GeographicLib  2.6
SphericalEngine.cpp
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1 /**
2  * \file SphericalEngine.cpp
3  * \brief Implementation for GeographicLib::SphericalEngine class
4  *
5  * Copyright (c) Charles Karney (2011-2022) <karney@alum.mit.edu> 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 
133 #include <GeographicLib/SphericalEngine.hpp>
134 #include <GeographicLib/CircularEngine.hpp>
135 #include <GeographicLib/Utility.hpp>
136 
137 #if defined(_MSC_VER)
138 // Squelch warnings about potentially uninitialized local variables
139 # pragma warning (disable: 4701)
140 #endif
141 
142 namespace GeographicLib {
143 
144  using namespace std;
145 
146  vector<Math::real>& SphericalEngine::sqrttable() {
147  static vector<real> sqrttable(0);
148  return sqrttable;
149  }
150 
151  template<bool gradp, SphericalEngine::normalization norm, int L>
152  Math::real SphericalEngine::Value(const coeff c[], const real f[],
153  real x, real y, real z, real a,
154  real& gradx, real& grady, real& gradz)
155  {
156  static_assert(L > 0, "L must be positive");
157  static_assert(norm == FULL || norm == SCHMIDT, "Unknown normalization");
158  int N = c[0].nmx(), M = c[0].mmx();
159 
160  real
161  p = hypot(x, y),
162  cl = p != 0 ? x / p : 1, // cos(lambda); at pole, pick lambda = 0
163  sl = p != 0 ? y / p : 0, // sin(lambda)
164  r = hypot(z, p),
165  t = r != 0 ? z / r : 0, // cos(theta); at origin, pick theta = pi/2
166  u = r != 0 ? fmax(p / r, eps()) : 1, // sin(theta); but avoid the pole
167  q = a / r;
168  real
169  q2 = Math::sq(q),
170  uq = u * q,
171  uq2 = Math::sq(uq),
172  tu = t / u;
173  // Initialize outer sum
174  real vc = 0, vc2 = 0, vs = 0, vs2 = 0; // v [N + 1], v [N + 2]
175  // vr, vt, vl and similar w variable accumulate the sums for the
176  // derivatives wrt r, theta, and lambda, respectively.
177  real vrc = 0, vrc2 = 0, vrs = 0, vrs2 = 0; // vr[N + 1], vr[N + 2]
178  real vtc = 0, vtc2 = 0, vts = 0, vts2 = 0; // vt[N + 1], vt[N + 2]
179  real vlc = 0, vlc2 = 0, vls = 0, vls2 = 0; // vl[N + 1], vl[N + 2]
180  int k[L];
181  const vector<real>& root( sqrttable() );
182  for (int m = M; m >= 0; --m) { // m = M .. 0
183  // Initialize inner sum
184  real
185  wc = 0, wc2 = 0, ws = 0, ws2 = 0, // w [N - m + 1], w [N - m + 2]
186  wrc = 0, wrc2 = 0, wrs = 0, wrs2 = 0, // wr[N - m + 1], wr[N - m + 2]
187  wtc = 0, wtc2 = 0, wts = 0, wts2 = 0; // wt[N - m + 1], wt[N - m + 2]
188  for (int l = 0; l < L; ++l)
189  k[l] = c[l].index(N, m) + 1;
190  for (int n = N; n >= m; --n) { // n = N .. m; l = N - m .. 0
191  real w, A, Ax, B, R; // alpha[l], beta[l + 1]
192  switch (norm) {
193  case FULL:
194  w = root[2 * n + 1] / (root[n - m + 1] * root[n + m + 1]);
195  Ax = q * w * root[2 * n + 3];
196  A = t * Ax;
197  B = - q2 * root[2 * n + 5] /
198  (w * root[n - m + 2] * root[n + m + 2]);
199  break;
200  case SCHMIDT:
201  w = root[n - m + 1] * root[n + m + 1];
202  Ax = q * (2 * n + 1) / w;
203  A = t * Ax;
204  B = - q2 * w / (root[n - m + 2] * root[n + m + 2]);
205  break;
206  default: break; // To suppress warning message from Visual Studio
207  }
208  R = c[0].Cv(--k[0]);
209  for (int l = 1; l < L; ++l)
210  R += c[l].Cv(--k[l], n, m, f[l]);
211  R *= scale();
212  w = A * wc + B * wc2 + R; wc2 = wc; wc = w;
213  if (gradp) {
214  w = A * wrc + B * wrc2 + (n + 1) * R; wrc2 = wrc; wrc = w;
215  w = A * wtc + B * wtc2 - u*Ax * wc2; wtc2 = wtc; wtc = w;
216  }
217  if (m) {
218  R = c[0].Sv(k[0]);
219  for (int l = 1; l < L; ++l)
220  R += c[l].Sv(k[l], n, m, f[l]);
221  R *= scale();
222  w = A * ws + B * ws2 + R; ws2 = ws; ws = w;
223  if (gradp) {
224  w = A * wrs + B * wrs2 + (n + 1) * R; wrs2 = wrs; wrs = w;
225  w = A * wts + B * wts2 - u*Ax * ws2; wts2 = wts; wts = w;
226  }
227  }
228  }
229  // Now Sc[m] = wc, Ss[m] = ws
230  // Sc'[m] = wtc, Ss'[m] = wtc
231  if (m) {
232  real v, A, B; // alpha[m], beta[m + 1]
233  switch (norm) {
234  case FULL:
235  v = root[2] * root[2 * m + 3] / root[m + 1];
236  A = cl * v * uq;
237  B = - v * root[2 * m + 5] / (root[8] * root[m + 2]) * uq2;
238  break;
239  case SCHMIDT:
240  v = root[2] * root[2 * m + 1] / root[m + 1];
241  A = cl * v * uq;
242  B = - v * root[2 * m + 3] / (root[8] * root[m + 2]) * uq2;
243  break;
244  default: break; // To suppress warning message from Visual Studio
245  }
246  v = A * vc + B * vc2 + wc ; vc2 = vc ; vc = v;
247  v = A * vs + B * vs2 + ws ; vs2 = vs ; vs = v;
248  if (gradp) {
249  // Include the terms Sc[m] * P'[m,m](t) and Ss[m] * P'[m,m](t)
250  wtc += m * tu * wc; wts += m * tu * ws;
251  v = A * vrc + B * vrc2 + wrc; vrc2 = vrc; vrc = v;
252  v = A * vrs + B * vrs2 + wrs; vrs2 = vrs; vrs = v;
253  v = A * vtc + B * vtc2 + wtc; vtc2 = vtc; vtc = v;
254  v = A * vts + B * vts2 + wts; vts2 = vts; vts = v;
255  v = A * vlc + B * vlc2 + m*ws; vlc2 = vlc; vlc = v;
256  v = A * vls + B * vls2 - m*wc; vls2 = vls; vls = v;
257  }
258  } else {
259  real A, B, qs;
260  switch (norm) {
261  case FULL:
262  A = root[3] * uq; // F[1]/(q*cl) or F[1]/(q*sl)
263  B = - root[15]/2 * uq2; // beta[1]/q
264  break;
265  case SCHMIDT:
266  A = uq;
267  B = - root[3]/2 * uq2;
268  break;
269  default: break; // To suppress warning message from Visual Studio
270  }
271  qs = q / scale();
272  vc = qs * (wc + A * (cl * vc + sl * vs ) + B * vc2);
273  if (gradp) {
274  qs /= r;
275  // The components of the gradient in spherical coordinates are
276  // r: dV/dr
277  // theta: 1/r * dV/dtheta
278  // lambda: 1/(r*u) * dV/dlambda
279  vrc = - qs * (wrc + A * (cl * vrc + sl * vrs) + B * vrc2);
280  vtc = qs * (wtc + A * (cl * vtc + sl * vts) + B * vtc2);
281  vlc = qs / u * ( A * (cl * vlc + sl * vls) + B * vlc2);
282  }
283  }
284  }
285 
286  if (gradp) {
287  // Rotate into cartesian (geocentric) coordinates
288  gradx = cl * (u * vrc + t * vtc) - sl * vlc;
289  grady = sl * (u * vrc + t * vtc) + cl * vlc;
290  gradz = t * vrc - u * vtc ;
291  }
292  return vc;
293  }
294 
295  template<bool gradp, SphericalEngine::normalization norm, int L>
296  CircularEngine SphericalEngine::Circle(const coeff c[], const real f[],
297  real p, real z, real a) {
298 
299  static_assert(L > 0, "L must be positive");
300  static_assert(norm == FULL || norm == SCHMIDT, "Unknown normalization");
301  int N = c[0].nmx(), M = c[0].mmx();
302 
303  real
304  r = hypot(z, p),
305  t = r != 0 ? z / r : 0, // cos(theta); at origin, pick theta = pi/2
306  u = r != 0 ? fmax(p / r, eps()) : 1, // sin(theta); but avoid the pole
307  q = a / r;
308  real
309  q2 = Math::sq(q),
310  tu = t / u;
311  CircularEngine circ(M, gradp, norm, a, r, u, t);
312  int k[L];
313  const vector<real>& root( sqrttable() );
314  for (int m = M; m >= 0; --m) { // m = M .. 0
315  // Initialize inner sum
316  real
317  wc = 0, wc2 = 0, ws = 0, ws2 = 0, // w [N - m + 1], w [N - m + 2]
318  wrc = 0, wrc2 = 0, wrs = 0, wrs2 = 0, // wr[N - m + 1], wr[N - m + 2]
319  wtc = 0, wtc2 = 0, wts = 0, wts2 = 0; // wt[N - m + 1], wt[N - m + 2]
320  for (int l = 0; l < L; ++l)
321  k[l] = c[l].index(N, m) + 1;
322  for (int n = N; n >= m; --n) { // n = N .. m; l = N - m .. 0
323  real w, A, Ax, B, R; // alpha[l], beta[l + 1]
324  switch (norm) {
325  case FULL:
326  w = root[2 * n + 1] / (root[n - m + 1] * root[n + m + 1]);
327  Ax = q * w * root[2 * n + 3];
328  A = t * Ax;
329  B = - q2 * root[2 * n + 5] /
330  (w * root[n - m + 2] * root[n + m + 2]);
331  break;
332  case SCHMIDT:
333  w = root[n - m + 1] * root[n + m + 1];
334  Ax = q * (2 * n + 1) / w;
335  A = t * Ax;
336  B = - q2 * w / (root[n - m + 2] * root[n + m + 2]);
337  break;
338  default: break; // To suppress warning message from Visual Studio
339  }
340  R = c[0].Cv(--k[0]);
341  for (int l = 1; l < L; ++l)
342  R += c[l].Cv(--k[l], n, m, f[l]);
343  R *= scale();
344  w = A * wc + B * wc2 + R; wc2 = wc; wc = w;
345  if (gradp) {
346  w = A * wrc + B * wrc2 + (n + 1) * R; wrc2 = wrc; wrc = w;
347  w = A * wtc + B * wtc2 - u*Ax * wc2; wtc2 = wtc; wtc = w;
348  }
349  if (m) {
350  R = c[0].Sv(k[0]);
351  for (int l = 1; l < L; ++l)
352  R += c[l].Sv(k[l], n, m, f[l]);
353  R *= scale();
354  w = A * ws + B * ws2 + R; ws2 = ws; ws = w;
355  if (gradp) {
356  w = A * wrs + B * wrs2 + (n + 1) * R; wrs2 = wrs; wrs = w;
357  w = A * wts + B * wts2 - u*Ax * ws2; wts2 = wts; wts = w;
358  }
359  }
360  }
361  if (!gradp)
362  circ.SetCoeff(m, wc, ws);
363  else {
364  // Include the terms Sc[m] * P'[m,m](t) and Ss[m] * P'[m,m](t)
365  wtc += m * tu * wc; wts += m * tu * ws;
366  circ.SetCoeff(m, wc, ws, wrc, wrs, wtc, wts);
367  }
368  }
369 
370  return circ;
371  }
372 
373  void SphericalEngine::RootTable(int N) {
374  // Need square roots up to max(2 * N + 5, 15).
375  vector<real>& root( sqrttable() );
376  int L = max(2 * N + 5, 15) + 1, oldL = int(root.size());
377  if (oldL >= L)
378  return;
379  root.resize(L);
380  for (int l = oldL; l < L; ++l)
381  root[l] = sqrt(real(l));
382  }
383 
384  void SphericalEngine::coeff::readcoeffs(istream& stream, int& N, int& M,
385  vector<real>& C,
386  vector<real>& S,
387  bool truncate) {
388  if (truncate) {
389  if (!((N >= M && M >= 0) || (N == -1 && M == -1)))
390  // The last condition is that M = -1 implies N = -1.
391  throw GeographicErr("Bad requested degree and order " +
392  Utility::str(N) + " " + Utility::str(M));
393  }
394  int nm[2];
395  Utility::readarray<int, int, false>(stream, nm, 2);
396  int N0 = nm[0], M0 = nm[1];
397  if (!((N0 >= M0 && M0 >= 0) || (N0 == -1 && M0 == -1)))
398  // The last condition is that M0 = -1 implies N0 = -1.
399  throw GeographicErr("Bad degree and order " +
400  Utility::str(N0) + " " + Utility::str(M0));
401  N = truncate ? min(N, N0) : N0;
402  M = truncate ? min(M, M0) : M0;
403  C.resize(SphericalEngine::coeff::Csize(N, M));
404  S.resize(SphericalEngine::coeff::Ssize(N, M));
405  int skip = (SphericalEngine::coeff::Csize(N0, M0) -
406  SphericalEngine::coeff::Csize(N0, M )) * sizeof(double);
407  if (N == N0) {
408  Utility::readarray<double, real, false>(stream, C);
409  if (skip) stream.seekg(streamoff(skip), ios::cur);
410  Utility::readarray<double, real, false>(stream, S);
411  if (skip) stream.seekg(streamoff(skip), ios::cur);
412  } else {
413  for (int m = 0, k = 0; m <= M; ++m) {
414  Utility::readarray<double, real, false>(stream, &C[k], N + 1 - m);
415  stream.seekg((N0 - N) * sizeof(double), ios::cur);
416  k += N + 1 - m;
417  }
418  if (skip) stream.seekg(streamoff(skip), ios::cur);
419  for (int m = 1, k = 0; m <= M; ++m) {
420  Utility::readarray<double, real, false>(stream, &S[k], N + 1 - m);
421  stream.seekg((N0 - N) * sizeof(double), ios::cur);
422  k += N + 1 - m;
423  }
424  if (skip) stream.seekg(streamoff(skip), ios::cur);
425  }
426  return;
427  }
428 
429  /// \cond SKIP
430  template Math::real GEOGRAPHICLIB_EXPORT
431  SphericalEngine::Value<true, SphericalEngine::FULL, 1>
432  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
433  template Math::real GEOGRAPHICLIB_EXPORT
434  SphericalEngine::Value<false, SphericalEngine::FULL, 1>
435  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
436  template Math::real GEOGRAPHICLIB_EXPORT
437  SphericalEngine::Value<true, SphericalEngine::SCHMIDT, 1>
438  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
439  template Math::real GEOGRAPHICLIB_EXPORT
440  SphericalEngine::Value<false, SphericalEngine::SCHMIDT, 1>
441  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
442 
443  template Math::real GEOGRAPHICLIB_EXPORT
444  SphericalEngine::Value<true, SphericalEngine::FULL, 2>
445  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
446  template Math::real GEOGRAPHICLIB_EXPORT
447  SphericalEngine::Value<false, SphericalEngine::FULL, 2>
448  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
449  template Math::real GEOGRAPHICLIB_EXPORT
450  SphericalEngine::Value<true, SphericalEngine::SCHMIDT, 2>
451  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
452  template Math::real GEOGRAPHICLIB_EXPORT
453  SphericalEngine::Value<false, SphericalEngine::SCHMIDT, 2>
454  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
455 
456  template Math::real GEOGRAPHICLIB_EXPORT
457  SphericalEngine::Value<true, SphericalEngine::FULL, 3>
458  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
459  template Math::real GEOGRAPHICLIB_EXPORT
460  SphericalEngine::Value<false, SphericalEngine::FULL, 3>
461  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
462  template Math::real GEOGRAPHICLIB_EXPORT
463  SphericalEngine::Value<true, SphericalEngine::SCHMIDT, 3>
464  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
465  template Math::real GEOGRAPHICLIB_EXPORT
466  SphericalEngine::Value<false, SphericalEngine::SCHMIDT, 3>
467  (const coeff[], const real[], real, real, real, real, real&, real&, real&);
468 
469  template CircularEngine GEOGRAPHICLIB_EXPORT
470  SphericalEngine::Circle<true, SphericalEngine::FULL, 1>
471  (const coeff[], const real[], real, real, real);
472  template CircularEngine GEOGRAPHICLIB_EXPORT
473  SphericalEngine::Circle<false, SphericalEngine::FULL, 1>
474  (const coeff[], const real[], real, real, real);
475  template CircularEngine GEOGRAPHICLIB_EXPORT
476  SphericalEngine::Circle<true, SphericalEngine::SCHMIDT, 1>
477  (const coeff[], const real[], real, real, real);
478  template CircularEngine GEOGRAPHICLIB_EXPORT
479  SphericalEngine::Circle<false, SphericalEngine::SCHMIDT, 1>
480  (const coeff[], const real[], real, real, real);
481 
482  template CircularEngine GEOGRAPHICLIB_EXPORT
483  SphericalEngine::Circle<true, SphericalEngine::FULL, 2>
484  (const coeff[], const real[], real, real, real);
485  template CircularEngine GEOGRAPHICLIB_EXPORT
486  SphericalEngine::Circle<false, SphericalEngine::FULL, 2>
487  (const coeff[], const real[], real, real, real);
488  template CircularEngine GEOGRAPHICLIB_EXPORT
489  SphericalEngine::Circle<true, SphericalEngine::SCHMIDT, 2>
490  (const coeff[], const real[], real, real, real);
491  template CircularEngine GEOGRAPHICLIB_EXPORT
492  SphericalEngine::Circle<false, SphericalEngine::SCHMIDT, 2>
493  (const coeff[], const real[], real, real, real);
494 
495  template CircularEngine GEOGRAPHICLIB_EXPORT
496  SphericalEngine::Circle<true, SphericalEngine::FULL, 3>
497  (const coeff[], const real[], real, real, real);
498  template CircularEngine GEOGRAPHICLIB_EXPORT
499  SphericalEngine::Circle<false, SphericalEngine::FULL, 3>
500  (const coeff[], const real[], real, real, real);
501  template CircularEngine GEOGRAPHICLIB_EXPORT
502  SphericalEngine::Circle<true, SphericalEngine::SCHMIDT, 3>
503  (const coeff[], const real[], real, real, real);
504  template CircularEngine GEOGRAPHICLIB_EXPORT
505  SphericalEngine::Circle<false, SphericalEngine::SCHMIDT, 3>
506  (const coeff[], const real[], real, real, real);
507  /// \endcond
508 
509 } // namespace GeographicLib
GeographicLib::SphericalEngine::coeff::readcoeffs
static void readcoeffs(std::istream &stream, int &N, int &M, std::vector< real > &C, std::vector< real > &S, bool truncate=false)
Definition: SphericalEngine.cpp:384
GEOGRAPHICLIB_EXPORT
#define GEOGRAPHICLIB_EXPORT
Definition: Constants.hpp:59
Utility.hpp
Header for GeographicLib::Utility class.
GeographicLib::SphericalEngine::Circle
static CircularEngine Circle(const coeff c[], const real f[], real p, real z, real a)
Definition: SphericalEngine.cpp:296
GeographicLib::SphericalEngine::coeff
Package up coefficients for SphericalEngine.
Definition: SphericalEngine.hpp:99
GeographicLib::Math::sq
static T sq(T x)
Definition: Math.hpp:209
GeographicLib
Namespace for GeographicLib.
Definition: Accumulator.cpp:12
GeographicLib::Utility::str
static std::string str(T x, int p=-1)
Definition: Utility.hpp:161
real
GeographicLib::Math::real real
Definition: Geod3Solve.cpp:25
CircularEngine.hpp
Header for GeographicLib::CircularEngine class.
GeographicLib::CircularEngine
Spherical harmonic sums for a circle.
Definition: CircularEngine.hpp:52
GeographicLib::GeographicErr
Exception handling for GeographicLib.
Definition: Constants.hpp:344
GeographicLib::SphericalEngine::Value
static Math::real Value(const coeff c[], const real f[], real x, real y, real z, real a, real &gradx, real &grady, real &gradz)
Definition: SphericalEngine.cpp:152
SphericalEngine.hpp
Header for GeographicLib::SphericalEngine class.