GeographicLib  2.0
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) <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 
17 namespace 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(fabs(_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 (!(fabs(stdlat) <= Math::qd))
43  throw GeographicErr("Standard latitude not in [-" + to_string(Math::qd)
44  + "d, " + to_string(Math::qd) + "d]");
45  real sphi, cphi;
46  Math::sincosd(stdlat, sphi, cphi);
47  Init(sphi, cphi, sphi, cphi, k0);
48  }
49 
50  AlbersEqualArea::AlbersEqualArea(real a, real f, real stdlat1, real stdlat2,
51  real k1)
52  : eps_(numeric_limits<real>::epsilon())
53  , epsx_(Math::sq(eps_))
54  , epsx2_(Math::sq(epsx_))
55  , tol_(sqrt(eps_))
56  , tol0_(tol_ * sqrt(sqrt(eps_)))
57  , _a(a)
58  , _f(f)
59  , _fm(1 - _f)
60  , _e2(_f * (2 - _f))
61  , _e(sqrt(fabs(_e2)))
62  , _e2m(1 - _e2)
63  , _qZ(1 + _e2m * atanhee(real(1)))
64  , _qx(_qZ / ( 2 * _e2m ))
65  {
66  if (!(isfinite(_a) && _a > 0))
67  throw GeographicErr("Equatorial radius is not positive");
68  if (!(isfinite(_f) && _f < 1))
69  throw GeographicErr("Polar semi-axis is not positive");
70  if (!(isfinite(k1) && k1 > 0))
71  throw GeographicErr("Scale is not positive");
72  if (!(fabs(stdlat1) <= Math::qd))
73  throw GeographicErr("Standard latitude 1 not in [-"
74  + to_string(Math::qd) + "d, "
75  + to_string(Math::qd) + "d]");
76  if (!(fabs(stdlat2) <= Math::qd))
77  throw GeographicErr("Standard latitude 2 not in [-"
78  + to_string(Math::qd) + "d, "
79  + to_string(Math::qd) + "d]");
80  real sphi1, cphi1, sphi2, cphi2;
81  Math::sincosd(stdlat1, sphi1, cphi1);
82  Math::sincosd(stdlat2, sphi2, cphi2);
83  Init(sphi1, cphi1, sphi2, cphi2, k1);
84  }
85 
87  real sinlat1, real coslat1,
88  real sinlat2, real coslat2,
89  real k1)
90  : eps_(numeric_limits<real>::epsilon())
91  , epsx_(Math::sq(eps_))
92  , epsx2_(Math::sq(epsx_))
93  , tol_(sqrt(eps_))
94  , tol0_(tol_ * sqrt(sqrt(eps_)))
95  , _a(a)
96  , _f(f)
97  , _fm(1 - _f)
98  , _e2(_f * (2 - _f))
99  , _e(sqrt(fabs(_e2)))
100  , _e2m(1 - _e2)
101  , _qZ(1 + _e2m * atanhee(real(1)))
102  , _qx(_qZ / ( 2 * _e2m ))
103  {
104  if (!(isfinite(_a) && _a > 0))
105  throw GeographicErr("Equatorial radius is not positive");
106  if (!(isfinite(_f) && _f < 1))
107  throw GeographicErr("Polar semi-axis is not positive");
108  if (!(isfinite(k1) && k1 > 0))
109  throw GeographicErr("Scale is not positive");
110  if (signbit(coslat1))
111  throw GeographicErr("Standard latitude 1 not in [-"
112  + to_string(Math::qd) + "d, "
113  + to_string(Math::qd) + "d]");
114  if (signbit(coslat2))
115  throw GeographicErr("Standard latitude 2 not in [-"
116  + to_string(Math::qd) + "d, "
117  + to_string(Math::qd) + "d]");
118  if (!(fabs(sinlat1) <= 1 && coslat1 <= 1) || (coslat1 == 0 && sinlat1 == 0))
119  throw GeographicErr("Bad sine/cosine of standard latitude 1");
120  if (!(fabs(sinlat2) <= 1 && coslat2 <= 1) || (coslat2 == 0 && sinlat2 == 0))
121  throw GeographicErr("Bad sine/cosine of standard latitude 2");
122  if (coslat1 == 0 && coslat2 == 0 && sinlat1 * sinlat2 <= 0)
123  throw GeographicErr
124  ("Standard latitudes cannot be opposite poles");
125  Init(sinlat1, coslat1, sinlat2, coslat2, k1);
126  }
127 
128  void AlbersEqualArea::Init(real sphi1, real cphi1,
129  real sphi2, real cphi2, real k1) {
130  {
131  real r;
132  r = hypot(sphi1, cphi1);
133  sphi1 /= r; cphi1 /= r;
134  r = hypot(sphi2, cphi2);
135  sphi2 /= r; cphi2 /= r;
136  }
137  bool polar = (cphi1 == 0);
138  cphi1 = fmax(epsx_, cphi1); // Avoid singularities at poles
139  cphi2 = fmax(epsx_, cphi2);
140  // Determine hemisphere of tangent latitude
141  _sign = sphi1 + sphi2 >= 0 ? 1 : -1;
142  // Internally work with tangent latitude positive
143  sphi1 *= _sign; sphi2 *= _sign;
144  if (sphi1 > sphi2) {
145  swap(sphi1, sphi2); swap(cphi1, cphi2); // Make phi1 < phi2
146  }
147  real
148  tphi1 = sphi1/cphi1, tphi2 = sphi2/cphi2;
149 
150  // q = (1-e^2)*(sphi/(1-e^2*sphi^2) - atanhee(sphi))
151  // qZ = q(pi/2) = (1 + (1-e^2)*atanhee(1))
152  // atanhee(x) = atanh(e*x)/e
153  // q = sxi * qZ
154  // dq/dphi = 2*(1-e^2)*cphi/(1-e^2*sphi^2)^2
155  //
156  // n = (m1^2-m2^2)/(q2-q1) -> sin(phi0) for phi1, phi2 -> phi0
157  // C = m1^2 + n*q1 = (m1^2*q2-m2^2*q1)/(q2-q1)
158  // let
159  // rho(pi/2)/rho(-pi/2) = (1-s)/(1+s)
160  // s = n*qZ/C
161  // = qZ * (m1^2-m2^2)/(m1^2*q2-m2^2*q1)
162  // = qZ * (scbet2^2 - scbet1^2)/(scbet2^2*q2 - scbet1^2*q1)
163  // = (scbet2^2 - scbet1^2)/(scbet2^2*sxi2 - scbet1^2*sxi1)
164  // = (tbet2^2 - tbet1^2)/(scbet2^2*sxi2 - scbet1^2*sxi1)
165  // 1-s = -((1-sxi2)*scbet2^2 - (1-sxi1)*scbet1^2)/
166  // (scbet2^2*sxi2 - scbet1^2*sxi1)
167  //
168  // Define phi0 to give same value of s, i.e.,
169  // s = sphi0 * qZ / (m0^2 + sphi0*q0)
170  // = sphi0 * scbet0^2 / (1/qZ + sphi0 * scbet0^2 * sxi0)
171 
172  real tphi0, C;
173  if (polar || tphi1 == tphi2) {
174  tphi0 = tphi2;
175  C = 1; // ignored
176  } else {
177  real
178  tbet1 = _fm * tphi1, scbet12 = 1 + Math::sq(tbet1),
179  tbet2 = _fm * tphi2, scbet22 = 1 + Math::sq(tbet2),
180  txi1 = txif(tphi1), cxi1 = 1/hyp(txi1), sxi1 = txi1 * cxi1,
181  txi2 = txif(tphi2), cxi2 = 1/hyp(txi2), sxi2 = txi2 * cxi2,
182  dtbet2 = _fm * (tbet1 + tbet2),
183  es1 = 1 - _e2 * Math::sq(sphi1), es2 = 1 - _e2 * Math::sq(sphi2),
184  /*
185  dsxi = ( (_e2 * sq(sphi2 + sphi1) + es2 + es1) / (2 * es2 * es1) +
186  Datanhee(sphi2, sphi1) ) * Dsn(tphi2, tphi1, sphi2, sphi1) /
187  ( 2 * _qx ),
188  */
189  dsxi = ( (1 + _e2 * sphi1 * sphi2) / (es2 * es1) +
190  Datanhee(sphi2, sphi1) ) * Dsn(tphi2, tphi1, sphi2, sphi1) /
191  ( 2 * _qx ),
192  den = (sxi2 + sxi1) * dtbet2 + (scbet22 + scbet12) * dsxi,
193  // s = (sq(tbet2) - sq(tbet1)) / (scbet22*sxi2 - scbet12*sxi1)
194  s = 2 * dtbet2 / den,
195  // 1-s = -(sq(scbet2)*(1-sxi2) - sq(scbet1)*(1-sxi1)) /
196  // (scbet22*sxi2 - scbet12*sxi1)
197  // Write
198  // sq(scbet)*(1-sxi) = sq(scbet)*(1-sphi) * (1-sxi)/(1-sphi)
199  sm1 = -Dsn(tphi2, tphi1, sphi2, sphi1) *
200  ( -( ((sphi2 <= 0 ? (1 - sxi2) / (1 - sphi2) :
201  Math::sq(cxi2/cphi2) * (1 + sphi2) / (1 + sxi2)) +
202  (sphi1 <= 0 ? (1 - sxi1) / (1 - sphi1) :
203  Math::sq(cxi1/cphi1) * (1 + sphi1) / (1 + sxi1))) ) *
204  (1 + _e2 * (sphi1 + sphi2 + sphi1 * sphi2)) /
205  (1 + (sphi1 + sphi2 + sphi1 * sphi2)) +
206  (scbet22 * (sphi2 <= 0 ? 1 - sphi2 :
207  Math::sq(cphi2) / ( 1 + sphi2)) +
208  scbet12 * (sphi1 <= 0 ? 1 - sphi1 : Math::sq(cphi1) / ( 1 + sphi1)))
209  * (_e2 * (1 + sphi1 + sphi2 + _e2 * sphi1 * sphi2)/(es1 * es2)
210  +_e2m * DDatanhee(sphi1, sphi2) ) / _qZ ) / den;
211  // C = (scbet22*sxi2 - scbet12*sxi1) / (scbet22 * scbet12 * (sx2 - sx1))
212  C = den / (2 * scbet12 * scbet22 * dsxi);
213  tphi0 = (tphi2 + tphi1)/2;
214  real stol = tol0_ * fmax(real(1), fabs(tphi0));
215  for (int i = 0; i < 2*numit0_ || GEOGRAPHICLIB_PANIC; ++i) {
216  // Solve (scbet0^2 * sphi0) / (1/qZ + scbet0^2 * sphi0 * sxi0) = s
217  // for tphi0 by Newton's method on
218  // v(tphi0) = (scbet0^2 * sphi0) - s * (1/qZ + scbet0^2 * sphi0 * sxi0)
219  // = 0
220  // Alt:
221  // (scbet0^2 * sphi0) / (1/qZ - scbet0^2 * sphi0 * (1-sxi0))
222  // = s / (1-s)
223  // w(tphi0) = (1-s) * (scbet0^2 * sphi0)
224  // - s * (1/qZ - scbet0^2 * sphi0 * (1-sxi0))
225  // = (1-s) * (scbet0^2 * sphi0)
226  // - S/qZ * (1 - scbet0^2 * sphi0 * (qZ-q0))
227  // Now
228  // qZ-q0 = (1+e2*sphi0)*(1-sphi0)/(1-e2*sphi0^2) +
229  // (1-e2)*atanhee((1-sphi0)/(1-e2*sphi0))
230  // In limit sphi0 -> 1, qZ-q0 -> 2*(1-sphi0)/(1-e2), so wrte
231  // qZ-q0 = 2*(1-sphi0)/(1-e2) + A + B
232  // A = (1-sphi0)*( (1+e2*sphi0)/(1-e2*sphi0^2) - (1+e2)/(1-e2) )
233  // = -e2 *(1-sphi0)^2 * (2+(1+e2)*sphi0) / ((1-e2)*(1-e2*sphi0^2))
234  // B = (1-e2)*atanhee((1-sphi0)/(1-e2*sphi0)) - (1-sphi0)
235  // = (1-sphi0)*(1-e2)/(1-e2*sphi0)*
236  // ((atanhee(x)/x-1) - e2*(1-sphi0)/(1-e2))
237  // x = (1-sphi0)/(1-e2*sphi0), atanhee(x)/x = atanh(e*x)/(e*x)
238  //
239  // 1 - scbet0^2 * sphi0 * (qZ-q0)
240  // = 1 - scbet0^2 * sphi0 * (2*(1-sphi0)/(1-e2) + A + B)
241  // = D - scbet0^2 * sphi0 * (A + B)
242  // D = 1 - scbet0^2 * sphi0 * 2*(1-sphi0)/(1-e2)
243  // = (1-sphi0)*(1-e2*(1+2*sphi0*(1+sphi0)))/((1-e2)*(1+sphi0))
244  // dD/dsphi0 = -2*(1-e2*sphi0^2*(2*sphi0+3))/((1-e2)*(1+sphi0)^2)
245  // d(A+B)/dsphi0 = 2*(1-sphi0^2)*e2*(2-e2*(1+sphi0^2))/
246  // ((1-e2)*(1-e2*sphi0^2)^2)
247 
248  real
249  scphi02 = 1 + Math::sq(tphi0), scphi0 = sqrt(scphi02),
250  // sphi0m = 1-sin(phi0) = 1/( sec(phi0) * (tan(phi0) + sec(phi0)) )
251  sphi0 = tphi0 / scphi0, sphi0m = 1/(scphi0 * (tphi0 + scphi0)),
252  // scbet0^2 * sphi0
253  g = (1 + Math::sq( _fm * tphi0 )) * sphi0,
254  // dg/dsphi0 = dg/dtphi0 * scphi0^3
255  dg = _e2m * scphi02 * (1 + 2 * Math::sq(tphi0)) + _e2,
256  D = sphi0m * (1 - _e2*(1 + 2*sphi0*(1+sphi0))) / (_e2m * (1+sphi0)),
257  // dD/dsphi0
258  dD = -2 * (1 - _e2*Math::sq(sphi0) * (2*sphi0+3)) /
259  (_e2m * Math::sq(1+sphi0)),
260  A = -_e2 * Math::sq(sphi0m) * (2+(1+_e2)*sphi0) /
261  (_e2m*(1-_e2*Math::sq(sphi0))),
262  B = (sphi0m * _e2m / (1 - _e2*sphi0) *
263  (atanhxm1(_e2 *
264  Math::sq(sphi0m / (1-_e2*sphi0))) - _e2*sphi0m/_e2m)),
265  // d(A+B)/dsphi0
266  dAB = (2 * _e2 * (2 - _e2 * (1 + Math::sq(sphi0))) /
267  (_e2m * Math::sq(1 - _e2*Math::sq(sphi0)) * scphi02)),
268  u = sm1 * g - s/_qZ * ( D - g * (A + B) ),
269  // du/dsphi0
270  du = sm1 * dg - s/_qZ * (dD - dg * (A + B) - g * dAB),
271  dtu = -u/du * (scphi0 * scphi02);
272  tphi0 += dtu;
273  if (!(fabs(dtu) >= stol))
274  break;
275  }
276  }
277  _txi0 = txif(tphi0); _scxi0 = hyp(_txi0); _sxi0 = _txi0 / _scxi0;
278  _n0 = tphi0/hyp(tphi0);
279  _m02 = 1 / (1 + Math::sq(_fm * tphi0));
280  _nrho0 = polar ? 0 : _a * sqrt(_m02);
281  _k0 = sqrt(tphi1 == tphi2 ? 1 : C / (_m02 + _n0 * _qZ * _sxi0)) * k1;
282  _k2 = Math::sq(_k0);
283  _lat0 = _sign * atan(tphi0)/Math::degree();
284  }
285 
287  static const AlbersEqualArea
288  cylindricalequalarea(Constants::WGS84_a(), Constants::WGS84_f(),
289  real(0), real(1), real(0), real(1), real(1));
290  return cylindricalequalarea;
291  }
292 
294  static const AlbersEqualArea
295  azimuthalequalareanorth(Constants::WGS84_a(), Constants::WGS84_f(),
296  real(1), real(0), real(1), real(0), real(1));
297  return azimuthalequalareanorth;
298  }
299 
301  static const AlbersEqualArea
302  azimuthalequalareasouth(Constants::WGS84_a(), Constants::WGS84_f(),
303  real(-1), real(0), real(-1), real(0), real(1));
304  return azimuthalequalareasouth;
305  }
306 
307  Math::real AlbersEqualArea::txif(real tphi) const {
308  // sxi = ( sphi/(1-e2*sphi^2) + atanhee(sphi) ) /
309  // ( 1/(1-e2) + atanhee(1) )
310  //
311  // txi = ( sphi/(1-e2*sphi^2) + atanhee(sphi) ) /
312  // sqrt( ( (1+e2*sphi)*(1-sphi)/( (1-e2*sphi^2) * (1-e2) ) +
313  // atanhee((1-sphi)/(1-e2*sphi)) ) *
314  // ( (1-e2*sphi)*(1+sphi)/( (1-e2*sphi^2) * (1-e2) ) +
315  // atanhee((1+sphi)/(1+e2*sphi)) ) )
316  // = ( tphi/(1-e2*sphi^2) + atanhee(sphi, e2)/cphi ) /
317  // sqrt(
318  // ( (1+e2*sphi)/( (1-e2*sphi^2) * (1-e2) ) + Datanhee(1, sphi) ) *
319  // ( (1-e2*sphi)/( (1-e2*sphi^2) * (1-e2) ) + Datanhee(1, -sphi) ) )
320  //
321  // This function maintains odd parity
322  real
323  cphi = 1 / sqrt(1 + Math::sq(tphi)),
324  sphi = tphi * cphi,
325  es1 = _e2 * sphi,
326  es2m1 = 1 - es1 * sphi, // 1 - e2 * sphi^2
327  es2m1a = _e2m * es2m1; // (1 - e2 * sphi^2) * (1 - e2)
328  return ( tphi / es2m1 + atanhee(sphi) / cphi ) /
329  sqrt( ( (1 + es1) / es2m1a + Datanhee(1, sphi) ) *
330  ( (1 - es1) / es2m1a + Datanhee(1, -sphi) ) );
331  }
332 
333  Math::real AlbersEqualArea::tphif(real txi) const {
334  real
335  tphi = txi,
336  stol = tol_ * fmax(real(1), fabs(txi));
337  // CHECK: min iterations = 1, max iterations = 2; mean = 1.99
338  for (int i = 0; i < numit_ || GEOGRAPHICLIB_PANIC; ++i) {
339  // dtxi/dtphi = (scxi/scphi)^3 * 2*(1-e^2)/(qZ*(1-e^2*sphi^2)^2)
340  real
341  txia = txif(tphi),
342  tphi2 = Math::sq(tphi),
343  scphi2 = 1 + tphi2,
344  scterm = scphi2/(1 + Math::sq(txia)),
345  dtphi = (txi - txia) * scterm * sqrt(scterm) *
346  _qx * Math::sq(1 - _e2 * tphi2 / scphi2);
347  tphi += dtphi;
348  if (!(fabs(dtphi) >= stol))
349  break;
350  }
351  return tphi;
352  }
353 
354  // return atanh(sqrt(x))/sqrt(x) - 1 = x/3 + x^2/5 + x^3/7 + ...
355  // typical x < e^2 = 2*f
356  Math::real AlbersEqualArea::atanhxm1(real x) {
357  real s = 0;
358  if (fabs(x) < real(0.5)) {
359  static const real lg2eps_ = -log2(numeric_limits<real>::epsilon() / 2);
360  int e;
361  frexp(x, &e);
362  e = -e;
363  // x = [0.5,1) * 2^(-e)
364  // estimate n s.t. x^n/(2*n+1) < x/3 * epsilon/2
365  // a stronger condition is x^(n-1) < epsilon/2
366  // taking log2 of both sides, a stronger condition is
367  // (n-1)*(-e) < -lg2eps or (n-1)*e > lg2eps or n > ceiling(lg2eps/e)+1
368  int n = x == 0 ? 1 : int(ceil(lg2eps_ / e)) + 1;
369  while (n--) // iterating from n-1 down to 0
370  s = x * s + (n ? 1 : 0)/Math::real(2*n + 1);
371  } else {
372  real xs = sqrt(fabs(x));
373  s = (x > 0 ? atanh(xs) : atan(xs)) / xs - 1;
374  }
375  return s;
376  }
377 
378  // return (Datanhee(1,y) - Datanhee(1,x))/(y-x)
379  Math::real AlbersEqualArea::DDatanhee(real x, real y) const {
380  // This function is called with x = sphi1, y = sphi2, phi1 <= phi2, sphi2
381  // >= 0, abs(sphi1) <= phi2. However for safety's sake we enforce x <= y.
382  if (y < x) swap(x, y); // ensure that x <= y
383  real q1 = fabs(_e2),
384  q2 = fabs(2 * _e / _e2m * (1 - x));
385  return
386  x <= 0 || !(fmin(q1, q2) < real(0.75)) ? DDatanhee0(x, y) :
387  (q1 < q2 ? DDatanhee1(x, y) : DDatanhee2(x, y));
388  }
389 
390  // Rearrange difference so that 1 - x is in the denominator, then do a
391  // straight divided difference.
392  Math::real AlbersEqualArea::DDatanhee0(real x, real y) const {
393  return (Datanhee(1, y) - Datanhee(x, y))/(1 - x);
394  }
395 
396  // The expansion for e2 small
397  Math::real AlbersEqualArea::DDatanhee1(real x, real y) const {
398  // The series in e2 is
399  // sum( c[l] * e2^l, l, 1, N)
400  // where
401  // c[l] = sum( x^i * y^j; i >= 0, j >= 0, i+j < 2*l) / (2*l + 1)
402  // = ( (x-y) - (1-y) * x^(2*l+1) + (1-x) * y^(2*l+1) ) /
403  // ( (2*l+1) * (x-y) * (1-y) * (1-x) )
404  // For x = y = 1, c[l] = l
405  //
406  // In the limit x,y -> 1,
407  //
408  // DDatanhee -> e2/(1-e2)^2 = sum(l * e2^l, l, 1, inf)
409  //
410  // Use if e2 is sufficiently small.
411  real s = 0;
412  real z = 1, k = 1, t = 0, c = 0, en = 1;
413  while (true) {
414  t = y * t + z; c += t; z *= x;
415  t = y * t + z; c += t; z *= x;
416  k += 2; en *= _e2;
417  // Here en[l] = e2^l, k[l] = 2*l + 1,
418  // c[l] = sum( x^i * y^j; i >= 0, j >= 0, i+j < 2*l) / (2*l + 1)
419  // Taylor expansion is
420  // s = sum( c[l] * e2^l, l, 1, N)
421  real ds = en * c / k;
422  s += ds;
423  if (!(fabs(ds) > fabs(s) * eps_/2))
424  break; // Iterate until the added term is sufficiently small
425  }
426  return s;
427  }
428 
429  // The expansion for x (and y) close to 1
430  Math::real AlbersEqualArea::DDatanhee2(real x, real y) const {
431  // If x and y are both close to 1, expand in Taylor series in dx = 1-x and
432  // dy = 1-y:
433  //
434  // DDatanhee = sum(C_m * (dx^(m+1) - dy^(m+1)) / (dx - dy), m, 0, inf)
435  //
436  // where
437  //
438  // C_m = sum( (m+2)!! / (m+2-2*k)!! *
439  // ((m+1)/2)! / ((m+1)/2-k)! /
440  // (k! * (2*k-1)!!) *
441  // e2^((m+1)/2+k),
442  // k, 0, (m+1)/2) * (-1)^m / ((m+2) * (1-e2)^(m+2))
443  // for m odd, and
444  //
445  // C_m = sum( 2 * (m+1)!! / (m+1-2*k)!! *
446  // (m/2+1)! / (m/2-k)! /
447  // (k! * (2*k+1)!!) *
448  // e2^(m/2+1+k),
449  // k, 0, m/2)) * (-1)^m / ((m+2) * (1-e2)^(m+2))
450  // for m even.
451  //
452  // Here i!! is the double factorial extended to negative i with
453  // i!! = (i+2)!!/(i+2).
454  //
455  // Note that
456  // (dx^(m+1) - dy^(m+1)) / (dx - dy) =
457  // dx^m + dx^(m-1)*dy ... + dx*dy^(m-1) + dy^m
458  //
459  // Leading (m = 0) term is e2 / (1 - e2)^2
460  //
461  // Magnitude of mth term relative to the leading term scales as
462  //
463  // 2*(2*e/(1-e2)*dx)^m
464  //
465  // So use series if (2*e/(1-e2)*dx) is sufficiently small
466  real s, dx = 1 - x, dy = 1 - y, xy = 1, yy = 1, ee = _e2 / Math::sq(_e2m);
467  s = ee;
468  for (int m = 1; ; ++m) {
469  real c = m + 2, t = c;
470  yy *= dy; // yy = dy^m
471  xy = dx * xy + yy;
472  // Now xy = dx^m + dx^(m-1)*dy ... + dx*dy^(m-1) + dy^m
473  // = (dx^(m+1) - dy^(m+1)) / (dx - dy)
474  // max value = (m+1) * max(dx,dy)^m
475  ee /= -_e2m;
476  if (m % 2 == 0) ee *= _e2;
477  // Now ee = (-1)^m * e2^(floor(m/2)+1) / (1-e2)^(m+2)
478  int kmax = (m+1)/2;
479  for (int k = kmax - 1; k >= 0; --k) {
480  // max coeff is less than 2^(m+1)
481  c *= (k + 1) * (2 * (k + m - 2*kmax) + 3);
482  c /= (kmax - k) * (2 * (kmax - k) + 1);
483  // Horner sum for inner _e2 series
484  t = _e2 * t + c;
485  }
486  // Straight sum for outer m series
487  real ds = t * ee * xy / (m + 2);
488  s = s + ds;
489  if (!(fabs(ds) > fabs(s) * eps_/2))
490  break; // Iterate until the added term is sufficiently small
491  }
492  return s;
493  }
494 
495  void AlbersEqualArea::Forward(real lon0, real lat, real lon,
496  real& x, real& y, real& gamma, real& k) const {
497  lon = Math::AngDiff(lon0, lon);
498  lat *= _sign;
499  real sphi, cphi;
500  Math::sincosd(Math::LatFix(lat) * _sign, sphi, cphi);
501  cphi = fmax(epsx_, cphi);
502  real
503  lam = lon * Math::degree(),
504  tphi = sphi/cphi, txi = txif(tphi), sxi = txi/hyp(txi),
505  dq = _qZ * Dsn(txi, _txi0, sxi, _sxi0) * (txi - _txi0),
506  drho = - _a * dq / (sqrt(_m02 - _n0 * dq) + _nrho0 / _a),
507  theta = _k2 * _n0 * lam, stheta = sin(theta), ctheta = cos(theta),
508  t = _nrho0 + _n0 * drho;
509  x = t * (_n0 != 0 ? stheta / _n0 : _k2 * lam) / _k0;
510  y = (_nrho0 *
511  (_n0 != 0 ?
512  (ctheta < 0 ? 1 - ctheta : Math::sq(stheta)/(1 + ctheta)) / _n0 :
513  0)
514  - drho * ctheta) / _k0;
515  k = _k0 * (t != 0 ? t * hyp(_fm * tphi) / _a : 1);
516  y *= _sign;
517  gamma = _sign * theta / Math::degree();
518  }
519 
520  void AlbersEqualArea::Reverse(real lon0, real x, real y,
521  real& lat, real& lon,
522  real& gamma, real& k) const {
523  y *= _sign;
524  real
525  nx = _k0 * _n0 * x, ny = _k0 * _n0 * y, y1 = _nrho0 - ny,
526  den = hypot(nx, y1) + _nrho0, // 0 implies origin with polar aspect
527  drho = den != 0 ? (_k0*x*nx - 2*_k0*y*_nrho0 + _k0*y*ny) / den : 0,
528  // dsxia = scxi0 * dsxi
529  dsxia = - _scxi0 * (2 * _nrho0 + _n0 * drho) * drho /
530  (Math::sq(_a) * _qZ),
531  txi = (_txi0 + dsxia) / sqrt(fmax(1 - dsxia * (2*_txi0 + dsxia), epsx2_)),
532  tphi = tphif(txi),
533  theta = atan2(nx, y1),
534  lam = _n0 != 0 ? theta / (_k2 * _n0) : x / (y1 * _k0);
535  gamma = _sign * theta / Math::degree();
536  lat = Math::atand(_sign * tphi);
537  lon = lam / Math::degree();
538  lon = Math::AngNormalize(lon + Math::AngNormalize(lon0));
539  k = _k0 * (den != 0 ? (_nrho0 + _n0 * drho) * hyp(_fm * tphi) / _a : 1);
540  }
541 
542  void AlbersEqualArea::SetScale(real lat, real k) {
543  if (!(isfinite(k) && k > 0))
544  throw GeographicErr("Scale is not positive");
545  if (!(fabs(lat) < Math::qd))
546  throw GeographicErr("Latitude for SetScale not in (-"
547  + to_string(Math::qd) + "d, "
548  + to_string(Math::qd) + "d)");
549  real x, y, gamma, kold;
550  Forward(0, lat, 0, x, y, gamma, kold);
551  k /= kold;
552  _k0 *= k;
553  _k2 = Math::sq(_k0);
554  }
555 
556 } // 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:316
Mathematical functions needed by GeographicLib.
Definition: Math.hpp:76
static T degree()
Definition: Math.hpp:200
static T LatFix(T x)
Definition: Math.hpp:299
static void sincosd(T x, T &sinx, T &cosx)
Definition: Math.cpp:106
static T sq(T x)
Definition: Math.hpp:212
static T AngNormalize(T x)
Definition: Math.cpp:71
static T atand(T x)
Definition: Math.cpp:202
static T AngDiff(T x, T y, T &e)
Definition: Math.cpp:82
@ qd
degrees per quarter turn
Definition: Math.hpp:141
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)