GeographicLib  2.6
AlbersEqualArea.cpp
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1 /**
2  * \file AlbersEqualArea.cpp
3  * \brief Implementation for GeographicLib::AlbersEqualArea class
4  *
5  * Copyright (c) Charles Karney (2010-2022) <karney@alum.mit.edu> and licensed
6  * under the MIT/X11 License. For more information, see
7  * https://geographiclib.sourceforge.io/
8  **********************************************************************/
9 
10 #include <GeographicLib/AlbersEqualArea.hpp>
11 
12 namespace GeographicLib {
13 
14  using namespace std;
15 
16  AlbersEqualArea::AlbersEqualArea(real a, real f, real stdlat, real k0)
17  : eps_(numeric_limits<real>::epsilon())
18  , epsx_(Math::sq(eps_))
19  , epsx2_(Math::sq(epsx_))
20  , tol_(sqrt(eps_))
21  , tol0_(tol_ * sqrt(sqrt(eps_)))
22  , _a(a)
23  , _f(f)
24  , _fm(1 - _f)
25  , _e2(_f * (2 - _f))
26  , _e(sqrt(fabs(_e2)))
27  , _e2m(1 - _e2)
28  , _qZ(1 + _e2m * atanhee(real(1)))
29  , _qx(_qZ / ( 2 * _e2m ))
30  {
31  if (!(isfinite(_a) && _a > 0))
32  throw GeographicErr("Equatorial radius is not positive");
33  if (!(isfinite(_f) && _f < 1))
34  throw GeographicErr("Polar semi-axis is not positive");
35  if (!(isfinite(k0) && k0 > 0))
36  throw GeographicErr("Scale is not positive");
37  if (!(fabs(stdlat) <= Math::qd))
38  throw GeographicErr("Standard latitude not in [-" + to_string(Math::qd)
39  + "d, " + to_string(Math::qd) + "d]");
40  real sphi, cphi;
41  Math::sincosd(stdlat, sphi, cphi);
42  Init(sphi, cphi, sphi, cphi, k0);
43  }
44 
45  AlbersEqualArea::AlbersEqualArea(real a, real f, real stdlat1, real stdlat2,
46  real k1)
47  : eps_(numeric_limits<real>::epsilon())
48  , epsx_(Math::sq(eps_))
49  , epsx2_(Math::sq(epsx_))
50  , tol_(sqrt(eps_))
51  , tol0_(tol_ * sqrt(sqrt(eps_)))
52  , _a(a)
53  , _f(f)
54  , _fm(1 - _f)
55  , _e2(_f * (2 - _f))
56  , _e(sqrt(fabs(_e2)))
57  , _e2m(1 - _e2)
58  , _qZ(1 + _e2m * atanhee(real(1)))
59  , _qx(_qZ / ( 2 * _e2m ))
60  {
61  if (!(isfinite(_a) && _a > 0))
62  throw GeographicErr("Equatorial radius is not positive");
63  if (!(isfinite(_f) && _f < 1))
64  throw GeographicErr("Polar semi-axis is not positive");
65  if (!(isfinite(k1) && k1 > 0))
66  throw GeographicErr("Scale is not positive");
67  if (!(fabs(stdlat1) <= Math::qd))
68  throw GeographicErr("Standard latitude 1 not in [-"
69  + to_string(Math::qd) + "d, "
70  + to_string(Math::qd) + "d]");
71  if (!(fabs(stdlat2) <= Math::qd))
72  throw GeographicErr("Standard latitude 2 not in [-"
73  + to_string(Math::qd) + "d, "
74  + to_string(Math::qd) + "d]");
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 
81  AlbersEqualArea::AlbersEqualArea(real a, real f,
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(fabs(_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 (signbit(coslat1))
106  throw GeographicErr("Standard latitude 1 not in [-"
107  + to_string(Math::qd) + "d, "
108  + to_string(Math::qd) + "d]");
109  if (signbit(coslat2))
110  throw GeographicErr("Standard latitude 2 not in [-"
111  + to_string(Math::qd) + "d, "
112  + to_string(Math::qd) + "d]");
113  if (!(fabs(sinlat1) <= 1 && coslat1 <= 1) || (coslat1 == 0 && sinlat1 == 0))
114  throw GeographicErr("Bad sine/cosine of standard latitude 1");
115  if (!(fabs(sinlat2) <= 1 && coslat2 <= 1) || (coslat2 == 0 && sinlat2 == 0))
116  throw GeographicErr("Bad sine/cosine of standard latitude 2");
117  if (coslat1 == 0 && coslat2 == 0 && sinlat1 * sinlat2 <= 0)
118  throw GeographicErr
119  ("Standard latitudes cannot be opposite poles");
120  Init(sinlat1, coslat1, sinlat2, coslat2, k1);
121  }
122 
123  void AlbersEqualArea::Init(real sphi1, real cphi1,
124  real sphi2, real cphi2, real k1) {
125  {
126  real r;
127  r = hypot(sphi1, cphi1);
128  sphi1 /= r; cphi1 /= r;
129  r = hypot(sphi2, cphi2);
130  sphi2 /= r; cphi2 /= r;
131  }
132  bool polar = (cphi1 == 0);
133  cphi1 = fmax(epsx_, cphi1); // Avoid singularities at poles
134  cphi2 = fmax(epsx_, cphi2);
135  // Determine hemisphere of tangent latitude
136  _sign = sphi1 + sphi2 >= 0 ? 1 : -1;
137  // Internally work with tangent latitude positive
138  sphi1 *= _sign; sphi2 *= _sign;
139  if (sphi1 > sphi2) {
140  swap(sphi1, sphi2); swap(cphi1, cphi2); // Make phi1 < phi2
141  }
142  real
143  tphi1 = sphi1/cphi1, tphi2 = sphi2/cphi2;
144 
145  // q = (1-e^2)*(sphi/(1-e^2*sphi^2) - atanhee(sphi))
146  // qZ = q(pi/2) = (1 + (1-e^2)*atanhee(1))
147  // atanhee(x) = atanh(e*x)/e
148  // q = sxi * qZ
149  // dq/dphi = 2*(1-e^2)*cphi/(1-e^2*sphi^2)^2
150  //
151  // n = (m1^2-m2^2)/(q2-q1) -> sin(phi0) for phi1, phi2 -> phi0
152  // C = m1^2 + n*q1 = (m1^2*q2-m2^2*q1)/(q2-q1)
153  // let
154  // rho(pi/2)/rho(-pi/2) = (1-s)/(1+s)
155  // s = n*qZ/C
156  // = qZ * (m1^2-m2^2)/(m1^2*q2-m2^2*q1)
157  // = qZ * (scbet2^2 - scbet1^2)/(scbet2^2*q2 - scbet1^2*q1)
158  // = (scbet2^2 - scbet1^2)/(scbet2^2*sxi2 - scbet1^2*sxi1)
159  // = (tbet2^2 - tbet1^2)/(scbet2^2*sxi2 - scbet1^2*sxi1)
160  // 1-s = -((1-sxi2)*scbet2^2 - (1-sxi1)*scbet1^2)/
161  // (scbet2^2*sxi2 - scbet1^2*sxi1)
162  //
163  // Define phi0 to give same value of s, i.e.,
164  // s = sphi0 * qZ / (m0^2 + sphi0*q0)
165  // = sphi0 * scbet0^2 / (1/qZ + sphi0 * scbet0^2 * sxi0)
166 
167  real tphi0, C;
168  if (polar || tphi1 == tphi2) {
169  tphi0 = tphi2;
170  C = 1; // ignored
171  } else {
172  real
173  tbet1 = _fm * tphi1, scbet12 = 1 + Math::sq(tbet1),
174  tbet2 = _fm * tphi2, scbet22 = 1 + Math::sq(tbet2),
175  txi1 = txif(tphi1), cxi1 = 1/hyp(txi1), sxi1 = txi1 * cxi1,
176  txi2 = txif(tphi2), cxi2 = 1/hyp(txi2), sxi2 = txi2 * cxi2,
177  dtbet2 = _fm * (tbet1 + tbet2),
178  es1 = 1 - _e2 * Math::sq(sphi1), es2 = 1 - _e2 * Math::sq(sphi2),
179  /*
180  dsxi = ( (_e2 * sq(sphi2 + sphi1) + es2 + es1) / (2 * es2 * es1) +
181  Datanhee(sphi2, sphi1) ) * Dsn(tphi2, tphi1, sphi2, sphi1) /
182  ( 2 * _qx ),
183  */
184  dsxi = ( (1 + _e2 * sphi1 * sphi2) / (es2 * es1) +
185  Datanhee(sphi2, sphi1) ) * Dsn(tphi2, tphi1, sphi2, sphi1) /
186  ( 2 * _qx ),
187  den = (sxi2 + sxi1) * dtbet2 + (scbet22 + scbet12) * dsxi,
188  // s = (sq(tbet2) - sq(tbet1)) / (scbet22*sxi2 - scbet12*sxi1)
189  s = 2 * dtbet2 / den,
190  // 1-s = -(sq(scbet2)*(1-sxi2) - sq(scbet1)*(1-sxi1)) /
191  // (scbet22*sxi2 - scbet12*sxi1)
192  // Write
193  // sq(scbet)*(1-sxi) = sq(scbet)*(1-sphi) * (1-sxi)/(1-sphi)
194  sm1 = -Dsn(tphi2, tphi1, sphi2, sphi1) *
195  ( -( ((sphi2 <= 0 ? (1 - sxi2) / (1 - sphi2) :
196  Math::sq(cxi2/cphi2) * (1 + sphi2) / (1 + sxi2)) +
197  (sphi1 <= 0 ? (1 - sxi1) / (1 - sphi1) :
198  Math::sq(cxi1/cphi1) * (1 + sphi1) / (1 + sxi1))) ) *
199  (1 + _e2 * (sphi1 + sphi2 + sphi1 * sphi2)) /
200  (1 + (sphi1 + sphi2 + sphi1 * sphi2)) +
201  (scbet22 * (sphi2 <= 0 ? 1 - sphi2 :
202  Math::sq(cphi2) / ( 1 + sphi2)) +
203  scbet12 * (sphi1 <= 0 ? 1 - sphi1 : Math::sq(cphi1) / ( 1 + sphi1)))
204  * (_e2 * (1 + sphi1 + sphi2 + _e2 * sphi1 * sphi2)/(es1 * es2)
205  +_e2m * DDatanhee(sphi1, sphi2) ) / _qZ ) / den;
206  // C = (scbet22*sxi2 - scbet12*sxi1) / (scbet22 * scbet12 * (sx2 - sx1))
207  C = den / (2 * scbet12 * scbet22 * dsxi);
208  tphi0 = (tphi2 + tphi1)/2;
209  real stol = tol0_ * fmax(real(1), fabs(tphi0));
210  for (int i = 0;
211  i < 2*numit0_ ||
212  GEOGRAPHICLIB_PANIC("Convergence failure in AlbersEqualArea");
213  ++i) {
214  // Solve (scbet0^2 * sphi0) / (1/qZ + scbet0^2 * sphi0 * sxi0) = s
215  // for tphi0 by Newton's method on
216  // v(tphi0) = (scbet0^2 * sphi0) - s * (1/qZ + scbet0^2 * sphi0 * sxi0)
217  // = 0
218  // Alt:
219  // (scbet0^2 * sphi0) / (1/qZ - scbet0^2 * sphi0 * (1-sxi0))
220  // = s / (1-s)
221  // w(tphi0) = (1-s) * (scbet0^2 * sphi0)
222  // - s * (1/qZ - scbet0^2 * sphi0 * (1-sxi0))
223  // = (1-s) * (scbet0^2 * sphi0)
224  // - S/qZ * (1 - scbet0^2 * sphi0 * (qZ-q0))
225  // Now
226  // qZ-q0 = (1+e2*sphi0)*(1-sphi0)/(1-e2*sphi0^2) +
227  // (1-e2)*atanhee((1-sphi0)/(1-e2*sphi0))
228  // In limit sphi0 -> 1, qZ-q0 -> 2*(1-sphi0)/(1-e2), so wrte
229  // qZ-q0 = 2*(1-sphi0)/(1-e2) + A + B
230  // A = (1-sphi0)*( (1+e2*sphi0)/(1-e2*sphi0^2) - (1+e2)/(1-e2) )
231  // = -e2 *(1-sphi0)^2 * (2+(1+e2)*sphi0) / ((1-e2)*(1-e2*sphi0^2))
232  // B = (1-e2)*atanhee((1-sphi0)/(1-e2*sphi0)) - (1-sphi0)
233  // = (1-sphi0)*(1-e2)/(1-e2*sphi0)*
234  // ((atanhee(x)/x-1) - e2*(1-sphi0)/(1-e2))
235  // x = (1-sphi0)/(1-e2*sphi0), atanhee(x)/x = atanh(e*x)/(e*x)
236  //
237  // 1 - scbet0^2 * sphi0 * (qZ-q0)
238  // = 1 - scbet0^2 * sphi0 * (2*(1-sphi0)/(1-e2) + A + B)
239  // = D - scbet0^2 * sphi0 * (A + B)
240  // D = 1 - scbet0^2 * sphi0 * 2*(1-sphi0)/(1-e2)
241  // = (1-sphi0)*(1-e2*(1+2*sphi0*(1+sphi0)))/((1-e2)*(1+sphi0))
242  // dD/dsphi0 = -2*(1-e2*sphi0^2*(2*sphi0+3))/((1-e2)*(1+sphi0)^2)
243  // d(A+B)/dsphi0 = 2*(1-sphi0^2)*e2*(2-e2*(1+sphi0^2))/
244  // ((1-e2)*(1-e2*sphi0^2)^2)
245 
246  real
247  scphi02 = 1 + Math::sq(tphi0), scphi0 = sqrt(scphi02),
248  // sphi0m = 1-sin(phi0) = 1/( sec(phi0) * (tan(phi0) + sec(phi0)) )
249  sphi0 = tphi0 / scphi0, sphi0m = 1/(scphi0 * (tphi0 + scphi0)),
250  // scbet0^2 * sphi0
251  g = (1 + Math::sq( _fm * tphi0 )) * sphi0,
252  // dg/dsphi0 = dg/dtphi0 * scphi0^3
253  dg = _e2m * scphi02 * (1 + 2 * Math::sq(tphi0)) + _e2,
254  D = sphi0m * (1 - _e2*(1 + 2*sphi0*(1+sphi0))) / (_e2m * (1+sphi0)),
255  // dD/dsphi0
256  dD = -2 * (1 - _e2*Math::sq(sphi0) * (2*sphi0+3)) /
257  (_e2m * Math::sq(1+sphi0)),
258  A = -_e2 * Math::sq(sphi0m) * (2+(1+_e2)*sphi0) /
259  (_e2m*(1-_e2*Math::sq(sphi0))),
260  B = (sphi0m * _e2m / (1 - _e2*sphi0) *
261  (atanhxm1(_e2 *
262  Math::sq(sphi0m / (1-_e2*sphi0))) - _e2*sphi0m/_e2m)),
263  // d(A+B)/dsphi0
264  dAB = (2 * _e2 * (2 - _e2 * (1 + Math::sq(sphi0))) /
265  (_e2m * Math::sq(1 - _e2*Math::sq(sphi0)) * scphi02)),
266  u = sm1 * g - s/_qZ * ( D - g * (A + B) ),
267  // du/dsphi0
268  du = sm1 * dg - s/_qZ * (dD - dg * (A + B) - g * dAB),
269  dtu = -u/du * (scphi0 * scphi02);
270  tphi0 += dtu;
271  if (!(fabs(dtu) >= stol))
272  break;
273  }
274  }
275  _txi0 = txif(tphi0); _scxi0 = hyp(_txi0); _sxi0 = _txi0 / _scxi0;
276  _n0 = tphi0/hyp(tphi0);
277  _m02 = 1 / (1 + Math::sq(_fm * tphi0));
278  _nrho0 = polar ? 0 : _a * sqrt(_m02);
279  _k0 = sqrt(tphi1 == tphi2 ? 1 : C / (_m02 + _n0 * _qZ * _sxi0)) * k1;
280  _k2 = Math::sq(_k0);
281  _lat0 = _sign * atan(tphi0)/Math::degree();
282  }
283 
284  const AlbersEqualArea& AlbersEqualArea::CylindricalEqualArea() {
285  static const AlbersEqualArea
286  cylindricalequalarea(Constants::WGS84_a(), Constants::WGS84_f(),
287  real(0), real(1), real(0), real(1), real(1));
288  return cylindricalequalarea;
289  }
290 
291  const AlbersEqualArea& AlbersEqualArea::AzimuthalEqualAreaNorth() {
292  static const AlbersEqualArea
293  azimuthalequalareanorth(Constants::WGS84_a(), Constants::WGS84_f(),
294  real(1), real(0), real(1), real(0), real(1));
295  return azimuthalequalareanorth;
296  }
297 
298  const AlbersEqualArea& AlbersEqualArea::AzimuthalEqualAreaSouth() {
299  static const AlbersEqualArea
300  azimuthalequalareasouth(Constants::WGS84_a(), Constants::WGS84_f(),
301  real(-1), real(0), real(-1), real(0), real(1));
302  return azimuthalequalareasouth;
303  }
304 
305  Math::real AlbersEqualArea::txif(real tphi) const {
306  // sxi = ( sphi/(1-e2*sphi^2) + atanhee(sphi) ) /
307  // ( 1/(1-e2) + atanhee(1) )
308  //
309  // txi = ( sphi/(1-e2*sphi^2) + atanhee(sphi) ) /
310  // sqrt( ( (1+e2*sphi)*(1-sphi)/( (1-e2*sphi^2) * (1-e2) ) +
311  // atanhee((1-sphi)/(1-e2*sphi)) ) *
312  // ( (1-e2*sphi)*(1+sphi)/( (1-e2*sphi^2) * (1-e2) ) +
313  // atanhee((1+sphi)/(1+e2*sphi)) ) )
314  // = ( tphi/(1-e2*sphi^2) + atanhee(sphi, e2)/cphi ) /
315  // sqrt(
316  // ( (1+e2*sphi)/( (1-e2*sphi^2) * (1-e2) ) + Datanhee(1, sphi) ) *
317  // ( (1-e2*sphi)/( (1-e2*sphi^2) * (1-e2) ) + Datanhee(1, -sphi) ) )
318  //
319  // This function maintains odd parity
320  real
321  cphi = 1 / sqrt(1 + Math::sq(tphi)),
322  sphi = tphi * cphi,
323  es1 = _e2 * sphi,
324  es2m1 = 1 - es1 * sphi, // 1 - e2 * sphi^2
325  es2m1a = _e2m * es2m1; // (1 - e2 * sphi^2) * (1 - e2)
326  return ( tphi / es2m1 + atanhee(sphi) / cphi ) /
327  sqrt( ( (1 + es1) / es2m1a + Datanhee(1, sphi) ) *
328  ( (1 - es1) / es2m1a + Datanhee(1, -sphi) ) );
329  }
330 
331  Math::real AlbersEqualArea::tphif(real txi) const {
332  real
333  tphi = txi,
334  stol = tol_ * fmax(real(1), fabs(txi));
335  // CHECK: min iterations = 1, max iterations = 2; mean = 1.99
336  for (int i = 0;
337  i < numit_ ||
338  GEOGRAPHICLIB_PANIC("Convergence failure in AlbersEqualArea");
339  ++i) {
340  // dtxi/dtphi = (scxi/scphi)^3 * 2*(1-e^2)/(qZ*(1-e^2*sphi^2)^2)
341  real
342  txia = txif(tphi),
343  tphi2 = Math::sq(tphi),
344  scphi2 = 1 + tphi2,
345  scterm = scphi2/(1 + Math::sq(txia)),
346  dtphi = (txi - txia) * scterm * sqrt(scterm) *
347  _qx * Math::sq(1 - _e2 * tphi2 / scphi2);
348  tphi += dtphi;
349  if (!(fabs(dtphi) >= stol))
350  break;
351  }
352  return tphi;
353  }
354 
355  // return atanh(sqrt(x))/sqrt(x) - 1 = x/3 + x^2/5 + x^3/7 + ...
356  // typical x < e^2 = 2*f
357  Math::real AlbersEqualArea::atanhxm1(real x) {
358  real s = 0;
359  if (fabs(x) < real(0.5)) {
360  static const real lg2eps_ = -log2(numeric_limits<real>::epsilon() / 2);
361  int e;
362  (void) frexp(x, &e);
363  e = -e;
364  // x = [0.5,1) * 2^(-e)
365  // estimate n s.t. x^n/(2*n+1) < x/3 * epsilon/2
366  // a stronger condition is x^(n-1) < epsilon/2
367  // taking log2 of both sides, a stronger condition is
368  // (n-1)*(-e) < -lg2eps or (n-1)*e > lg2eps or n > ceiling(lg2eps/e)+1
369  int n = x == 0 ? 1 : int(ceil(lg2eps_ / e)) + 1;
370  while (n--) // iterating from n-1 down to 0
371  s = x * s + (n ? 1 : 0)/Math::real(2*n + 1);
372  } else {
373  real xs = sqrt(fabs(x));
374  s = (x > 0 ? atanh(xs) : atan(xs)) / xs - 1;
375  }
376  return s;
377  }
378 
379  // return (Datanhee(1,y) - Datanhee(1,x))/(y-x)
380  Math::real AlbersEqualArea::DDatanhee(real x, real y) const {
381  // This function is called with x = sphi1, y = sphi2, phi1 <= phi2, sphi2
382  // >= 0, abs(sphi1) <= phi2. However for safety's sake we enforce x <= y.
383  if (y < x) swap(x, y); // ensure that x <= y
384  real q1 = fabs(_e2),
385  q2 = fabs(2 * _e / _e2m * (1 - x));
386  return
387  x <= 0 || !(fmin(q1, q2) < real(0.75)) ? DDatanhee0(x, y) :
388  (q1 < q2 ? DDatanhee1(x, y) : DDatanhee2(x, y));
389  }
390 
391  // Rearrange difference so that 1 - x is in the denominator, then do a
392  // straight divided difference.
393  Math::real AlbersEqualArea::DDatanhee0(real x, real y) const {
394  return (Datanhee(1, y) - Datanhee(x, y))/(1 - x);
395  }
396 
397  // The expansion for e2 small
398  Math::real AlbersEqualArea::DDatanhee1(real x, real y) const {
399  // The series in e2 is
400  // sum( c[l] * e2^l, l, 1, N)
401  // where
402  // c[l] = sum( x^i * y^j; i >= 0, j >= 0, i+j < 2*l) / (2*l + 1)
403  // = ( (x-y) - (1-y) * x^(2*l+1) + (1-x) * y^(2*l+1) ) /
404  // ( (2*l+1) * (x-y) * (1-y) * (1-x) )
405  // For x = y = 1, c[l] = l
406  //
407  // In the limit x,y -> 1,
408  //
409  // DDatanhee -> e2/(1-e2)^2 = sum(l * e2^l, l, 1, inf)
410  //
411  // Use if e2 is sufficiently small.
412  real s = 0;
413  real z = 1, k = 1, t = 0, c = 0, en = 1;
414  while (true) {
415  t = y * t + z; c += t; z *= x;
416  t = y * t + z; c += t; z *= x;
417  k += 2; en *= _e2;
418  // Here en[l] = e2^l, k[l] = 2*l + 1,
419  // c[l] = sum( x^i * y^j; i >= 0, j >= 0, i+j < 2*l) / (2*l + 1)
420  // Taylor expansion is
421  // s = sum( c[l] * e2^l, l, 1, N)
422  real ds = en * c / k;
423  s += ds;
424  if (!(fabs(ds) > fabs(s) * eps_/2))
425  break; // Iterate until the added term is sufficiently small
426  }
427  return s;
428  }
429 
430  // The expansion for x (and y) close to 1
431  Math::real AlbersEqualArea::DDatanhee2(real x, real y) const {
432  // If x and y are both close to 1, expand in Taylor series in dx = 1-x and
433  // dy = 1-y:
434  //
435  // DDatanhee = sum(C_m * (dx^(m+1) - dy^(m+1)) / (dx - dy), m, 0, inf)
436  //
437  // where
438  //
439  // C_m = sum( (m+2)!! / (m+2-2*k)!! *
440  // ((m+1)/2)! / ((m+1)/2-k)! /
441  // (k! * (2*k-1)!!) *
442  // e2^((m+1)/2+k),
443  // k, 0, (m+1)/2) * (-1)^m / ((m+2) * (1-e2)^(m+2))
444  // for m odd, and
445  //
446  // C_m = sum( 2 * (m+1)!! / (m+1-2*k)!! *
447  // (m/2+1)! / (m/2-k)! /
448  // (k! * (2*k+1)!!) *
449  // e2^(m/2+1+k),
450  // k, 0, m/2)) * (-1)^m / ((m+2) * (1-e2)^(m+2))
451  // for m even.
452  //
453  // Here i!! is the double factorial extended to negative i with
454  // i!! = (i+2)!!/(i+2).
455  //
456  // Note that
457  // (dx^(m+1) - dy^(m+1)) / (dx - dy) =
458  // dx^m + dx^(m-1)*dy ... + dx*dy^(m-1) + dy^m
459  //
460  // Leading (m = 0) term is e2 / (1 - e2)^2
461  //
462  // Magnitude of mth term relative to the leading term scales as
463  //
464  // 2*(2*e/(1-e2)*dx)^m
465  //
466  // So use series if (2*e/(1-e2)*dx) is sufficiently small
467  real s, dx = 1 - x, dy = 1 - y, xy = 1, yy = 1, ee = _e2 / Math::sq(_e2m);
468  s = ee;
469  for (int m = 1; ; ++m) {
470  real c = m + 2, t = c;
471  yy *= dy; // yy = dy^m
472  xy = dx * xy + yy;
473  // Now xy = dx^m + dx^(m-1)*dy ... + dx*dy^(m-1) + dy^m
474  // = (dx^(m+1) - dy^(m+1)) / (dx - dy)
475  // max value = (m+1) * max(dx,dy)^m
476  ee /= -_e2m;
477  if (m % 2 == 0) ee *= _e2;
478  // Now ee = (-1)^m * e2^(floor(m/2)+1) / (1-e2)^(m+2)
479  int kmax = (m+1)/2;
480  for (int k = kmax - 1; k >= 0; --k) {
481  // max coeff is less than 2^(m+1)
482  c *= (k + 1) * (2 * (k + m - 2*kmax) + 3);
483  c /= (kmax - k) * (2 * (kmax - k) + 1);
484  // Horner sum for inner _e2 series
485  t = _e2 * t + c;
486  }
487  // Straight sum for outer m series
488  real ds = t * ee * xy / (m + 2);
489  s = s + ds;
490  if (!(fabs(ds) > fabs(s) * eps_/2))
491  break; // Iterate until the added term is sufficiently small
492  }
493  return s;
494  }
495 
496  void AlbersEqualArea::Forward(real lon0, real lat, real lon,
497  real& x, real& y, real& gamma, real& k) const {
498  lon = Math::AngDiff(lon0, lon);
499  lat *= _sign;
500  real sphi, cphi;
501  Math::sincosd(Math::LatFix(lat) * _sign, sphi, cphi);
502  cphi = fmax(epsx_, cphi);
503  real
504  lam = lon * Math::degree(),
505  tphi = sphi/cphi, txi = txif(tphi), sxi = txi/hyp(txi),
506  dq = _qZ * Dsn(txi, _txi0, sxi, _sxi0) * (txi - _txi0),
507  drho = - _a * dq / (sqrt(_m02 - _n0 * dq) + _nrho0 / _a),
508  theta = _k2 * _n0 * lam, stheta = sin(theta), ctheta = cos(theta),
509  t = _nrho0 + _n0 * drho;
510  x = t * (_n0 != 0 ? stheta / _n0 : _k2 * lam) / _k0;
511  y = (_nrho0 *
512  (_n0 != 0 ?
513  (ctheta < 0 ? 1 - ctheta : Math::sq(stheta)/(1 + ctheta)) / _n0 :
514  0)
515  - drho * ctheta) / _k0;
516  k = _k0 * (t != 0 ? t * hyp(_fm * tphi) / _a : 1);
517  y *= _sign;
518  gamma = _sign * theta / Math::degree();
519  }
520 
521  void AlbersEqualArea::Reverse(real lon0, real x, real y,
522  real& lat, real& lon,
523  real& gamma, real& k) const {
524  y *= _sign;
525  real
526  nx = _k0 * _n0 * x, ny = _k0 * _n0 * y, y1 = _nrho0 - ny,
527  den = hypot(nx, y1) + _nrho0, // 0 implies origin with polar aspect
528  drho = den != 0 ? (_k0*x*nx - 2*_k0*y*_nrho0 + _k0*y*ny) / den : 0,
529  // dsxia = scxi0 * dsxi
530  dsxia = - _scxi0 * (2 * _nrho0 + _n0 * drho) * drho /
531  (Math::sq(_a) * _qZ),
532  txi = (_txi0 + dsxia) / sqrt(fmax(1 - dsxia * (2*_txi0 + dsxia), epsx2_)),
533  tphi = tphif(txi),
534  theta = atan2(nx, y1),
535  lam = _n0 != 0 ? theta / (_k2 * _n0) : x / (y1 * _k0);
536  gamma = _sign * theta / Math::degree();
537  lat = Math::atand(_sign * tphi);
538  lon = lam / Math::degree();
539  lon = Math::AngNormalize(lon + Math::AngNormalize(lon0));
540  k = _k0 * (den != 0 ? (_nrho0 + _n0 * drho) * hyp(_fm * tphi) / _a : 1);
541  }
542 
543  void AlbersEqualArea::SetScale(real lat, real k) {
544  if (!(isfinite(k) && k > 0))
545  throw GeographicErr("Scale is not positive");
546  if (!(fabs(lat) < Math::qd))
547  throw GeographicErr("Latitude for SetScale not in (-"
548  + to_string(Math::qd) + "d, "
549  + to_string(Math::qd) + "d)");
550  real x, y, gamma, kold;
551  Forward(0, lat, 0, x, y, gamma, kold);
552  k /= kold;
553  _k0 *= k;
554  _k2 = Math::sq(_k0);
555  }
556 
557 } // namespace GeographicLib
GeographicLib::Math::atand
static T atand(T x)
Definition: Math.cpp:234
GeographicLib::AlbersEqualArea::AlbersEqualArea
AlbersEqualArea(real a, real f, real stdlat, real k0)
Definition: AlbersEqualArea.cpp:16
GeographicLib::AlbersEqualArea::CylindricalEqualArea
static const AlbersEqualArea & CylindricalEqualArea()
Definition: AlbersEqualArea.cpp:284
GeographicLib::Math::LatFix
static T LatFix(T x)
Definition: Math.hpp:303
GeographicLib::Math
Mathematical functions needed by GeographicLib.
Definition: Math.hpp:82
AlbersEqualArea.hpp
Header for GeographicLib::AlbersEqualArea class.
GeographicLib::AlbersEqualArea
Albers equal area conic projection.
Definition: AlbersEqualArea.hpp:60
GeographicLib::AlbersEqualArea::Forward
void Forward(real lon0, real lat, real lon, real &x, real &y, real &gamma, real &k) const
Definition: AlbersEqualArea.cpp:496
GeographicLib::Math::AngNormalize
static T AngNormalize(T x)
Definition: Math.cpp:69
GeographicLib::AlbersEqualArea::AzimuthalEqualAreaNorth
static const AlbersEqualArea & AzimuthalEqualAreaNorth()
Definition: AlbersEqualArea.cpp:291
GeographicLib::Math::sq
static T sq(T x)
Definition: Math.hpp:209
GeographicLib::Math::qd
static constexpr int qd
degrees per quarter turn
Definition: Math.hpp:142
GeographicLib
Namespace for GeographicLib.
Definition: Accumulator.cpp:12
GeographicLib::Math::AngDiff
static T AngDiff(T x, T y, T &e)
Definition: Math.cpp:80
GeographicLib::Math::degree
static T degree()
Definition: Math.hpp:197
real
GeographicLib::Math::real real
Definition: Geod3Solve.cpp:25
std::swap
void swap(GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &a, GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &b)
Definition: NearestNeighbor.hpp:819
GEOGRAPHICLIB_PANIC
#define GEOGRAPHICLIB_PANIC(msg)
Definition: Math.hpp:67
GeographicLib::GeographicErr
Exception handling for GeographicLib.
Definition: Constants.hpp:344
GeographicLib::AlbersEqualArea::AzimuthalEqualAreaSouth
static const AlbersEqualArea & AzimuthalEqualAreaSouth()
Definition: AlbersEqualArea.cpp:298
GeographicLib::AlbersEqualArea::Reverse
void Reverse(real lon0, real x, real y, real &lat, real &lon, real &gamma, real &k) const
Definition: AlbersEqualArea.cpp:521
GeographicLib::Math::sincosd
static void sincosd(T x, T &sinx, T &cosx)
Definition: Math.cpp:104
GeographicLib::AlbersEqualArea::SetScale
void SetScale(real lat, real k=real(1))
Definition: AlbersEqualArea.cpp:543