GeographicLib  1.51
SphericalEngine.cpp
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1 /**
2  * \file SphericalEngine.cpp
3  * \brief Implementation for GeographicLib::SphericalEngine class
4  *
5  * Copyright (c) Charles Karney (2011-2020) <charles@karney.com> and licensed
6  * under the MIT/X11 License. For more information, see
7  * https://geographiclib.sourceforge.io/
8  *
9  * The general sum is\verbatim
10  V(r, theta, lambda) = sum(n = 0..N) sum(m = 0..n)
11  q^(n+1) * (C[n,m] * cos(m*lambda) + S[n,m] * sin(m*lambda)) * P[n,m](t)
12 \endverbatim
13  * where <tt>t = cos(theta)</tt>, <tt>q = a/r</tt>. In addition write <tt>u =
14  * sin(theta)</tt>.
15  *
16  * <tt>P[n,m]</tt> is a normalized associated Legendre function of degree
17  * <tt>n</tt> and order <tt>m</tt>. Here the formulas are given for full
18  * normalized functions (usually denoted <tt>Pbar</tt>).
19  *
20  * Rewrite outer sum\verbatim
21  V(r, theta, lambda) = sum(m = 0..N) * P[m,m](t) * q^(m+1) *
22  [Sc[m] * cos(m*lambda) + Ss[m] * sin(m*lambda)]
23 \endverbatim
24  * where the inner sums are\verbatim
25  Sc[m] = sum(n = m..N) q^(n-m) * C[n,m] * P[n,m](t)/P[m,m](t)
26  Ss[m] = sum(n = m..N) q^(n-m) * S[n,m] * P[n,m](t)/P[m,m](t)
27 \endverbatim
28  * Evaluate sums via Clenshaw method. The overall framework is similar to
29  * Deakin with the following changes:
30  * - Clenshaw summation is used to roll the computation of
31  * <tt>cos(m*lambda)</tt> and <tt>sin(m*lambda)</tt> into the evaluation of
32  * the outer sum (rather than independently computing an array of these
33  * trigonometric terms).
34  * - Scale the coefficients to guard against overflow when <tt>N</tt> is large.
35  * .
36  * For the general framework of Clenshaw, see
37  * http://mathworld.wolfram.com/ClenshawRecurrenceFormula.html
38  *
39  * Let\verbatim
40  S = sum(k = 0..N) c[k] * F[k](x)
41  F[n+1](x) = alpha[n](x) * F[n](x) + beta[n](x) * F[n-1](x)
42 \endverbatim
43  * Evaluate <tt>S</tt> with\verbatim
44  y[N+2] = y[N+1] = 0
45  y[k] = alpha[k] * y[k+1] + beta[k+1] * y[k+2] + c[k]
46  S = c[0] * F[0] + y[1] * F[1] + beta[1] * F[0] * y[2]
47 \endverbatim
48  * \e IF <tt>F[0](x) = 1</tt> and <tt>beta(0,x) = 0</tt>, then <tt>F[1](x) =
49  * alpha(0,x)</tt> and we can continue the recursion for <tt>y[k]</tt> until
50  * <tt>y[0]</tt>, giving\verbatim
51  S = y[0]
52 \endverbatim
53  *
54  * Evaluating the inner sum\verbatim
55  l = n-m; n = l+m
56  Sc[m] = sum(l = 0..N-m) C[l+m,m] * q^l * P[l+m,m](t)/P[m,m](t)
57  F[l] = q^l * P[l+m,m](t)/P[m,m](t)
58 \endverbatim
59  * Holmes + Featherstone, Eq. (11), give\verbatim
60  P[n,m] = sqrt((2*n-1)*(2*n+1)/((n-m)*(n+m))) * t * P[n-1,m] -
61  sqrt((2*n+1)*(n+m-1)*(n-m-1)/((n-m)*(n+m)*(2*n-3))) * P[n-2,m]
62 \endverbatim
63  * thus\verbatim
64  alpha[l] = t * q * sqrt(((2*n+1)*(2*n+3))/
65  ((n-m+1)*(n+m+1)))
66  beta[l+1] = - q^2 * sqrt(((n-m+1)*(n+m+1)*(2*n+5))/
67  ((n-m+2)*(n+m+2)*(2*n+1)))
68 \endverbatim
69  * In this case, <tt>F[0] = 1</tt> and <tt>beta[0] = 0</tt>, so the <tt>Sc[m]
70  * = y[0]</tt>.
71  *
72  * Evaluating the outer sum\verbatim
73  V = sum(m = 0..N) Sc[m] * q^(m+1) * cos(m*lambda) * P[m,m](t)
74  + sum(m = 0..N) Ss[m] * q^(m+1) * cos(m*lambda) * P[m,m](t)
75  F[m] = q^(m+1) * cos(m*lambda) * P[m,m](t) [or sin(m*lambda)]
76 \endverbatim
77  * Holmes + Featherstone, Eq. (13), give\verbatim
78  P[m,m] = u * sqrt((2*m+1)/((m>1?2:1)*m)) * P[m-1,m-1]
79 \endverbatim
80  * also, we have\verbatim
81  cos((m+1)*lambda) = 2*cos(lambda)*cos(m*lambda) - cos((m-1)*lambda)
82 \endverbatim
83  * thus\verbatim
84  alpha[m] = 2*cos(lambda) * sqrt((2*m+3)/(2*(m+1))) * u * q
85  = cos(lambda) * sqrt( 2*(2*m+3)/(m+1) ) * u * q
86  beta[m+1] = -sqrt((2*m+3)*(2*m+5)/(4*(m+1)*(m+2))) * u^2 * q^2
87  * (m == 0 ? sqrt(2) : 1)
88 \endverbatim
89  * Thus\verbatim
90  F[0] = q [or 0]
91  F[1] = cos(lambda) * sqrt(3) * u * q^2 [or sin(lambda)]
92  beta[1] = - sqrt(15/4) * u^2 * q^2
93 \endverbatim
94  *
95  * Here is how the various components of the gradient are computed
96  *
97  * Differentiate wrt <tt>r</tt>\verbatim
98  d q^(n+1) / dr = (-1/r) * (n+1) * q^(n+1)
99 \endverbatim
100  * so multiply <tt>C[n,m]</tt> by <tt>n+1</tt> in inner sum and multiply the
101  * sum by <tt>-1/r</tt>.
102  *
103  * Differentiate wrt <tt>lambda</tt>\verbatim
104  d cos(m*lambda) = -m * sin(m*lambda)
105  d sin(m*lambda) = m * cos(m*lambda)
106 \endverbatim
107  * so multiply terms by <tt>m</tt> in outer sum and swap sine and cosine
108  * variables.
109  *
110  * Differentiate wrt <tt>theta</tt>\verbatim
111  dV/dtheta = V' = -u * dV/dt = -u * V'
112 \endverbatim
113  * here <tt>'</tt> denotes differentiation wrt <tt>theta</tt>.\verbatim
114  d/dtheta (Sc[m] * P[m,m](t)) = Sc'[m] * P[m,m](t) + Sc[m] * P'[m,m](t)
115 \endverbatim
116  * Now <tt>P[m,m](t) = const * u^m</tt>, so <tt>P'[m,m](t) = m * t/u *
117  * P[m,m](t)</tt>, thus\verbatim
118  d/dtheta (Sc[m] * P[m,m](t)) = (Sc'[m] + m * t/u * Sc[m]) * P[m,m](t)
119 \endverbatim
120  * Clenshaw recursion for <tt>Sc[m]</tt> reads\verbatim
121  y[k] = alpha[k] * y[k+1] + beta[k+1] * y[k+2] + c[k]
122 \endverbatim
123  * Substituting <tt>alpha[k] = const * t</tt>, <tt>alpha'[k] = -u/t *
124  * alpha[k]</tt>, <tt>beta'[k] = c'[k] = 0</tt> gives\verbatim
125  y'[k] = alpha[k] * y'[k+1] + beta[k+1] * y'[k+2] - u/t * alpha[k] * y[k+1]
126 \endverbatim
127  *
128  * Finally, given the derivatives of <tt>V</tt>, we can compute the components
129  * of the gradient in spherical coordinates and transform the result into
130  * cartesian coordinates.
131  **********************************************************************/
132 
135 #include <GeographicLib/Utility.hpp>
136 
137 #if defined(_MSC_VER)
138 // Squelch warnings about constant conditional expressions and potentially
139 // uninitialized local variables
140 # pragma warning (disable: 4127 4701)
141 #endif
142 
143 namespace GeographicLib {
144 
145  using namespace std;
146 
147  vector<Math::real>& SphericalEngine::sqrttable() {
148  static vector<real> sqrttable(0);
149  return sqrttable;
150  }
151 
152  template<bool gradp, SphericalEngine::normalization norm, int L>
153  Math::real SphericalEngine::Value(const coeff c[], const real f[],
154  real x, real y, real z, real a,
155  real& gradx, real& grady, real& gradz)
156  {
157  static_assert(L > 0, "L must be positive");
158  static_assert(norm == FULL || norm == SCHMIDT, "Unknown normalization");
159  int N = c[0].nmx(), M = c[0].mmx();
160 
161  real
162  p = hypot(x, y),
163  cl = p != 0 ? x / p : 1, // cos(lambda); at pole, pick lambda = 0
164  sl = p != 0 ? y / p : 0, // sin(lambda)
165  r = hypot(z, p),
166  t = r != 0 ? z / r : 0, // cos(theta); at origin, pick theta = pi/2
167  u = r != 0 ? max(p / r, eps()) : 1, // sin(theta); but avoid the pole
168  q = a / r;
169  real
170  q2 = Math::sq(q),
171  uq = u * q,
172  uq2 = Math::sq(uq),
173  tu = t / u;
174  // Initialize outer sum
175  real vc = 0, vc2 = 0, vs = 0, vs2 = 0; // v [N + 1], v [N + 2]
176  // vr, vt, vl and similar w variable accumulate the sums for the
177  // derivatives wrt r, theta, and lambda, respectively.
178  real vrc = 0, vrc2 = 0, vrs = 0, vrs2 = 0; // vr[N + 1], vr[N + 2]
179  real vtc = 0, vtc2 = 0, vts = 0, vts2 = 0; // vt[N + 1], vt[N + 2]
180  real vlc = 0, vlc2 = 0, vls = 0, vls2 = 0; // vl[N + 1], vl[N + 2]
181  int k[L];
182  const vector<real>& root( sqrttable() );
183  for (int m = M; m >= 0; --m) { // m = M .. 0
184  // Initialize inner sum
185  real
186  wc = 0, wc2 = 0, ws = 0, ws2 = 0, // w [N - m + 1], w [N - m + 2]
187  wrc = 0, wrc2 = 0, wrs = 0, wrs2 = 0, // wr[N - m + 1], wr[N - m + 2]
188  wtc = 0, wtc2 = 0, wts = 0, wts2 = 0; // wt[N - m + 1], wt[N - m + 2]
189  for (int l = 0; l < L; ++l)
190  k[l] = c[l].index(N, m) + 1;
191  for (int n = N; n >= m; --n) { // n = N .. m; l = N - m .. 0
192  real w, A, Ax, B, R; // alpha[l], beta[l + 1]
193  switch (norm) {
194  case FULL:
195  w = root[2 * n + 1] / (root[n - m + 1] * root[n + m + 1]);
196  Ax = q * w * root[2 * n + 3];
197  A = t * Ax;
198  B = - q2 * root[2 * n + 5] /
199  (w * root[n - m + 2] * root[n + m + 2]);
200  break;
201  case SCHMIDT:
202  w = root[n - m + 1] * root[n + m + 1];
203  Ax = q * (2 * n + 1) / w;
204  A = t * Ax;
205  B = - q2 * w / (root[n - m + 2] * root[n + m + 2]);
206  break;
207  default: break; // To suppress warning message from Visual Studio
208  }
209  R = c[0].Cv(--k[0]);
210  for (int l = 1; l < L; ++l)
211  R += c[l].Cv(--k[l], n, m, f[l]);
212  R *= scale();
213  w = A * wc + B * wc2 + R; wc2 = wc; wc = w;
214  if (gradp) {
215  w = A * wrc + B * wrc2 + (n + 1) * R; wrc2 = wrc; wrc = w;
216  w = A * wtc + B * wtc2 - u*Ax * wc2; wtc2 = wtc; wtc = w;
217  }
218  if (m) {
219  R = c[0].Sv(k[0]);
220  for (int l = 1; l < L; ++l)
221  R += c[l].Sv(k[l], n, m, f[l]);
222  R *= scale();
223  w = A * ws + B * ws2 + R; ws2 = ws; ws = w;
224  if (gradp) {
225  w = A * wrs + B * wrs2 + (n + 1) * R; wrs2 = wrs; wrs = w;
226  w = A * wts + B * wts2 - u*Ax * ws2; wts2 = wts; wts = w;
227  }
228  }
229  }
230  // Now Sc[m] = wc, Ss[m] = ws
231  // Sc'[m] = wtc, Ss'[m] = wtc
232  if (m) {
233  real v, A, B; // alpha[m], beta[m + 1]
234  switch (norm) {
235  case FULL:
236  v = root[2] * root[2 * m + 3] / root[m + 1];
237  A = cl * v * uq;
238  B = - v * root[2 * m + 5] / (root[8] * root[m + 2]) * uq2;
239  break;
240  case SCHMIDT:
241  v = root[2] * root[2 * m + 1] / root[m + 1];
242  A = cl * v * uq;
243  B = - v * root[2 * m + 3] / (root[8] * root[m + 2]) * uq2;
244  break;
245  default: break; // To suppress warning message from Visual Studio
246  }
247  v = A * vc + B * vc2 + wc ; vc2 = vc ; vc = v;
248  v = A * vs + B * vs2 + ws ; vs2 = vs ; vs = v;
249  if (gradp) {
250  // Include the terms Sc[m] * P'[m,m](t) and Ss[m] * P'[m,m](t)
251  wtc += m * tu * wc; wts += m * tu * ws;
252  v = A * vrc + B * vrc2 + wrc; vrc2 = vrc; vrc = v;
253  v = A * vrs + B * vrs2 + wrs; vrs2 = vrs; vrs = v;
254  v = A * vtc + B * vtc2 + wtc; vtc2 = vtc; vtc = v;
255  v = A * vts + B * vts2 + wts; vts2 = vts; vts = v;
256  v = A * vlc + B * vlc2 + m*ws; vlc2 = vlc; vlc = v;
257  v = A * vls + B * vls2 - m*wc; vls2 = vls; vls = v;
258  }
259  } else {
260  real A, B, qs;
261  switch (norm) {
262  case FULL:
263  A = root[3] * uq; // F[1]/(q*cl) or F[1]/(q*sl)
264  B = - root[15]/2 * uq2; // beta[1]/q
265  break;
266  case SCHMIDT:
267  A = uq;
268  B = - root[3]/2 * uq2;
269  break;
270  default: break; // To suppress warning message from Visual Studio
271  }
272  qs = q / scale();
273  vc = qs * (wc + A * (cl * vc + sl * vs ) + B * vc2);
274  if (gradp) {
275  qs /= r;
276  // The components of the gradient in spherical coordinates are
277  // r: dV/dr
278  // theta: 1/r * dV/dtheta
279  // lambda: 1/(r*u) * dV/dlambda
280  vrc = - qs * (wrc + A * (cl * vrc + sl * vrs) + B * vrc2);
281  vtc = qs * (wtc + A * (cl * vtc + sl * vts) + B * vtc2);
282  vlc = qs / u * ( A * (cl * vlc + sl * vls) + B * vlc2);
283  }
284  }
285  }
286 
287  if (gradp) {
288  // Rotate into cartesian (geocentric) coordinates
289  gradx = cl * (u * vrc + t * vtc) - sl * vlc;
290  grady = sl * (u * vrc + t * vtc) + cl * vlc;
291  gradz = t * vrc - u * vtc ;
292  }
293  return vc;
294  }
295 
296  template<bool gradp, SphericalEngine::normalization norm, int L>
297  CircularEngine SphericalEngine::Circle(const coeff c[], const real f[],
298  real p, real z, real a) {
299 
300  static_assert(L > 0, "L must be positive");
301  static_assert(norm == FULL || norm == SCHMIDT, "Unknown normalization");
302  int N = c[0].nmx(), M = c[0].mmx();
303 
304  real
305  r = hypot(z, p),
306  t = r != 0 ? z / r : 0, // cos(theta); at origin, pick theta = pi/2
307  u = r != 0 ? max(p / r, eps()) : 1, // sin(theta); but avoid the pole
308  q = a / r;
309  real
310  q2 = Math::sq(q),
311  tu = t / u;
312  CircularEngine circ(M, gradp, norm, a, r, u, t);
313  int k[L];
314  const vector<real>& root( sqrttable() );
315  for (int m = M; m >= 0; --m) { // m = M .. 0
316  // Initialize inner sum
317  real
318  wc = 0, wc2 = 0, ws = 0, ws2 = 0, // w [N - m + 1], w [N - m + 2]
319  wrc = 0, wrc2 = 0, wrs = 0, wrs2 = 0, // wr[N - m + 1], wr[N - m + 2]
320  wtc = 0, wtc2 = 0, wts = 0, wts2 = 0; // wt[N - m + 1], wt[N - m + 2]
321  for (int l = 0; l < L; ++l)
322  k[l] = c[l].index(N, m) + 1;
323  for (int n = N; n >= m; --n) { // n = N .. m; l = N - m .. 0
324  real w, A, Ax, B, R; // alpha[l], beta[l + 1]
325  switch (norm) {
326  case FULL:
327  w = root[2 * n + 1] / (root[n - m + 1] * root[n + m + 1]);
328  Ax = q * w * root[2 * n + 3];
329  A = t * Ax;
330  B = - q2 * root[2 * n + 5] /
331  (w * root[n - m + 2] * root[n + m + 2]);
332  break;
333  case SCHMIDT:
334  w = root[n - m + 1] * root[n + m + 1];
335  Ax = q * (2 * n + 1) / w;
336  A = t * Ax;
337  B = - q2 * w / (root[n - m + 2] * root[n + m + 2]);
338  break;
339  default: break; // To suppress warning message from Visual Studio
340  }
341  R = c[0].Cv(--k[0]);
342  for (int l = 1; l < L; ++l)
343  R += c[l].Cv(--k[l], n, m, f[l]);
344  R *= scale();
345  w = A * wc + B * wc2 + R; wc2 = wc; wc = w;
346  if (gradp) {
347  w = A * wrc + B * wrc2 + (n + 1) * R; wrc2 = wrc; wrc = w;
348  w = A * wtc + B * wtc2 - u*Ax * wc2; wtc2 = wtc; wtc = w;
349  }
350  if (m) {
351  R = c[0].Sv(k[0]);
352  for (int l = 1; l < L; ++l)
353  R += c[l].Sv(k[l], n, m, f[l]);
354  R *= scale();
355  w = A * ws + B * ws2 + R; ws2 = ws; ws = w;
356  if (gradp) {
357  w = A * wrs + B * wrs2 + (n + 1) * R; wrs2 = wrs; wrs = w;
358  w = A * wts + B * wts2 - u*Ax * ws2; wts2 = wts; wts = w;
359  }
360  }
361  }
362  if (!gradp)
363  circ.SetCoeff(m, wc, ws);
364  else {
365  // Include the terms Sc[m] * P'[m,m](t) and Ss[m] * P'[m,m](t)
366  wtc += m * tu * wc; wts += m * tu * ws;
367  circ.SetCoeff(m, wc, ws, wrc, wrs, wtc, wts);
368  }
369  }
370 
371  return circ;
372  }
373 
375  // Need square roots up to max(2 * N + 5, 15).
376  vector<real>& root( sqrttable() );
377  int L = max(2 * N + 5, 15) + 1, oldL = int(root.size());
378  if (oldL >= L)
379  return;
380  root.resize(L);
381  for (int l = oldL; l < L; ++l)
382  root[l] = sqrt(real(l));
383  }
384 
385  void SphericalEngine::coeff::readcoeffs(istream& stream, int& N, int& M,
386  vector<real>& C,
387  vector<real>& S,
388  bool truncate) {
389  if (truncate) {
390  if (!((N >= M && M >= 0) || (N == -1 && M == -1)))
391  // The last condition is that M = -1 implies N = -1.
392  throw GeographicErr("Bad requested degree and order " +
393  Utility::str(N) + " " + Utility::str(M));
394  }
395  int nm[2];
396  Utility::readarray<int, int, false>(stream, nm, 2);
397  int N0 = nm[0], M0 = nm[1];
398  if (!((N0 >= M0 && M0 >= 0) || (N0 == -1 && M0 == -1)))
399  // The last condition is that M0 = -1 implies N0 = -1.
400  throw GeographicErr("Bad degree and order " +
401  Utility::str(N0) + " " + Utility::str(M0));
402  N = truncate ? min(N, N0) : N0;
403  M = truncate ? min(M, M0) : M0;
404  C.resize(SphericalEngine::coeff::Csize(N, M));
405  S.resize(SphericalEngine::coeff::Ssize(N, M));
406  int skip = (SphericalEngine::coeff::Csize(N0, M0) -
407  SphericalEngine::coeff::Csize(N0, M )) * sizeof(double);
408  if (N == N0) {
409  Utility::readarray<double, real, false>(stream, C);
410  if (skip) stream.seekg(streamoff(skip), ios::cur);
411  Utility::readarray<double, real, false>(stream, S);
412  if (skip) stream.seekg(streamoff(skip), ios::cur);
413  } else {
414  for (int m = 0, k = 0; m <= M; ++m) {
415  Utility::readarray<double, real, false>(stream, &C[k], N + 1 - m);
416  stream.seekg((N0 - N) * sizeof(double), ios::cur);
417  k += N + 1 - m;
418  }
419  if (skip) stream.seekg(streamoff(skip), ios::cur);
420  for (int m = 1, k = 0; m <= M; ++m) {
421  Utility::readarray<double, real, false>(stream, &S[k], N + 1 - m);
422  stream.seekg((N0 - N) * sizeof(double), ios::cur);
423  k += N + 1 - m;
424  }
425  if (skip) stream.seekg(streamoff(skip), ios::cur);
426  }
427  return;
428  }
429 
430  /// \cond SKIP
432  SphericalEngine::Value<true, SphericalEngine::FULL, 1>
433  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
435  SphericalEngine::Value<false, SphericalEngine::FULL, 1>
436  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
438  SphericalEngine::Value<true, SphericalEngine::SCHMIDT, 1>
439  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
441  SphericalEngine::Value<false, SphericalEngine::SCHMIDT, 1>
442  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
443 
445  SphericalEngine::Value<true, SphericalEngine::FULL, 2>
446  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
448  SphericalEngine::Value<false, SphericalEngine::FULL, 2>
449  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
451  SphericalEngine::Value<true, SphericalEngine::SCHMIDT, 2>
452  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
454  SphericalEngine::Value<false, SphericalEngine::SCHMIDT, 2>
455  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
456 
458  SphericalEngine::Value<true, SphericalEngine::FULL, 3>
459  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
461  SphericalEngine::Value<false, SphericalEngine::FULL, 3>
462  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
464  SphericalEngine::Value<true, SphericalEngine::SCHMIDT, 3>
465  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
467  SphericalEngine::Value<false, SphericalEngine::SCHMIDT, 3>
468  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
469 
471  SphericalEngine::Circle<true, SphericalEngine::FULL, 1>
472  (const coeff[], const real[], real, real, real);
474  SphericalEngine::Circle<false, SphericalEngine::FULL, 1>
475  (const coeff[], const real[], real, real, real);
477  SphericalEngine::Circle<true, SphericalEngine::SCHMIDT, 1>
478  (const coeff[], const real[], real, real, real);
480  SphericalEngine::Circle<false, SphericalEngine::SCHMIDT, 1>
481  (const coeff[], const real[], real, real, real);
482 
484  SphericalEngine::Circle<true, SphericalEngine::FULL, 2>
485  (const coeff[], const real[], real, real, real);
487  SphericalEngine::Circle<false, SphericalEngine::FULL, 2>
488  (const coeff[], const real[], real, real, real);
490  SphericalEngine::Circle<true, SphericalEngine::SCHMIDT, 2>
491  (const coeff[], const real[], real, real, real);
493  SphericalEngine::Circle<false, SphericalEngine::SCHMIDT, 2>
494  (const coeff[], const real[], real, real, real);
495 
497  SphericalEngine::Circle<true, SphericalEngine::FULL, 3>
498  (const coeff[], const real[], real, real, real);
500  SphericalEngine::Circle<false, SphericalEngine::FULL, 3>
501  (const coeff[], const real[], real, real, real);
503  SphericalEngine::Circle<true, SphericalEngine::SCHMIDT, 3>
504  (const coeff[], const real[], real, real, real);
506  SphericalEngine::Circle<false, SphericalEngine::SCHMIDT, 3>
507  (const coeff[], const real[], real, real, real);
508  /// \endcond
509 
510 } // namespace GeographicLib
Header for GeographicLib::CircularEngine class.
#define GEOGRAPHICLIB_EXPORT
Definition: Constants.hpp:66
GeographicLib::Math::real real
Definition: GeodSolve.cpp:31
Header for GeographicLib::SphericalEngine class.
Header for GeographicLib::Utility class.
Spherical harmonic sums for a circle.
Exception handling for GeographicLib.
Definition: Constants.hpp:315
static T sq(T x)
Definition: Math.hpp:171
Package up coefficients for SphericalEngine.
static void readcoeffs(std::istream &stream, int &N, int &M, std::vector< real > &C, std::vector< real > &S, bool truncate=false)
static Math::real Value(const coeff c[], const real f[], real x, real y, real z, real a, real &gradx, real &grady, real &gradz)
static CircularEngine Circle(const coeff c[], const real f[], real p, real z, real a)
static std::string str(T x, int p=-1)
Definition: Utility.hpp:276
Namespace for GeographicLib.
Definition: Accumulator.cpp:12