GeographicLib 1.52
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AlbersEqualArea.cpp
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1/**
2 * \file AlbersEqualArea.cpp
3 * \brief Implementation for GeographicLib::AlbersEqualArea class
4 *
5 * Copyright (c) Charles Karney (2010-2021) <charles@karney.com> and licensed
6 * under the MIT/X11 License. For more information, see
7 * https://geographiclib.sourceforge.io/
8 **********************************************************************/
9
11
12#if defined(_MSC_VER)
13// Squelch warnings about constant conditional expressions
14# pragma warning (disable: 4127)
15#endif
16
17namespace GeographicLib {
18
19 using namespace std;
20
21 AlbersEqualArea::AlbersEqualArea(real a, real f, real stdlat, real k0)
22 : eps_(numeric_limits<real>::epsilon())
23 , epsx_(Math::sq(eps_))
24 , epsx2_(Math::sq(epsx_))
25 , tol_(sqrt(eps_))
26 , tol0_(tol_ * sqrt(sqrt(eps_)))
27 , _a(a)
28 , _f(f)
29 , _fm(1 - _f)
30 , _e2(_f * (2 - _f))
31 , _e(sqrt(abs(_e2)))
32 , _e2m(1 - _e2)
33 , _qZ(1 + _e2m * atanhee(real(1)))
34 , _qx(_qZ / ( 2 * _e2m ))
35 {
36 if (!(isfinite(_a) && _a > 0))
37 throw GeographicErr("Equatorial radius is not positive");
38 if (!(isfinite(_f) && _f < 1))
39 throw GeographicErr("Polar semi-axis is not positive");
40 if (!(isfinite(k0) && k0 > 0))
41 throw GeographicErr("Scale is not positive");
42 if (!(abs(stdlat) <= 90))
43 throw GeographicErr("Standard latitude not in [-90d, 90d]");
44 real sphi, cphi;
45 Math::sincosd(stdlat, sphi, cphi);
46 Init(sphi, cphi, sphi, cphi, k0);
47 }
48
49 AlbersEqualArea::AlbersEqualArea(real a, real f, real stdlat1, real stdlat2,
50 real k1)
51 : eps_(numeric_limits<real>::epsilon())
52 , epsx_(Math::sq(eps_))
53 , epsx2_(Math::sq(epsx_))
54 , tol_(sqrt(eps_))
55 , tol0_(tol_ * sqrt(sqrt(eps_)))
56 , _a(a)
57 , _f(f)
58 , _fm(1 - _f)
59 , _e2(_f * (2 - _f))
60 , _e(sqrt(abs(_e2)))
61 , _e2m(1 - _e2)
62 , _qZ(1 + _e2m * atanhee(real(1)))
63 , _qx(_qZ / ( 2 * _e2m ))
64 {
65 if (!(isfinite(_a) && _a > 0))
66 throw GeographicErr("Equatorial radius is not positive");
67 if (!(isfinite(_f) && _f < 1))
68 throw GeographicErr("Polar semi-axis is not positive");
69 if (!(isfinite(k1) && k1 > 0))
70 throw GeographicErr("Scale is not positive");
71 if (!(abs(stdlat1) <= 90))
72 throw GeographicErr("Standard latitude 1 not in [-90d, 90d]");
73 if (!(abs(stdlat2) <= 90))
74 throw GeographicErr("Standard latitude 2 not in [-90d, 90d]");
75 real sphi1, cphi1, sphi2, cphi2;
76 Math::sincosd(stdlat1, sphi1, cphi1);
77 Math::sincosd(stdlat2, sphi2, cphi2);
78 Init(sphi1, cphi1, sphi2, cphi2, k1);
79 }
80
82 real sinlat1, real coslat1,
83 real sinlat2, real coslat2,
84 real k1)
85 : eps_(numeric_limits<real>::epsilon())
86 , epsx_(Math::sq(eps_))
87 , epsx2_(Math::sq(epsx_))
88 , tol_(sqrt(eps_))
89 , tol0_(tol_ * sqrt(sqrt(eps_)))
90 , _a(a)
91 , _f(f)
92 , _fm(1 - _f)
93 , _e2(_f * (2 - _f))
94 , _e(sqrt(abs(_e2)))
95 , _e2m(1 - _e2)
96 , _qZ(1 + _e2m * atanhee(real(1)))
97 , _qx(_qZ / ( 2 * _e2m ))
98 {
99 if (!(isfinite(_a) && _a > 0))
100 throw GeographicErr("Equatorial radius is not positive");
101 if (!(isfinite(_f) && _f < 1))
102 throw GeographicErr("Polar semi-axis is not positive");
103 if (!(isfinite(k1) && k1 > 0))
104 throw GeographicErr("Scale is not positive");
105 if (!(coslat1 >= 0))
106 throw GeographicErr("Standard latitude 1 not in [-90d, 90d]");
107 if (!(coslat2 >= 0))
108 throw GeographicErr("Standard latitude 2 not in [-90d, 90d]");
109 if (!(abs(sinlat1) <= 1 && coslat1 <= 1) || (coslat1 == 0 && sinlat1 == 0))
110 throw GeographicErr("Bad sine/cosine of standard latitude 1");
111 if (!(abs(sinlat2) <= 1 && coslat2 <= 1) || (coslat2 == 0 && sinlat2 == 0))
112 throw GeographicErr("Bad sine/cosine of standard latitude 2");
113 if (coslat1 == 0 && coslat2 == 0 && sinlat1 * sinlat2 <= 0)
114 throw GeographicErr
115 ("Standard latitudes cannot be opposite poles");
116 Init(sinlat1, coslat1, sinlat2, coslat2, k1);
117 }
118
119 void AlbersEqualArea::Init(real sphi1, real cphi1,
120 real sphi2, real cphi2, real k1) {
121 {
122 real r;
123 r = hypot(sphi1, cphi1);
124 sphi1 /= r; cphi1 /= r;
125 r = hypot(sphi2, cphi2);
126 sphi2 /= r; cphi2 /= r;
127 }
128 bool polar = (cphi1 == 0);
129 cphi1 = max(epsx_, cphi1); // Avoid singularities at poles
130 cphi2 = max(epsx_, cphi2);
131 // Determine hemisphere of tangent latitude
132 _sign = sphi1 + sphi2 >= 0 ? 1 : -1;
133 // Internally work with tangent latitude positive
134 sphi1 *= _sign; sphi2 *= _sign;
135 if (sphi1 > sphi2) {
136 swap(sphi1, sphi2); swap(cphi1, cphi2); // Make phi1 < phi2
137 }
138 real
139 tphi1 = sphi1/cphi1, tphi2 = sphi2/cphi2;
140
141 // q = (1-e^2)*(sphi/(1-e^2*sphi^2) - atanhee(sphi))
142 // qZ = q(pi/2) = (1 + (1-e^2)*atanhee(1))
143 // atanhee(x) = atanh(e*x)/e
144 // q = sxi * qZ
145 // dq/dphi = 2*(1-e^2)*cphi/(1-e^2*sphi^2)^2
146 //
147 // n = (m1^2-m2^2)/(q2-q1) -> sin(phi0) for phi1, phi2 -> phi0
148 // C = m1^2 + n*q1 = (m1^2*q2-m2^2*q1)/(q2-q1)
149 // let
150 // rho(pi/2)/rho(-pi/2) = (1-s)/(1+s)
151 // s = n*qZ/C
152 // = qZ * (m1^2-m2^2)/(m1^2*q2-m2^2*q1)
153 // = qZ * (scbet2^2 - scbet1^2)/(scbet2^2*q2 - scbet1^2*q1)
154 // = (scbet2^2 - scbet1^2)/(scbet2^2*sxi2 - scbet1^2*sxi1)
155 // = (tbet2^2 - tbet1^2)/(scbet2^2*sxi2 - scbet1^2*sxi1)
156 // 1-s = -((1-sxi2)*scbet2^2 - (1-sxi1)*scbet1^2)/
157 // (scbet2^2*sxi2 - scbet1^2*sxi1)
158 //
159 // Define phi0 to give same value of s, i.e.,
160 // s = sphi0 * qZ / (m0^2 + sphi0*q0)
161 // = sphi0 * scbet0^2 / (1/qZ + sphi0 * scbet0^2 * sxi0)
162
163 real tphi0, C;
164 if (polar || tphi1 == tphi2) {
165 tphi0 = tphi2;
166 C = 1; // ignored
167 } else {
168 real
169 tbet1 = _fm * tphi1, scbet12 = 1 + Math::sq(tbet1),
170 tbet2 = _fm * tphi2, scbet22 = 1 + Math::sq(tbet2),
171 txi1 = txif(tphi1), cxi1 = 1/hyp(txi1), sxi1 = txi1 * cxi1,
172 txi2 = txif(tphi2), cxi2 = 1/hyp(txi2), sxi2 = txi2 * cxi2,
173 dtbet2 = _fm * (tbet1 + tbet2),
174 es1 = 1 - _e2 * Math::sq(sphi1), es2 = 1 - _e2 * Math::sq(sphi2),
175 /*
176 dsxi = ( (_e2 * sq(sphi2 + sphi1) + es2 + es1) / (2 * es2 * es1) +
177 Datanhee(sphi2, sphi1) ) * Dsn(tphi2, tphi1, sphi2, sphi1) /
178 ( 2 * _qx ),
179 */
180 dsxi = ( (1 + _e2 * sphi1 * sphi2) / (es2 * es1) +
181 Datanhee(sphi2, sphi1) ) * Dsn(tphi2, tphi1, sphi2, sphi1) /
182 ( 2 * _qx ),
183 den = (sxi2 + sxi1) * dtbet2 + (scbet22 + scbet12) * dsxi,
184 // s = (sq(tbet2) - sq(tbet1)) / (scbet22*sxi2 - scbet12*sxi1)
185 s = 2 * dtbet2 / den,
186 // 1-s = -(sq(scbet2)*(1-sxi2) - sq(scbet1)*(1-sxi1)) /
187 // (scbet22*sxi2 - scbet12*sxi1)
188 // Write
189 // sq(scbet)*(1-sxi) = sq(scbet)*(1-sphi) * (1-sxi)/(1-sphi)
190 sm1 = -Dsn(tphi2, tphi1, sphi2, sphi1) *
191 ( -( ((sphi2 <= 0 ? (1 - sxi2) / (1 - sphi2) :
192 Math::sq(cxi2/cphi2) * (1 + sphi2) / (1 + sxi2)) +
193 (sphi1 <= 0 ? (1 - sxi1) / (1 - sphi1) :
194 Math::sq(cxi1/cphi1) * (1 + sphi1) / (1 + sxi1))) ) *
195 (1 + _e2 * (sphi1 + sphi2 + sphi1 * sphi2)) /
196 (1 + (sphi1 + sphi2 + sphi1 * sphi2)) +
197 (scbet22 * (sphi2 <= 0 ? 1 - sphi2 :
198 Math::sq(cphi2) / ( 1 + sphi2)) +
199 scbet12 * (sphi1 <= 0 ? 1 - sphi1 : Math::sq(cphi1) / ( 1 + sphi1)))
200 * (_e2 * (1 + sphi1 + sphi2 + _e2 * sphi1 * sphi2)/(es1 * es2)
201 +_e2m * DDatanhee(sphi1, sphi2) ) / _qZ ) / den;
202 // C = (scbet22*sxi2 - scbet12*sxi1) / (scbet22 * scbet12 * (sx2 - sx1))
203 C = den / (2 * scbet12 * scbet22 * dsxi);
204 tphi0 = (tphi2 + tphi1)/2;
205 real stol = tol0_ * max(real(1), abs(tphi0));
206 for (int i = 0; i < 2*numit0_ || GEOGRAPHICLIB_PANIC; ++i) {
207 // Solve (scbet0^2 * sphi0) / (1/qZ + scbet0^2 * sphi0 * sxi0) = s
208 // for tphi0 by Newton's method on
209 // v(tphi0) = (scbet0^2 * sphi0) - s * (1/qZ + scbet0^2 * sphi0 * sxi0)
210 // = 0
211 // Alt:
212 // (scbet0^2 * sphi0) / (1/qZ - scbet0^2 * sphi0 * (1-sxi0))
213 // = s / (1-s)
214 // w(tphi0) = (1-s) * (scbet0^2 * sphi0)
215 // - s * (1/qZ - scbet0^2 * sphi0 * (1-sxi0))
216 // = (1-s) * (scbet0^2 * sphi0)
217 // - S/qZ * (1 - scbet0^2 * sphi0 * (qZ-q0))
218 // Now
219 // qZ-q0 = (1+e2*sphi0)*(1-sphi0)/(1-e2*sphi0^2) +
220 // (1-e2)*atanhee((1-sphi0)/(1-e2*sphi0))
221 // In limit sphi0 -> 1, qZ-q0 -> 2*(1-sphi0)/(1-e2), so wrte
222 // qZ-q0 = 2*(1-sphi0)/(1-e2) + A + B
223 // A = (1-sphi0)*( (1+e2*sphi0)/(1-e2*sphi0^2) - (1+e2)/(1-e2) )
224 // = -e2 *(1-sphi0)^2 * (2+(1+e2)*sphi0) / ((1-e2)*(1-e2*sphi0^2))
225 // B = (1-e2)*atanhee((1-sphi0)/(1-e2*sphi0)) - (1-sphi0)
226 // = (1-sphi0)*(1-e2)/(1-e2*sphi0)*
227 // ((atanhee(x)/x-1) - e2*(1-sphi0)/(1-e2))
228 // x = (1-sphi0)/(1-e2*sphi0), atanhee(x)/x = atanh(e*x)/(e*x)
229 //
230 // 1 - scbet0^2 * sphi0 * (qZ-q0)
231 // = 1 - scbet0^2 * sphi0 * (2*(1-sphi0)/(1-e2) + A + B)
232 // = D - scbet0^2 * sphi0 * (A + B)
233 // D = 1 - scbet0^2 * sphi0 * 2*(1-sphi0)/(1-e2)
234 // = (1-sphi0)*(1-e2*(1+2*sphi0*(1+sphi0)))/((1-e2)*(1+sphi0))
235 // dD/dsphi0 = -2*(1-e2*sphi0^2*(2*sphi0+3))/((1-e2)*(1+sphi0)^2)
236 // d(A+B)/dsphi0 = 2*(1-sphi0^2)*e2*(2-e2*(1+sphi0^2))/
237 // ((1-e2)*(1-e2*sphi0^2)^2)
238
239 real
240 scphi02 = 1 + Math::sq(tphi0), scphi0 = sqrt(scphi02),
241 // sphi0m = 1-sin(phi0) = 1/( sec(phi0) * (tan(phi0) + sec(phi0)) )
242 sphi0 = tphi0 / scphi0, sphi0m = 1/(scphi0 * (tphi0 + scphi0)),
243 // scbet0^2 * sphi0
244 g = (1 + Math::sq( _fm * tphi0 )) * sphi0,
245 // dg/dsphi0 = dg/dtphi0 * scphi0^3
246 dg = _e2m * scphi02 * (1 + 2 * Math::sq(tphi0)) + _e2,
247 D = sphi0m * (1 - _e2*(1 + 2*sphi0*(1+sphi0))) / (_e2m * (1+sphi0)),
248 // dD/dsphi0
249 dD = -2 * (1 - _e2*Math::sq(sphi0) * (2*sphi0+3)) /
250 (_e2m * Math::sq(1+sphi0)),
251 A = -_e2 * Math::sq(sphi0m) * (2+(1+_e2)*sphi0) /
252 (_e2m*(1-_e2*Math::sq(sphi0))),
253 B = (sphi0m * _e2m / (1 - _e2*sphi0) *
254 (atanhxm1(_e2 *
255 Math::sq(sphi0m / (1-_e2*sphi0))) - _e2*sphi0m/_e2m)),
256 // d(A+B)/dsphi0
257 dAB = (2 * _e2 * (2 - _e2 * (1 + Math::sq(sphi0))) /
258 (_e2m * Math::sq(1 - _e2*Math::sq(sphi0)) * scphi02)),
259 u = sm1 * g - s/_qZ * ( D - g * (A + B) ),
260 // du/dsphi0
261 du = sm1 * dg - s/_qZ * (dD - dg * (A + B) - g * dAB),
262 dtu = -u/du * (scphi0 * scphi02);
263 tphi0 += dtu;
264 if (!(abs(dtu) >= stol))
265 break;
266 }
267 }
268 _txi0 = txif(tphi0); _scxi0 = hyp(_txi0); _sxi0 = _txi0 / _scxi0;
269 _n0 = tphi0/hyp(tphi0);
270 _m02 = 1 / (1 + Math::sq(_fm * tphi0));
271 _nrho0 = polar ? 0 : _a * sqrt(_m02);
272 _k0 = sqrt(tphi1 == tphi2 ? 1 : C / (_m02 + _n0 * _qZ * _sxi0)) * k1;
273 _k2 = Math::sq(_k0);
274 _lat0 = _sign * atan(tphi0)/Math::degree();
275 }
276
278 static const AlbersEqualArea
279 cylindricalequalarea(Constants::WGS84_a(), Constants::WGS84_f(),
280 real(0), real(1), real(0), real(1), real(1));
281 return cylindricalequalarea;
282 }
283
285 static const AlbersEqualArea
286 azimuthalequalareanorth(Constants::WGS84_a(), Constants::WGS84_f(),
287 real(1), real(0), real(1), real(0), real(1));
288 return azimuthalequalareanorth;
289 }
290
292 static const AlbersEqualArea
293 azimuthalequalareasouth(Constants::WGS84_a(), Constants::WGS84_f(),
294 real(-1), real(0), real(-1), real(0), real(1));
295 return azimuthalequalareasouth;
296 }
297
298 Math::real AlbersEqualArea::txif(real tphi) const {
299 // sxi = ( sphi/(1-e2*sphi^2) + atanhee(sphi) ) /
300 // ( 1/(1-e2) + atanhee(1) )
301 //
302 // txi = ( sphi/(1-e2*sphi^2) + atanhee(sphi) ) /
303 // sqrt( ( (1+e2*sphi)*(1-sphi)/( (1-e2*sphi^2) * (1-e2) ) +
304 // atanhee((1-sphi)/(1-e2*sphi)) ) *
305 // ( (1-e2*sphi)*(1+sphi)/( (1-e2*sphi^2) * (1-e2) ) +
306 // atanhee((1+sphi)/(1+e2*sphi)) ) )
307 // = ( tphi/(1-e2*sphi^2) + atanhee(sphi, e2)/cphi ) /
308 // sqrt(
309 // ( (1+e2*sphi)/( (1-e2*sphi^2) * (1-e2) ) + Datanhee(1, sphi) ) *
310 // ( (1-e2*sphi)/( (1-e2*sphi^2) * (1-e2) ) + Datanhee(1, -sphi) ) )
311 //
312 // This function maintains odd parity
313 real
314 cphi = 1 / sqrt(1 + Math::sq(tphi)),
315 sphi = tphi * cphi,
316 es1 = _e2 * sphi,
317 es2m1 = 1 - es1 * sphi, // 1 - e2 * sphi^2
318 es2m1a = _e2m * es2m1; // (1 - e2 * sphi^2) * (1 - e2)
319 return ( tphi / es2m1 + atanhee(sphi) / cphi ) /
320 sqrt( ( (1 + es1) / es2m1a + Datanhee(1, sphi) ) *
321 ( (1 - es1) / es2m1a + Datanhee(1, -sphi) ) );
322 }
323
324 Math::real AlbersEqualArea::tphif(real txi) const {
325 real
326 tphi = txi,
327 stol = tol_ * max(real(1), abs(txi));
328 // CHECK: min iterations = 1, max iterations = 2; mean = 1.99
329 for (int i = 0; i < numit_ || GEOGRAPHICLIB_PANIC; ++i) {
330 // dtxi/dtphi = (scxi/scphi)^3 * 2*(1-e^2)/(qZ*(1-e^2*sphi^2)^2)
331 real
332 txia = txif(tphi),
333 tphi2 = Math::sq(tphi),
334 scphi2 = 1 + tphi2,
335 scterm = scphi2/(1 + Math::sq(txia)),
336 dtphi = (txi - txia) * scterm * sqrt(scterm) *
337 _qx * Math::sq(1 - _e2 * tphi2 / scphi2);
338 tphi += dtphi;
339 if (!(abs(dtphi) >= stol))
340 break;
341 }
342 return tphi;
343 }
344
345 // return atanh(sqrt(x))/sqrt(x) - 1 = x/3 + x^2/5 + x^3/7 + ...
346 // typical x < e^2 = 2*f
347 Math::real AlbersEqualArea::atanhxm1(real x) {
348 real s = 0;
349 if (abs(x) < real(0.5)) {
350 static const real lg2eps_ = -log2(numeric_limits<real>::epsilon() / 2);
351 int e;
352 frexp(x, &e);
353 e = -e;
354 // x = [0.5,1) * 2^(-e)
355 // estimate n s.t. x^n/(2*n+1) < x/3 * epsilon/2
356 // a stronger condition is x^(n-1) < epsilon/2
357 // taking log2 of both sides, a stronger condition is
358 // (n-1)*(-e) < -lg2eps or (n-1)*e > lg2eps or n > ceiling(lg2eps/e)+1
359 int n = x == 0 ? 1 : int(ceil(lg2eps_ / e)) + 1;
360 while (n--) // iterating from n-1 down to 0
361 s = x * s + (n ? 1 : 0)/Math::real(2*n + 1);
362 } else {
363 real xs = sqrt(abs(x));
364 s = (x > 0 ? atanh(xs) : atan(xs)) / xs - 1;
365 }
366 return s;
367 }
368
369 // return (Datanhee(1,y) - Datanhee(1,x))/(y-x)
370 Math::real AlbersEqualArea::DDatanhee(real x, real y) const {
371 // This function is called with x = sphi1, y = sphi2, phi1 <= phi2, sphi2
372 // >= 0, abs(sphi1) <= phi2. However for safety's sake we enforce x <= y.
373 if (y < x) swap(x, y); // ensure that x <= y
374 real q1 = abs(_e2),
375 q2 = abs(2 * _e / _e2m * (1 - x));
376 return
377 x <= 0 || !(min(q1, q2) < real(0.75)) ? DDatanhee0(x, y) :
378 (q1 < q2 ? DDatanhee1(x, y) : DDatanhee2(x, y));
379 }
380
381 // Rearrange difference so that 1 - x is in the denominator, then do a
382 // straight divided difference.
383 Math::real AlbersEqualArea::DDatanhee0(real x, real y) const {
384 return (Datanhee(1, y) - Datanhee(x, y))/(1 - x);
385 }
386
387 // The expansion for e2 small
388 Math::real AlbersEqualArea::DDatanhee1(real x, real y) const {
389 // The series in e2 is
390 // sum( c[l] * e2^l, l, 1, N)
391 // where
392 // c[l] = sum( x^i * y^j; i >= 0, j >= 0, i+j < 2*l) / (2*l + 1)
393 // = ( (x-y) - (1-y) * x^(2*l+1) + (1-x) * y^(2*l+1) ) /
394 // ( (2*l+1) * (x-y) * (1-y) * (1-x) )
395 // For x = y = 1, c[l] = l
396 //
397 // In the limit x,y -> 1,
398 //
399 // DDatanhee -> e2/(1-e2)^2 = sum(l * e2^l, l, 1, inf)
400 //
401 // Use if e2 is sufficiently small.
402 real s = 0;
403 real z = 1, k = 1, t = 0, c = 0, en = 1;
404 while (true) {
405 t = y * t + z; c += t; z *= x;
406 t = y * t + z; c += t; z *= x;
407 k += 2; en *= _e2;
408 // Here en[l] = e2^l, k[l] = 2*l + 1,
409 // c[l] = sum( x^i * y^j; i >= 0, j >= 0, i+j < 2*l) / (2*l + 1)
410 // Taylor expansion is
411 // s = sum( c[l] * e2^l, l, 1, N)
412 real ds = en * c / k;
413 s += ds;
414 if (!(abs(ds) > abs(s) * eps_/2))
415 break; // Iterate until the added term is sufficiently small
416 }
417 return s;
418 }
419
420 // The expansion for x (and y) close to 1
421 Math::real AlbersEqualArea::DDatanhee2(real x, real y) const {
422 // If x and y are both close to 1, expand in Taylor series in dx = 1-x and
423 // dy = 1-y:
424 //
425 // DDatanhee = sum(C_m * (dx^(m+1) - dy^(m+1)) / (dx - dy), m, 0, inf)
426 //
427 // where
428 //
429 // C_m = sum( (m+2)!! / (m+2-2*k)!! *
430 // ((m+1)/2)! / ((m+1)/2-k)! /
431 // (k! * (2*k-1)!!) *
432 // e2^((m+1)/2+k),
433 // k, 0, (m+1)/2) * (-1)^m / ((m+2) * (1-e2)^(m+2))
434 // for m odd, and
435 //
436 // C_m = sum( 2 * (m+1)!! / (m+1-2*k)!! *
437 // (m/2+1)! / (m/2-k)! /
438 // (k! * (2*k+1)!!) *
439 // e2^(m/2+1+k),
440 // k, 0, m/2)) * (-1)^m / ((m+2) * (1-e2)^(m+2))
441 // for m even.
442 //
443 // Here i!! is the double factorial extended to negative i with
444 // i!! = (i+2)!!/(i+2).
445 //
446 // Note that
447 // (dx^(m+1) - dy^(m+1)) / (dx - dy) =
448 // dx^m + dx^(m-1)*dy ... + dx*dy^(m-1) + dy^m
449 //
450 // Leading (m = 0) term is e2 / (1 - e2)^2
451 //
452 // Magnitude of mth term relative to the leading term scales as
453 //
454 // 2*(2*e/(1-e2)*dx)^m
455 //
456 // So use series if (2*e/(1-e2)*dx) is sufficiently small
457 real s, dx = 1 - x, dy = 1 - y, xy = 1, yy = 1, ee = _e2 / Math::sq(_e2m);
458 s = ee;
459 for (int m = 1; ; ++m) {
460 real c = m + 2, t = c;
461 yy *= dy; // yy = dy^m
462 xy = dx * xy + yy;
463 // Now xy = dx^m + dx^(m-1)*dy ... + dx*dy^(m-1) + dy^m
464 // = (dx^(m+1) - dy^(m+1)) / (dx - dy)
465 // max value = (m+1) * max(dx,dy)^m
466 ee /= -_e2m;
467 if (m % 2 == 0) ee *= _e2;
468 // Now ee = (-1)^m * e2^(floor(m/2)+1) / (1-e2)^(m+2)
469 int kmax = (m+1)/2;
470 for (int k = kmax - 1; k >= 0; --k) {
471 // max coeff is less than 2^(m+1)
472 c *= (k + 1) * (2 * (k + m - 2*kmax) + 3);
473 c /= (kmax - k) * (2 * (kmax - k) + 1);
474 // Horner sum for inner _e2 series
475 t = _e2 * t + c;
476 }
477 // Straight sum for outer m series
478 real ds = t * ee * xy / (m + 2);
479 s = s + ds;
480 if (!(abs(ds) > abs(s) * eps_/2))
481 break; // Iterate until the added term is sufficiently small
482 }
483 return s;
484 }
485
486 void AlbersEqualArea::Forward(real lon0, real lat, real lon,
487 real& x, real& y, real& gamma, real& k) const {
488 lon = Math::AngDiff(lon0, lon);
489 lat *= _sign;
490 real sphi, cphi;
491 Math::sincosd(Math::LatFix(lat) * _sign, sphi, cphi);
492 cphi = max(epsx_, cphi);
493 real
494 lam = lon * Math::degree(),
495 tphi = sphi/cphi, txi = txif(tphi), sxi = txi/hyp(txi),
496 dq = _qZ * Dsn(txi, _txi0, sxi, _sxi0) * (txi - _txi0),
497 drho = - _a * dq / (sqrt(_m02 - _n0 * dq) + _nrho0 / _a),
498 theta = _k2 * _n0 * lam, stheta = sin(theta), ctheta = cos(theta),
499 t = _nrho0 + _n0 * drho;
500 x = t * (_n0 != 0 ? stheta / _n0 : _k2 * lam) / _k0;
501 y = (_nrho0 *
502 (_n0 != 0 ?
503 (ctheta < 0 ? 1 - ctheta : Math::sq(stheta)/(1 + ctheta)) / _n0 :
504 0)
505 - drho * ctheta) / _k0;
506 k = _k0 * (t != 0 ? t * hyp(_fm * tphi) / _a : 1);
507 y *= _sign;
508 gamma = _sign * theta / Math::degree();
509 }
510
511 void AlbersEqualArea::Reverse(real lon0, real x, real y,
512 real& lat, real& lon,
513 real& gamma, real& k) const {
514 y *= _sign;
515 real
516 nx = _k0 * _n0 * x, ny = _k0 * _n0 * y, y1 = _nrho0 - ny,
517 den = hypot(nx, y1) + _nrho0, // 0 implies origin with polar aspect
518 drho = den != 0 ? (_k0*x*nx - 2*_k0*y*_nrho0 + _k0*y*ny) / den : 0,
519 // dsxia = scxi0 * dsxi
520 dsxia = - _scxi0 * (2 * _nrho0 + _n0 * drho) * drho /
521 (Math::sq(_a) * _qZ),
522 txi = (_txi0 + dsxia) / sqrt(max(1 - dsxia * (2*_txi0 + dsxia), epsx2_)),
523 tphi = tphif(txi),
524 theta = atan2(nx, y1),
525 lam = _n0 != 0 ? theta / (_k2 * _n0) : x / (y1 * _k0);
526 gamma = _sign * theta / Math::degree();
527 lat = Math::atand(_sign * tphi);
528 lon = lam / Math::degree();
529 lon = Math::AngNormalize(lon + Math::AngNormalize(lon0));
530 k = _k0 * (den != 0 ? (_nrho0 + _n0 * drho) * hyp(_fm * tphi) / _a : 1);
531 }
532
533 void AlbersEqualArea::SetScale(real lat, real k) {
534 if (!(isfinite(k) && k > 0))
535 throw GeographicErr("Scale is not positive");
536 if (!(abs(lat) < 90))
537 throw GeographicErr("Latitude for SetScale not in (-90d, 90d)");
538 real x, y, gamma, kold;
539 Forward(0, lat, 0, x, y, gamma, kold);
540 k /= kold;
541 _k0 *= k;
542 _k2 = Math::sq(_k0);
543 }
544
545} // namespace GeographicLib
Header for GeographicLib::AlbersEqualArea class.
GeographicLib::Math::real real
Definition: GeodSolve.cpp:31
#define GEOGRAPHICLIB_PANIC
Definition: Math.hpp:61
Albers equal area conic projection.
void Reverse(real lon0, real x, real y, real &lat, real &lon, real &gamma, real &k) const
AlbersEqualArea(real a, real f, real stdlat, real k0)
void SetScale(real lat, real k=real(1))
static const AlbersEqualArea & CylindricalEqualArea()
static const AlbersEqualArea & AzimuthalEqualAreaNorth()
static const AlbersEqualArea & AzimuthalEqualAreaSouth()
void Forward(real lon0, real lat, real lon, real &x, real &y, real &gamma, real &k) const
Exception handling for GeographicLib.
Definition: Constants.hpp:315
Mathematical functions needed by GeographicLib.
Definition: Math.hpp:76
static T AngNormalize(T x)
Definition: Math.hpp:420
static T degree()
Definition: Math.hpp:159
static T LatFix(T x)
Definition: Math.hpp:433
static void sincosd(T x, T &sinx, T &cosx)
Definition: Math.cpp:126
static T sq(T x)
Definition: Math.hpp:171
static T atand(T x)
Definition: Math.cpp:205
static T AngDiff(T x, T y, T &e)
Definition: Math.hpp:452
Namespace for GeographicLib.
Definition: Accumulator.cpp:12
void swap(GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &a, GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &b)