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
#include "erfa.h"
int eraTpors(double xi, double eta, double a, double b,
double *a01, double *b01, double *a02, double *b02)
/*
** - - - - - - - - -
** e r a T p o r s
** - - - - - - - - -
**
** In the tangent plane projection, given the rectangular coordinates
** of a star and its spherical coordinates, determine the spherical
** coordinates of the tangent point.
**
** Given:
** xi,eta double rectangular coordinates of star image (Note 2)
** a,b double star's spherical coordinates (Note 3)
**
** Returned:
** *a01,*b01 double tangent point's spherical coordinates, Soln. 1
** *a02,*b02 double tangent point's spherical coordinates, Soln. 2
**
** Returned (function value):
** int number of solutions:
** 0 = no solutions returned (Note 5)
** 1 = only the first solution is useful (Note 6)
** 2 = both solutions are useful (Note 6)
**
** Notes:
**
** 1) The tangent plane projection is also called the "gnomonic
** projection" and the "central projection".
**
** 2) The eta axis points due north in the adopted coordinate system.
** If the spherical coordinates are observed (RA,Dec), the tangent
** plane coordinates (xi,eta) are conventionally called the
** "standard coordinates". If the spherical coordinates are with
** respect to a right-handed triad, (xi,eta) are also right-handed.
** The units of (xi,eta) are, effectively, radians at the tangent
** point.
**
** 3) All angular arguments are in radians.
**
** 4) The angles a01 and a02 are returned in the range 0-2pi. The
** angles b01 and b02 are returned in the range +/-pi, but in the
** usual, non-pole-crossing, case, the range is +/-pi/2.
**
** 5) Cases where there is no solution can arise only near the poles.
** For example, it is clearly impossible for a star at the pole
** itself to have a non-zero xi value, and hence it is meaningless
** to ask where the tangent point would have to be to bring about
** this combination of xi and dec.
**
** 6) Also near the poles, cases can arise where there are two useful
** solutions. The return value indicates whether the second of the
** two solutions returned is useful; 1 indicates only one useful
** solution, the usual case.
**
** 7) The basis of the algorithm is to solve the spherical triangle PSC,
** where P is the north celestial pole, S is the star and C is the
** tangent point. The spherical coordinates of the tangent point are
** [a0,b0]; writing rho^2 = (xi^2+eta^2) and r^2 = (1+rho^2), side c
** is then (pi/2-b), side p is sqrt(xi^2+eta^2) and side s (to be
** found) is (pi/2-b0). Angle C is given by sin(C) = xi/rho and
** cos(C) = eta/rho. Angle P (to be found) is the longitude
** difference between star and tangent point (a-a0).
**
** 8) This function is a member of the following set:
**
** spherical vector solve for
**
** eraTpxes eraTpxev xi,eta
** eraTpsts eraTpstv star
** > eraTpors < eraTporv origin
**
** Called:
** eraAnp normalize angle into range 0 to 2pi
**
** References:
**
** Calabretta M.R. & Greisen, E.W., 2002, "Representations of
** celestial coordinates in FITS", Astron.Astrophys. 395, 1077
**
** Green, R.M., "Spherical Astronomy", Cambridge University Press,
** 1987, Chapter 13.
**
** This revision: 2018 January 2
**
** Copyright (C) 2013-2021, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
double xi2, r, sb, cb, rsb, rcb, w2, w, s, c;
xi2 = xi*xi;
r = sqrt(1.0 + xi2 + eta*eta);
sb = sin(b);
cb = cos(b);
rsb = r*sb;
rcb = r*cb;
w2 = rcb*rcb - xi2;
if ( w2 >= 0.0 ) {
w = sqrt(w2);
s = rsb - eta*w;
c = rsb*eta + w;
if ( xi == 0.0 && w == 0.0 ) w = 1.0;
*a01 = eraAnp(a - atan2(xi,w));
*b01 = atan2(s,c);
w = -w;
s = rsb - eta*w;
c = rsb*eta + w;
*a02 = eraAnp(a - atan2(xi,w));
*b02 = atan2(s,c);
return (fabs(rsb) < 1.0) ? 1 : 2;
} else {
return 0;
}
/* Finished. */
}
/*----------------------------------------------------------------------
**
**
** Copyright (C) 2013-2021, NumFOCUS Foundation.
** All rights reserved.
**
** This library is derived, with permission, from the International
** Astronomical Union's "Standards of Fundamental Astronomy" library,
** available from http://www.iausofa.org.
**
** The ERFA version is intended to retain identical functionality to
** the SOFA library, but made distinct through different function and
** file names, as set out in the SOFA license conditions. The SOFA
** original has a role as a reference standard for the IAU and IERS,
** and consequently redistribution is permitted only in its unaltered
** state. The ERFA version is not subject to this restriction and
** therefore can be included in distributions which do not support the
** concept of "read only" software.
**
** Although the intent is to replicate the SOFA API (other than
** replacement of prefix names) and results (with the exception of
** bugs; any that are discovered will be fixed), SOFA is not
** responsible for any errors found in this version of the library.
**
** If you wish to acknowledge the SOFA heritage, please acknowledge
** that you are using a library derived from SOFA, rather than SOFA
** itself.
**
**
** TERMS AND CONDITIONS
**
** Redistribution and use in source and binary forms, with or without
** modification, are permitted provided that the following conditions
** are met:
**
** 1 Redistributions of source code must retain the above copyright
** notice, this list of conditions and the following disclaimer.
**
** 2 Redistributions in binary form must reproduce the above copyright
** notice, this list of conditions and the following disclaimer in
** the documentation and/or other materials provided with the
** distribution.
**
** 3 Neither the name of the Standards Of Fundamental Astronomy Board,
** the International Astronomical Union nor the names of its
** contributors may be used to endorse or promote products derived
** from this software without specific prior written permission.
**
** THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
** "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
** LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
** FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
** COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
** INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
** BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
** LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
** CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
** LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
** ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
** POSSIBILITY OF SUCH DAMAGE.
**
*/