);
               P->origin = handles->dest;
               P->ending = handles->link->dest;
               KillPath(handles);
               return((struct xobject *)P);
       }
 
       if (ISPATHTYPE(obj->type)) {
               struct XYspace pseudo;  /* local temporary space              */
               PseudoSpace(&pseudo, M);
               return((struct xobject *) PathTransform((struct segment *)obj, 
						       &pseudo));
       }
 
 
       if (obj->type == SPACETYPE) {
               register struct XYspace *S = (struct XYspace *) obj;
 
/* replaced ISPERMANENT(S->flag) with S->references > 1 3-26-91 PNM */
               if (S->references > 1)
                       S = CopySpace(S);
               else
                       S->ID = NEXTID;
 
               MatrixMultiply(S->tofract.normal, M, S->tofract.normal);
               /*
               * mark inverted matrix invalid:
               */
               S->flag &= ~HASINVERSE(ON);
 
               FillOutFcns(S);
               return((struct xobject *) S);
       }
 
       return(ArgErr("Untransformable object", obj, obj));
}
 
/*
:h3.Transform() - Transform an Object
 
This is the external user's entry point.
*/
struct xobject *
t1_Transform(struct xobject *obj, 
	     double cxx, double cyx, /* 2x2 transform matrix elements        */
	     double cxy, double cyy) /* in row order                         */
{
       double M[2][2];
 
       M[0][0] = cxx;
       M[0][1] = cyx;
       M[1][0] = cxy;
       M[1][1] = cyy;
       ConsiderContext(obj, M);
       return(Xform(obj, M));
}
/*
:h3.Scale() - Special Case of Transform()
 
This is a user operator.
*/
 
struct xobject *
t1_Scale(struct xobject *obj,  /* object to scale                            */
	 double sx, double sy) /* scale factors in x and y                   */
{
       double M[2][2];

       M[0][0] = sx;
       M[1][1] = sy;
       M[1][0] = M[0][1] = 0.0;
       ConsiderContext(obj, M);
       return(Xform(obj, M));
}
 
/*
:h3 id=rotate.Rotate() - Special Case of Transform()
 
We special-case different settings of 'degrees' for performance
and accuracy within the DegreeSin() and DegreeCos() routines themselves.
*/
 
#ifdef notdef
struct xobject *
xiRotate(struct xobject *obj, /* object to be transformed                    */
	 double degrees)      /* degrees of COUNTER-clockwise rotation       */
{
       double M[2][2];
 
       M[0][0] = M[1][1] = DegreeCos(degrees);
       M[1][0] = - (M[0][1] = DegreeSin(degrees));
       ConsiderContext(obj, M);
       return(Xform(obj, M));
}
#endif
 
/*
:h3.PseudoSpace() - Build a Coordinate Space from a Matrix
 
Since we have built all this optimized code that, given an (x,y) and
a coordinate space, yield transformed (x,y), it seems a shame not to
use the same logic when we need to multiply an (x,y) by an arbitrary
matrix that is not (initially) part of a coordinate space.  This
subroutine takes the arbitrary matrix and builds a coordinate
space, with all its nifty function pointers.
*/
 
void 
PseudoSpace(struct XYspace *S, /* coordinate space structure to fill out     */
	    double M[2][2])    /* matrix that will become 'tofract.normal'   */
{
       S->type = SPACETYPE;
       S->flag = ISPERMANENT(ON) + ISIMMORTAL(ON);
       S->references = 2;   /* 3-26-91 added PNM  */
       S->tofract.normal[0][0] = M[0][0];
       S->tofract.normal[1][0] = M[1][0];
       S->tofract.normal[0][1] = M[0][1];
       S->tofract.normal[1][1] = M[1][1];
 
       FillOutFcns(S);
}
 
/*
:h2 id=matrixa.Matrix Arithmetic
 
Following the convention in Newman and Sproull, :hp1/Interactive
Computer Graphics/,
matrices are organized:
:xmp.
       | cxx   cyx |
       | cxy   cyy |
:exmp.
A point is horizontal, for example:
:xmp.
       [ x y ]
:exmp.
This means that:
:formula/x prime = cxx times x + cxy times y/
:formula/y prime = cyx times x + cyy times y/
I've seen the other convention, where transform matrices are
transposed, equally often in the literature.
*/
 
/*
:h3.MatrixMultiply() - Implements Multiplication of Two Matrices
 
Implements matrix multiplication, A * B = C.
 
To remind myself, matrix multiplication goes rows of A times columns
of B.
The output matrix may be the same as one of the input matrices.
*/
void 
MatrixMultiply(double A[2][2], double B[2][2], /* input matrices             */
	       double C[2][2])                 /* output matrix              */
{
       register double txx,txy,tyx,tyy;
 
       txx = A[0][0] * B[0][0] + A[0][1] * B[1][0];
       txy = A[1][0] * B[0][0] + A[1][1] * B[1][0];
       tyx = A[0][0] * B[0][1] + A[0][1] * B[1][1];
       tyy = A[1][0] * B[0][1] + A[1][1] * B[1][1];
 
       C[0][0] = txx;
       C[1][0] = txy;
       C[0][1] = tyx;
       C[1][1] = tyy;
}
/*
:h3.MatrixInvert() - Invert a Matrix
 
My reference for matrix inversion was :hp1/Elementary Linear Algebra/
by Paul C. Shields, Worth Publishers, Inc., 1968.
*/
void 
MatrixInvert(double M[2][2],      /* input matrix                            */
	     double Mprime[2][2]) /* output inverted matrix                  */
{
       register double D;    /* determinant of matrix M                      */
       register double txx,txy,tyx,tyy;
 
       txx = M[0][0];
       txy = M[1][0];
       tyx = M[0][1];
       tyy = M[1][1];
 
       D = M[1][1] * M[0][0] - M[1][0] * M[0][1];
       if (D == 0.0)
               Abort("MatrixInvert:  can't");
 
       Mprime[0][0] = tyy / D;
       Mprime[1][0] = -txy / D;
       Mprime[0][1] = -tyx / D;
       Mprime[1][1] = txx / D;
}
/*
:h2.Initialization, Queries, and Debug
*/
/*
:h3.InitSpaces() - Initialize Constant Spaces
 
For compatibility, we initialize a coordinate space called USER which
maps 72nds of an inch to pels on the default device.
*/
 
struct XYspace *USER = &identity;
 
void 
InitSpaces(void)
{
       IDENTITY->type = SPACETYPE;
       FillOutFcns(IDENTITY);
 
       contexts[NULLCONTEXT].normal[1][0]
             = contexts[NULLCONTEXT].normal[0][1]
             = contexts[NULLCONTEXT].inverse[1][0]
             = contexts[NULLCONTEXT].inverse[0][1] = 0.0;
       contexts[NULLCONTEXT].normal[0][0]
             = contexts[NULLCONTEXT].normal[1][1]
             = contexts[NULLCONTEXT].inverse[0][0]
             = contexts[NULLCONTEXT].inverse[1][1] = 1.0;
 
       USER->flag |= ISIMMORTAL(ON);
       CoerceInverse(USER);
}
/*
:h3.QuerySpace() - Returns the Transformation Matrix of a Space
 
Since the tofract matrix of an XYspace includes the scale factor
necessary to produce fractpel results (i.e., FRACTFLOAT), this
must be taken out before we return the matrix to the user.  Fortunately,
this is simple:  just multiply by the inverse of IDENTITY!
*/
 
void 
QuerySpace(struct XYspace *S,          /* space asked about                  */
	   double *cxxP, double *cyxP, /* where to put answer                */
	   double *cxyP, double *cyyP)
{
       double M[2][2];       /* temp matrix to build user's answer           */
 
       if (S->type != SPACETYPE) {
               ArgErr("QuerySpace: not a space", S, NULL);
               return;
       }
       MatrixMultiply(S->tofract.normal, IDENTITY->tofract.inverse, M);
       *cxxP = M[0][0];
       *cxyP = M[1][0];
       *cyxP = M[0][1];
       *cyyP = M[1][1];
}
 
/*
:h3.FormatFP() - Format a Fixed Point Pel
 
We format the pel as "dddd.XXXX", where XX's are hexidecimal digits,
and the dd's are decimal digits.  This might be a little confusing
mixing hexidecimal and decimal like that, but it is convenient
to use for debug.
 
We make sure we have N (FRACTBITS/4) digits past the decimal point.
*/
#define  FRACTMASK   ((1<<FRACTBITS)-1)  /* mask for fractional part         */
 
void 
FormatFP(char *string,         /* output string                              */
	 fractpel fpel)        /* fractional pel input                       */
{
       char temp[8];
       register char *s;
       register char *sign;
 
       if (fpel < 0) {
               sign = "-";
               fpel = -fpel;
       }
       else
               sign = "";
 
       sprintf(temp, "000%lx", fpel & FRACTMASK);
       s = temp + strlen(temp) - (FRACTBITS/4);
 
       sprintf(string, "%s%d.%sx", sign, (int)(fpel >> FRACTBITS), s);
}
 
/*
:h3.DumpSpace() - Display a Coordinate Space
*/
/*ARGSUSED*/
void 
DumpSpace(struct XYspace *S)
{
}