proj.c 81.8 KB
Newer Older
Emmanuel Bertin's avatar
Emmanuel Bertin committed
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
/*============================================================================
*
*   WCSLIB - an implementation of the FITS WCS proposal.
*   Copyright (C) 1995-1999, Mark Calabretta
*
*   This library is free software; you can redistribute it and/or modify it
*   under the terms of the GNU Library General Public License as published
*   by the Free Software Foundation; either version 2 of the License, or (at
*   your option) any later version.
*
*   This 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 Library
*   General Public License for more details.
*
*   You should have received a copy of the GNU Library General Public License
*   along with this library; if not, write to the Free Software Foundation,
*   Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
*
*   Correspondence concerning WCSLIB may be directed to:
*      Internet email: mcalabre@atnf.csiro.au
*      Postal address: Dr. Mark Calabretta,
*                      Australia Telescope National Facility,
*                      P.O. Box 76,
*                      Epping, NSW, 2121,
*                      AUSTRALIA
*
*=============================================================================
*
*   C implementation of the spherical map projections recognized by the FITS
*   "World Coordinate System" (WCS) convention.
*
*   Summary of routines
*   -------------------
*   Each projection is implemented via separate functions for the forward,
*   *fwd(), and reverse, *rev(), transformation.
*
*   Initialization routines, *set(), compute intermediate values from the
*   projection parameters but need not be called explicitly - see the
*   explanation of prj.flag below.
*
*      azpset azpfwd azprev   AZP: zenithal/azimuthal perspective
*      tanset tanfwd tanrev   TAN: gnomonic
*      sinset sinfwd sinrev   SIN: orthographic/synthesis
*      stgset stgfwd stgrev   STG: stereographic
*      arcset arcfwd arcrev   ARC: zenithal/azimuthal equidistant
*      zpnset zpnfwd zpnrev   ZPN: zenithal/azimuthal polynomial
*      zeaset zeafwd zearev   ZEA: zenithal/azimuthal equal area
*      airset airfwd airrev   AIR: Airy
*      cypset cypfwd cyprev   CYP: cylindrical perspective
*      carset carfwd carrev   CAR: Cartesian
*      merset merfwd merrev   MER: Mercator
*      ceaset ceafwd cearev   CEA: cylindrical equal area
*      copset copfwd coprev   COP: conic perspective
*      codset codfwd codrev   COD: conic equidistant
*      coeset coefwd coerev   COE: conic equal area
*      cooset coofwd coorev   COO: conic orthomorphic
*      bonset bonfwd bonrev   BON: Bonne
*      pcoset pcofwd pcorev   PCO: polyconic
*      glsset glsfwd glsrev   GLS: Sanson-Flamsteed (global sinusoidal)
*      parset parfwd parrev   PAR: parabolic
*      aitset aitfwd aitrev   AIT: Hammer-Aitoff
*      molset molfwd molrev   MOL: Mollweide
*      cscset cscfwd cscrev   CSC: COBE quadrilateralized spherical cube
*      qscset qscfwd qscrev   QSC: quadrilateralized spherical cube
*      tscset tscfwd tscrev   TSC: tangential spherical cube
*      tnxset tnxfwd tnxrev   TNX: IRAF's gnomonic
*
*
*   Initialization routine; *set()
*   ------------------------------
*   Initializes members of a prjprm data structure which hold intermediate
*   values.  Note that this routine need not be called directly; it will be
*   invoked by prjfwd() and prjrev() if the "flag" structure member is
*   anything other than a predefined magic value.
*
*   Given and/or returned:
*      prj      prjprm*  Projection parameters (see below).
*
*   Function return value:
*               int      Error status
*                           0: Success.
*                           1: Invalid projection parameters.
*
*   Forward transformation; *fwd()
*   -----------------------------
*   Compute (x,y) coordinates in the plane of projection from native spherical
*   coordinates (phi,theta).
*
*   Given:
*      phi,     const double
*      theta             Longitude and latitude of the projected point in
*                        native spherical coordinates, in degrees.
*
*   Given and returned:
*      prj      prjprm*  Projection parameters (see below).
*
*   Returned:
*      x,y      double*  Projected coordinates.
*
*   Function return value:
*               int      Error status
*                           0: Success.
*                           1: Invalid projection parameters.
*                           2: Invalid value of (phi,theta).
*
*   Reverse transformation; *rev()
*   -----------------------------
*   Compute native spherical coordinates (phi,theta) from (x,y) coordinates in
*   the plane of projection.
*
*   Given:
*      x,y      const double
*                        Projected coordinates.
*
*   Given and returned:
*      prj      prjprm*  Projection parameters (see below).
*
*   Returned:
*      phi,     double*  Longitude and latitude of the projected point in
*      theta             native spherical coordinates, in degrees.
*
*   Function return value:
*               int      Error status
*                           0: Success.
*                           1: Invalid projection parameters.
*                           2: Invalid value of (x,y).
*
*   Projection parameters
*   ---------------------
*   The prjprm struct consists of the following:
*
*      int flag
*         This flag must be set to zero whenever any of p[10] or r0 are set
*         or changed.  This signals the initialization routine to recompute
*         intermediaries.  flag may also be set to -1 to disable strict bounds
*         checking for the AZP, TAN, SIN, ZPN, and COP projections.
*      double r0
*         r0; The radius of the generating sphere for the projection, a linear
*         scaling parameter.  If this is zero, it will be reset to the default
*         value of 180/pi (the value for FITS WCS).
*      double p[10]
*         The first 10 elements contain projection parameters which correspond
*         to the PROJPn keywords in FITS, so p[0] is PROJP0, and p[9] is
*         PROJP9.  Many projections use p[1] (PROJP1) and some also use p[2]
*         (PROJP2).  ZPN is the only projection which uses any of the others.
*
*   The remaining members of the prjprm struct are maintained by the
*   initialization routines and should not be modified.  This is done for the
*   sake of efficiency and to allow an arbitrary number of contexts to be
*   maintained simultaneously.
*
*      int n
*      double w[10]
*         Intermediate values derived from the projection parameters.
*
*   Usage of the p[] array as it applies to each projection is described in
*   the prologue to each trio of projection routines.
*
*   Argument checking
*   -----------------
*   Forward routines:
*
*      The values of phi and theta (the native longitude and latitude)
*      normally lie in the range [-180,180] for phi, and [-90,90] for theta.
*      However, all forward projections will accept any value of phi and will
*      not normalize it.
*
*      The forward projection routines do not explicitly check that theta lies
*      within the range [-90,90].  They do check for any value of theta which
*      produces an invalid argument to the projection equations (e.g. leading
*      to division by zero).  The forward routines for AZP, TAN, SIN, ZPN, and
*      COP also return error 2 if (phi,theta) corresponds to the overlapped
*      (far) side of the projection but also return the corresponding value of
*      (x,y).  This strict bounds checking may be relaxed by setting prj->flag
*      to -1 (rather than 0) when these projections are initialized.
*
*   Reverse routines:
*
*      Error checking on the projected coordinates (x,y) is limited to that
*      required to ascertain whether a solution exists.  Where a solution does
*      exist no check is made that the value of phi and theta obtained lie
*      within the ranges [-180,180] for phi, and [-90,90] for theta.
*
*   Accuracy
*   --------
*   Closure to a precision of at least 1E-10 degree of longitude and latitude
*   has been verified for typical projection parameters on the 1 degree grid
*   of native longitude and latitude (to within 5 degrees of any latitude
*   where the projection may diverge).
*
*   Author: Mark Calabretta, Australia Telescope National Facility
*   IRAF's TNX added by E.Bertin 2000/08/23
Emmanuel Bertin's avatar
Emmanuel Bertin committed
194
195
*   Behaviour of Cartesian-like projections modified by E.Bertin 2005/08/29
*   $Id: proj.c,v 1.1.1.1 2008/01/10 18:00:00 bertin Exp $
Emmanuel Bertin's avatar
Emmanuel Bertin committed
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
996
997
998
999
1000
*===========================================================================*/

#ifdef HAVE_CONFIG_H
#include	"config.h"
#endif

#ifdef HAVE_MATHIMF_H
#include <mathimf.h>
#else
#include <math.h>
#endif
#include <stdio.h>
#include <stdlib.h>
#include "poly.h"
#include "proj.h"
#include "tnx.h"
#include "wcsmath.h"
#include "wcstrig.h"

/* Map error number to error message for each function. */
const char *prjset_errmsg[] = {
   0,
   "Invalid projection parameters"};

const char *prjfwd_errmsg[] = {
   0,
   "Invalid projection parameters",
   "Invalid value of (phi,theta)"};

const char *prjrev_errmsg[] = {
   0,
   "Invalid projection parameters",
   "Invalid value of (x,y)"};

#define wcs_copysign(X, Y) ((Y) < 0.0 ? -fabs(X) : fabs(X))

/*============================================================================
*   AZP: zenithal/azimuthal perspective projection.
*
*   Given:
*      prj->p[1]   AZP distance parameter, mu in units of r0.
*
*   Given and/or returned:
*      prj->r0     r0; reset to 180/pi if 0.
*      prj->w[0]   r0*(mu+1)
*      prj->w[1]   1/prj->w[0]
*      prj->w[2]   Boundary parameter, -mu    for |mu| <= 1,
*                                      -1/mu  for |mu| >= 1.
*===========================================================================*/

int azpset(prj)

struct prjprm *prj;

{
   if (prj->r0 == 0.0) prj->r0 = R2D;

   prj->w[0] = prj->r0*(prj->p[1] + 1.0);
   if (prj->w[0] == 0.0) {
      return 1;
   }

   prj->w[1] = 1.0/prj->w[0];
   if (fabs(prj->p[1]) <= 1.0) {
      prj->w[2] = -prj->p[1];
   } else {
      prj->w[2] = -1.0/prj->p[1];
   }

   if (prj->flag == -1) {
      prj->flag = -PRJSET;
   } else {
      prj->flag = PRJSET;
   }

   return 0;
}

/*--------------------------------------------------------------------------*/

int azpfwd(phi, theta, prj, x, y)

const double phi, theta;
struct prjprm *prj;
double *x, *y;

{
   double r, s, sthe;

   if (abs(prj->flag) != PRJSET) {
      if (azpset(prj)) return 1;
   }

   sthe = wcs_sind(theta);

   s = prj->p[1] + sthe;
   if (s == 0.0) {
      return 2;
   }

   r =  prj->w[0]*wcs_cosd(theta)/s;
   *x =  r*wcs_sind(phi);
   *y = -r*wcs_cosd(phi);

   if (prj->flag == PRJSET && sthe < prj->w[2]) {
      return 2;
   }

   return 0;
}

/*--------------------------------------------------------------------------*/

int azprev(x, y, prj, phi, theta)

const double x, y;
struct prjprm *prj;
double *phi, *theta;

{
   double r, rho, s;
   const double tol = 1.0e-13;

   if (abs(prj->flag) != PRJSET) {
      if (azpset(prj)) return 1;
   }

   r = sqrt(x*x + y*y);
   if (r == 0.0) {
      *phi = 0.0;
   } else {
      *phi = wcs_atan2d(x, -y);
   }

   rho = r*prj->w[1];
   s = rho*prj->p[1]/sqrt(rho*rho+1.0);
   if (fabs(s) > 1.0) {
      if (fabs(s) > 1.0+tol) {
         return 2;
      }
      *theta = wcs_atan2d(1.0,rho) - wcs_copysign(90.0,s);
   } else {
      *theta = wcs_atan2d(1.0,rho) - wcs_asind(s);
   }

   return 0;
}

/*============================================================================
*   TAN: gnomonic projection.
*
*   Given and/or returned:
*      prj->r0     r0; reset to 180/pi if 0.
*===========================================================================*/

int tanset(prj)

struct prjprm *prj;

{
   int	k;

   if (prj->r0 == 0.0) prj->r0 = R2D;

   if (prj->flag == -1) {
      prj->flag = -PRJSET;
   } else {
      prj->flag = PRJSET;
   } 

   for (k = 99; k >= 0 && prj->p[k] == 0.0 && prj->p[k+100] == 0.0; k--);
   if (k < 0)
     k = 0;

   prj->n = k;

   return 0;
}

/*--------------------------------------------------------------------------*/

int tanfwd(phi, theta, prj, x, y)

const double phi, theta;
struct prjprm *prj;
double *x, *y;

{
   double r, s, xp[2];

   if (abs(prj->flag) != PRJSET) {
      if(tanset(prj)) return 1;
   }

   s = wcs_sind(theta);
   if (s == 0.0) return 2;

   r =  prj->r0*wcs_cosd(theta)/s;
   xp[0] =  r*wcs_sind(phi);
   xp[1] = -r*wcs_cosd(phi);
   *x = prj->inv_x? poly_func(prj->inv_x, xp) : xp[0];
   *y = prj->inv_y? poly_func(prj->inv_y, xp) : xp[1];

   if (prj->flag == PRJSET && s < 0.0) {
      return 2;
   }

   return 0;
}

/*--------------------------------------------------------------------------*/

int tanrev(x, y, prj, phi, theta)

const double x, y;
struct prjprm *prj;
double *phi, *theta;

{
   double	xp,yp, rp;

   if (abs(prj->flag) != PRJSET) {
      if (tanset(prj)) return 1;
   }

   if (prj->n)
     raw_to_pv(prj, x,y, &xp, &yp);
   else
     {
     xp = x;
     yp = y;
     }
   rp = sqrt(xp*xp+yp*yp);
   if (rp == 0.0) {
      *phi = 0.0;
   } else {
      *phi = wcs_atan2d(xp, -yp);
   }
   *theta = wcs_atan2d(prj->r0, rp);

   return 0;
}

/*============================================================================
*   SIN: orthographic/synthesis projection.
*
*   Given:
*      prj->p[1:2] SIN obliqueness parameters, alpha and beta.
*
*   Given and/or returned:
*      prj->r0     r0; reset to 180/pi if 0.
*      prj->w[0]   1/r0
*      prj->w[1]   alpha**2 + beta**2
*      prj->w[2]   2*(alpha**2 + beta**2)
*      prj->w[3]   2*(alpha**2 + beta**2 + 1)
*      prj->w[4]   alpha**2 + beta**2 - 1
*===========================================================================*/

int sinset(prj)

struct prjprm *prj;

{
   if (prj->r0 == 0.0) prj->r0 = R2D;

   prj->w[0] = 1.0/prj->r0;
   prj->w[1] = prj->p[1]*prj->p[1] + prj->p[2]*prj->p[2];
   prj->w[2] = 2.0*prj->w[1];
   prj->w[3] = prj->w[2] + 2.0;
   prj->w[4] = prj->w[1] - 1.0;

   if (prj->flag == -1) {
      prj->flag = -PRJSET;
   } else {
      prj->flag = PRJSET;
   }

   return 0;
}

/*--------------------------------------------------------------------------*/

int sinfwd(phi, theta, prj, x, y)

const double phi, theta;
struct prjprm *prj;
double *x, *y;

{
   double cphi, cthe, sphi, t, z;

   if (abs(prj->flag) != PRJSET) {
      if (sinset(prj)) return 1;
   }

   t = (90.0 - fabs(theta))*D2R;
   if (t < 1.0e-5) {
      if (theta > 0.0) {
         z = -t*t/2.0;
      } else {
         z = -2.0 + t*t/2.0;
      }
      cthe = t;
   } else {
      z =  wcs_sind(theta) - 1.0;
      cthe = wcs_cosd(theta);
   }

   cphi = wcs_cosd(phi);
   sphi = wcs_sind(phi);
   *x =  prj->r0*(cthe*sphi + prj->p[1]*z);
   *y = -prj->r0*(cthe*cphi + prj->p[2]*z);
   /* Validate this solution. */
   if (prj->flag == PRJSET) {
      if (prj->w[1] == 0.0) {
         /* Orthographic projection. */
         if (theta < 0.0) {
            return 2;
         }
      } else {
         /* "Synthesis" projection. */
         t = wcs_atand(prj->p[1]*sphi + prj->p[2]*cphi);
         if (theta < t) {
            return 2;
         }
      }
   }

   return 0;
}

/*--------------------------------------------------------------------------*/

int sinrev (x, y, prj, phi, theta)

const double x, y;
struct prjprm *prj;
double *phi, *theta;

{
   const double tol = 1.0e-13;
   double a, b, c, d, r2, sth, sth1, sth2, sxy, x0, xp, y0, yp, z;

   if (abs(prj->flag) != PRJSET) {
      if (sinset(prj)) return 1;
   }

   /* Compute intermediaries. */
   x0 = x*prj->w[0];
   y0 = y*prj->w[0];
   r2 = x0*x0 + y0*y0;

   if (prj->w[1] == 0.0) {
      /* Orthographic projection. */
      if (r2 != 0.0) {
         *phi = wcs_atan2d(x0, -y0);
      } else {
         *phi = 0.0;
      }

      if (r2 < 0.5) {
         *theta = wcs_acosd(sqrt(r2));
      } else if (r2 <= 1.0) {
         *theta = wcs_asind(sqrt(1.0 - r2));
      } else {
         return 2;
      }

   } else {
      /* "Synthesis" projection. */
      if (r2 < 1.0e-10) {
         /* Use small angle formula. */
         z = -r2/2.0;
         *theta = 90.0 - R2D*sqrt(r2/(1.0 - x0*prj->p[1] + y0*prj->p[2]));

      } else {
         sxy = 2.0*(prj->p[1]*x0 - prj->p[2]*y0);

         a = prj->w[3];
         b = -(sxy + prj->w[2]);
         c = r2 + sxy + prj->w[4];
         d = b*b - 2.0*a*c;

         /* Check for a solution. */
         if (d < 0.0) {
            return 2;
         }
         d = sqrt(d);

         /* Choose solution closest to pole. */
         sth1 = (-b + d)/a;
         sth2 = (-b - d)/a;
         sth = (sth1>sth2) ? sth1 : sth2;
         if (sth > 1.0) {
            if (sth-1.0 < tol) {
               sth = 1.0;
            } else {
               sth = (sth1<sth2) ? sth1 : sth2;
            }
         }
         if (sth > 1.0 || sth < -1.0) {
            return 2;
         }

         *theta = wcs_asind(sth);
         z = sth - 1.0;
      }

      xp = -y0 - prj->p[2]*z;
      yp =  x0 - prj->p[1]*z;
      if (xp == 0.0 && yp == 0.0) {
         *phi = 0.0;
      } else {
         *phi = wcs_atan2d(yp,xp);
      }
   }

   return 0;
}

/*============================================================================
*   STG: stereographic projection.
*
*   Given and/or returned:
*      prj->r0     r0; reset to 180/pi if 0.
*      prj->w[0]   2*r0
*      prj->w[1]   1/(2*r0)
*===========================================================================*/

int stgset(prj)

struct prjprm *prj;

{
   if (prj->r0 == 0.0) {
      prj->r0 = R2D;
      prj->w[0] = 360.0/PI;
      prj->w[1] = PI/360.0;
   } else {
      prj->w[0] = 2.0*prj->r0;
      prj->w[1] = 1.0/prj->w[0];
   }

   prj->flag = PRJSET;
   return 0;
}

/*--------------------------------------------------------------------------*/

int stgfwd(phi, theta, prj, x, y)

const double phi, theta;
struct prjprm *prj;
double *x, *y;

{
   double r, s;

   if (prj->flag != PRJSET) {
      if (stgset(prj)) return 1;
   }

   s = 1.0 + wcs_sind(theta);
   if (s == 0.0) {
      return 2;
   }

   r =  prj->w[0]*wcs_cosd(theta)/s;
   *x =  r*wcs_sind(phi);
   *y = -r*wcs_cosd(phi);

   return 0;
}

/*--------------------------------------------------------------------------*/

int stgrev(x, y, prj, phi, theta)

const double x, y;
struct prjprm *prj;
double *phi, *theta;

{
   double r;

   if (prj->flag != PRJSET) {
      if (stgset(prj)) return 1;
   }

   r = sqrt(x*x + y*y);
   if (r == 0.0) {
      *phi = 0.0;
   } else {
      *phi = wcs_atan2d(x, -y);
   }
   *theta = 90.0 - 2.0*wcs_atand(r*prj->w[1]);

   return 0;
}

/*============================================================================
*   ARC: zenithal/azimuthal equidistant projection.
*
*   Given and/or returned:
*      prj->r0     r0; reset to 180/pi if 0.
*      prj->w[0]   r0*(pi/180)
*      prj->w[1]   (180/pi)/r0
*===========================================================================*/

int arcset(prj)

struct prjprm *prj;

{
   if (prj->r0 == 0.0) {
      prj->r0 = R2D;
      prj->w[0] = 1.0;
      prj->w[1] = 1.0;
   } else {
      prj->w[0] = prj->r0*D2R;
      prj->w[1] = 1.0/prj->w[0];
   }

   prj->flag = PRJSET;

   return 0;
}

/*--------------------------------------------------------------------------*/

int arcfwd(phi, theta, prj, x, y)

const double phi, theta;
struct prjprm *prj;
double *x, *y;

{
   double r;

   if (prj->flag != PRJSET) {
      if (arcset(prj)) return 1;
   }

   r =  prj->w[0]*(90.0 - theta);
   *x =  r*wcs_sind(phi);
   *y = -r*wcs_cosd(phi);

   return 0;
}

/*--------------------------------------------------------------------------*/

int arcrev(x, y, prj, phi, theta)

const double x, y;
struct prjprm *prj;
double *phi, *theta;

{
   double r;

   if (prj->flag != PRJSET) {
      if (arcset(prj)) return 1;
   }

   r = sqrt(x*x + y*y);
   if (r == 0.0) {
      *phi = 0.0;
   } else {
      *phi = wcs_atan2d(x, -y);
   }
   *theta = 90.0 - r*prj->w[1];

   return 0;
}

/*============================================================================
*   ZPN: zenithal/azimuthal polynomial projection.
*
*   Given:
*      prj->p[0:99] Polynomial coefficients.
*
*   Given and/or returned:
*      prj->r0     r0; reset to 180/pi if 0.
*      prj->n      Degree of the polynomial, N.
*      prj->w[0]   Co-latitude of the first point of inflection (N > 2).
*      prj->w[1]   Radius of the first point of inflection (N > 2).
*===========================================================================*/

int zpnset(prj)

struct prjprm *prj;

{
   int   i, j, k;
   double d, d1, d2, r, zd, zd1, zd2;
   const double tol = 1.0e-13;

   if (prj->r0 == 0.0) prj->r0 = R2D;

   /* Find the highest non-zero coefficient. */
   for (k = 99; k >= 0 && prj->p[k] == 0.0; k--);
   if (k < 0) return 1;

   prj->n = k;

   if (k >= 3) {
      /* Find the point of inflection closest to the pole. */
      zd1 = 0.0;
      d1  = prj->p[1];
      if (d1 <= 0.0) {
         return 1;
      }

      /* Find the point where the derivative first goes negative. */
      for (i = 0; i < 180; i++) {
         zd2 = i*D2R;
         d2  = 0.0;
         for (j = k; j > 0; j--) {
            d2 = d2*zd2 + j*prj->p[j];
         }

         if (d2 <= 0.0) break;
         zd1 = zd2;
         d1  = d2;
      }

      if (i == 180) {
         /* No negative derivative -> no point of inflection. */
         zd = PI;
      } else {
         /* Find where the derivative is zero. */
         for (i = 1; i <= 10; i++) {
            zd = zd1 - d1*(zd2-zd1)/(d2-d1);

            d = 0.0;
            for (j = k; j > 0; j--) {
               d = d*zd + j*prj->p[j];
            }

            if (fabs(d) < tol) break;

            if (d < 0.0) {
               zd2 = zd;
               d2  = d;
            } else {
               zd1 = zd;
               d1  = d;
            }
         }
      }

      r = 0.0;
      for (j = k; j >= 0; j--) {
         r = r*zd + prj->p[j];
      }
      prj->w[0] = zd;
      prj->w[1] = r;
   }

   if (prj->flag == -1) {
      prj->flag = -PRJSET;
   } else {
      prj->flag = PRJSET;
   }

   return 0;
}

/*--------------------------------------------------------------------------*/

int zpnfwd(phi, theta, prj, x, y)

const double phi, theta;
struct prjprm *prj;
double *x, *y;

{
   int   j;
   double r, s;

   if (abs(prj->flag) != PRJSET) {
      if (zpnset(prj)) return 1;
   }

   s = (90.0 - theta)*D2R;
   r = 0.0;
   for (j = prj->n; j >= 0; j--) {
      r = r*s + prj->p[j];
   }
   r = prj->r0*r;

   *x =  r*wcs_sind(phi);
   *y = -r*wcs_cosd(phi);

   if (prj->flag == PRJSET && s > prj->w[0]) {
      return 2;
   }

   return 0;
}

/*--------------------------------------------------------------------------*/

int zpnrev(x, y, prj, phi, theta)

const double x, y;
struct prjprm *prj;
double *phi, *theta;

{
   int   i, j, k;
   double a, b, c, d, lambda, r, r1, r2, rt, zd, zd1, zd2;
   const double tol = 1.0e-13;

   if (abs(prj->flag) != PRJSET) {
      if (zpnset(prj)) return 1;
   }

   k = prj->n;

   r = sqrt(x*x + y*y)/prj->r0;

   if (k < 1) {
      /* Constant - no solution. */
      return 1;
   } else if (k == 1) {
      /* Linear. */
      zd = (r - prj->p[0])/prj->p[1];
   } else if (k == 2) {
      /* Quadratic. */
      a = prj->p[2];
      b = prj->p[1];
      c = prj->p[0] - r;

      d = b*b - 4.0*a*c;
      if (d < 0.0) {
         return 2;
      }
      d = sqrt(d);

      /* Choose solution closest to pole. */
      zd1 = (-b + d)/(2.0*a);
      zd2 = (-b - d)/(2.0*a);
      zd  = (zd1<zd2) ? zd1 : zd2;
      if (zd < -tol) zd = (zd1>zd2) ? zd1 : zd2;
      if (zd < 0.0) {
         if (zd < -tol) {
            return 2;
         }
         zd = 0.0;
      } else if (zd > PI) {
         if (zd > PI+tol) {
            return 2;
         }
         zd = PI;
      }
   } else {
      /* Higher order - solve iteratively. */
      zd1 = 0.0;
      r1  = prj->p[0];
      zd2 = prj->w[0];
      r2  = prj->w[1];

      if (r < r1) {
         if (r < r1-tol) {
            return 2;
         }
         zd = zd1;
      } else if (r > r2) {
         if (r > r2+tol) {
            return 2;
         }
         zd = zd2;
      } else {
         /* Disect the interval. */
         for (j = 0; j < 100; j++) {
            lambda = (r2 - r)/(r2 - r1);
            if (lambda < 0.1) {
               lambda = 0.1;
            } else if (lambda > 0.9) {
               lambda = 0.9;
            }

            zd = zd2 - lambda*(zd2 - zd1);

            rt = 0.0;
            for (i = k; i >= 0; i--) {
                rt = (rt * zd) + prj->p[i];
            }

            if (rt < r) {
                if (r-rt < tol) break;
                r1 = rt;
                zd1 = zd;
            } else {
                if (rt-r < tol) break;
                r2 = rt;
                zd2 = zd;
            }

            if (fabs(zd2-zd1) < tol) break;
         }
      }
   }
For faster browsing, not all history is shown. View entire blame