GeographicLib  1.51
GravityModel.hpp
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1 /**
2  * \file GravityModel.hpp
3  * \brief Header for GeographicLib::GravityModel 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 
10 #if !defined(GEOGRAPHICLIB_GRAVITYMODEL_HPP)
11 #define GEOGRAPHICLIB_GRAVITYMODEL_HPP 1
12 
17 
18 #if defined(_MSC_VER)
19 // Squelch warnings about dll vs vector
20 # pragma warning (push)
21 # pragma warning (disable: 4251)
22 #endif
23 
24 namespace GeographicLib {
25 
26  class GravityCircle;
27 
28  /**
29  * \brief Model of the earth's gravity field
30  *
31  * Evaluate the earth's gravity field according to a model. The supported
32  * models treat only the gravitational field exterior to the mass of the
33  * earth. When computing the field at points near (but above) the surface of
34  * the earth a small correction can be applied to account for the mass of the
35  * atmosphere above the point in question; see \ref gravityatmos.
36  * Determining the height of the geoid above the ellipsoid entails correcting
37  * for the mass of the earth above the geoid. The egm96 and egm2008 include
38  * separate correction terms to account for this mass.
39  *
40  * Definitions and terminology (from Heiskanen and Moritz, Sec 2-13):
41  * - \e V = gravitational potential;
42  * - &Phi; = rotational potential;
43  * - \e W = \e V + &Phi; = \e T + \e U = total potential;
44  * - <i>V</i><sub>0</sub> = normal gravitation potential;
45  * - \e U = <i>V</i><sub>0</sub> + &Phi; = total normal potential;
46  * - \e T = \e W &minus; \e U = \e V &minus; <i>V</i><sub>0</sub> = anomalous
47  * or disturbing potential;
48  * - <b>g</b> = &nabla;\e W = <b>&gamma;</b> + <b>&delta;</b>;
49  * - <b>f</b> = &nabla;&Phi;;
50  * - <b>&Gamma;</b> = &nabla;<i>V</i><sub>0</sub>;
51  * - <b>&gamma;</b> = &nabla;\e U;
52  * - <b>&delta;</b> = &nabla;\e T = gravity disturbance vector
53  * = <b>g</b><sub><i>P</i></sub> &minus; <b>&gamma;</b><sub><i>P</i></sub>;
54  * - &delta;\e g = gravity disturbance = <i>g</i><sub><i>P</i></sub> &minus;
55  * &gamma;<sub><i>P</i></sub>;
56  * - &Delta;<b>g</b> = gravity anomaly vector = <b>g</b><sub><i>P</i></sub>
57  * &minus; <b>&gamma;</b><sub><i>Q</i></sub>; here the line \e PQ is
58  * perpendicular to ellipsoid and the potential at \e P equals the normal
59  * potential at \e Q;
60  * - &Delta;\e g = gravity anomaly = <i>g</i><sub><i>P</i></sub> &minus;
61  * &gamma;<sub><i>Q</i></sub>;
62  * - (&xi;, &eta;) deflection of the vertical, the difference in
63  * directions of <b>g</b><sub><i>P</i></sub> and
64  * <b>&gamma;</b><sub><i>Q</i></sub>, &xi; = NS, &eta; = EW.
65  * - \e X, \e Y, \e Z, geocentric coordinates;
66  * - \e x, \e y, \e z, local cartesian coordinates used to denote the east,
67  * north and up directions.
68  *
69  * See \ref gravity for details of how to install the gravity models and the
70  * data format.
71  *
72  * References:
73  * - W. A. Heiskanen and H. Moritz, Physical Geodesy (Freeman, San
74  * Francisco, 1967).
75  *
76  * Example of use:
77  * \include example-GravityModel.cpp
78  *
79  * <a href="Gravity.1.html">Gravity</a> is a command-line utility providing
80  * access to the functionality of GravityModel and GravityCircle.
81  **********************************************************************/
82 
84  private:
85  typedef Math::real real;
86  friend class GravityCircle;
87  static const int idlength_ = 8;
88  std::string _name, _dir, _description, _date, _filename, _id;
89  real _amodel, _GMmodel, _zeta0, _corrmult;
90  int _nmx, _mmx;
92  NormalGravity _earth;
93  std::vector<real> _Cx, _Sx, _CC, _CS, _zonal;
94  real _dzonal0; // A left over contribution to _zonal.
95  SphericalHarmonic _gravitational;
96  SphericalHarmonic1 _disturbing;
97  SphericalHarmonic _correction;
98  void ReadMetadata(const std::string& name);
99  Math::real InternalT(real X, real Y, real Z,
100  real& deltaX, real& deltaY, real& deltaZ,
101  bool gradp, bool correct) const;
102  GravityModel(const GravityModel&); // copy constructor not allowed
103  GravityModel& operator=(const GravityModel&); // nor copy assignment
104 
105  enum captype {
106  CAP_NONE = 0U,
107  CAP_G = 1U<<0, // implies potentials W and V
108  CAP_T = 1U<<1,
109  CAP_DELTA = 1U<<2 | CAP_T, // delta implies T?
110  CAP_C = 1U<<3,
111  CAP_GAMMA0 = 1U<<4,
112  CAP_GAMMA = 1U<<5,
113  CAP_ALL = 0x3FU,
114  };
115 
116  public:
117 
118  /**
119  * Bit masks for the capabilities to be given to the GravityCircle object
120  * produced by Circle.
121  **********************************************************************/
122  enum mask {
123  /**
124  * No capabilities.
125  * @hideinitializer
126  **********************************************************************/
127  NONE = 0U,
128  /**
129  * Allow calls to GravityCircle::Gravity, GravityCircle::W, and
130  * GravityCircle::V.
131  * @hideinitializer
132  **********************************************************************/
133  GRAVITY = CAP_G,
134  /**
135  * Allow calls to GravityCircle::Disturbance and GravityCircle::T.
136  * @hideinitializer
137  **********************************************************************/
138  DISTURBANCE = CAP_DELTA,
139  /**
140  * Allow calls to GravityCircle::T(real lon) (i.e., computing the
141  * disturbing potential and not the gravity disturbance vector).
142  * @hideinitializer
143  **********************************************************************/
144  DISTURBING_POTENTIAL = CAP_T,
145  /**
146  * Allow calls to GravityCircle::SphericalAnomaly.
147  * @hideinitializer
148  **********************************************************************/
149  SPHERICAL_ANOMALY = CAP_DELTA | CAP_GAMMA,
150  /**
151  * Allow calls to GravityCircle::GeoidHeight.
152  * @hideinitializer
153  **********************************************************************/
154  GEOID_HEIGHT = CAP_T | CAP_C | CAP_GAMMA0,
155  /**
156  * All capabilities.
157  * @hideinitializer
158  **********************************************************************/
159  ALL = CAP_ALL,
160  };
161  /** \name Setting up the gravity model
162  **********************************************************************/
163  ///@{
164  /**
165  * Construct a gravity model.
166  *
167  * @param[in] name the name of the model.
168  * @param[in] path (optional) directory for data file.
169  * @param[in] Nmax (optional) if non-negative, truncate the degree of the
170  * model this value.
171  * @param[in] Mmax (optional) if non-negative, truncate the order of the
172  * model this value.
173  * @exception GeographicErr if the data file cannot be found, is
174  * unreadable, or is corrupt, or if \e Mmax > \e Nmax.
175  * @exception std::bad_alloc if the memory necessary for storing the model
176  * can't be allocated.
177  *
178  * A filename is formed by appending ".egm" (World Gravity Model) to the
179  * name. If \e path is specified (and is non-empty), then the file is
180  * loaded from directory, \e path. Otherwise the path is given by
181  * DefaultGravityPath().
182  *
183  * This file contains the metadata which specifies the properties of the
184  * model. The coefficients for the spherical harmonic sums are obtained
185  * from a file obtained by appending ".cof" to metadata file (so the
186  * filename ends in ".egm.cof").
187  *
188  * If \e Nmax &ge; 0 and \e Mmax < 0, then \e Mmax is set to \e Nmax.
189  * After the model is loaded, the maximum degree and order of the model can
190  * be found by the Degree() and Order() methods.
191  **********************************************************************/
192  explicit GravityModel(const std::string& name,
193  const std::string& path = "",
194  int Nmax = -1, int Mmax = -1);
195  ///@}
196 
197  /** \name Compute gravity in geodetic coordinates
198  **********************************************************************/
199  ///@{
200  /**
201  * Evaluate the gravity at an arbitrary point above (or below) the
202  * ellipsoid.
203  *
204  * @param[in] lat the geographic latitude (degrees).
205  * @param[in] lon the geographic longitude (degrees).
206  * @param[in] h the height above the ellipsoid (meters).
207  * @param[out] gx the easterly component of the acceleration
208  * (m s<sup>&minus;2</sup>).
209  * @param[out] gy the northerly component of the acceleration
210  * (m s<sup>&minus;2</sup>).
211  * @param[out] gz the upward component of the acceleration
212  * (m s<sup>&minus;2</sup>); this is usually negative.
213  * @return \e W the sum of the gravitational and centrifugal potentials
214  * (m<sup>2</sup> s<sup>&minus;2</sup>).
215  *
216  * The function includes the effects of the earth's rotation.
217  **********************************************************************/
218  Math::real Gravity(real lat, real lon, real h,
219  real& gx, real& gy, real& gz) const;
220 
221  /**
222  * Evaluate the gravity disturbance vector at an arbitrary point above (or
223  * below) the ellipsoid.
224  *
225  * @param[in] lat the geographic latitude (degrees).
226  * @param[in] lon the geographic longitude (degrees).
227  * @param[in] h the height above the ellipsoid (meters).
228  * @param[out] deltax the easterly component of the disturbance vector
229  * (m s<sup>&minus;2</sup>).
230  * @param[out] deltay the northerly component of the disturbance vector
231  * (m s<sup>&minus;2</sup>).
232  * @param[out] deltaz the upward component of the disturbance vector
233  * (m s<sup>&minus;2</sup>).
234  * @return \e T the corresponding disturbing potential
235  * (m<sup>2</sup> s<sup>&minus;2</sup>).
236  **********************************************************************/
237  Math::real Disturbance(real lat, real lon, real h,
238  real& deltax, real& deltay, real& deltaz)
239  const;
240 
241  /**
242  * Evaluate the geoid height.
243  *
244  * @param[in] lat the geographic latitude (degrees).
245  * @param[in] lon the geographic longitude (degrees).
246  * @return \e N the height of the geoid above the ReferenceEllipsoid()
247  * (meters).
248  *
249  * This calls NormalGravity::U for ReferenceEllipsoid(). Some
250  * approximations are made in computing the geoid height so that the
251  * results of the NGA codes are reproduced accurately. Details are given
252  * in \ref gravitygeoid.
253  **********************************************************************/
254  Math::real GeoidHeight(real lat, real lon) const;
255 
256  /**
257  * Evaluate the components of the gravity anomaly vector using the
258  * spherical approximation.
259  *
260  * @param[in] lat the geographic latitude (degrees).
261  * @param[in] lon the geographic longitude (degrees).
262  * @param[in] h the height above the ellipsoid (meters).
263  * @param[out] Dg01 the gravity anomaly (m s<sup>&minus;2</sup>).
264  * @param[out] xi the northerly component of the deflection of the vertical
265  * (degrees).
266  * @param[out] eta the easterly component of the deflection of the vertical
267  * (degrees).
268  *
269  * The spherical approximation (see Heiskanen and Moritz, Sec 2-14) is used
270  * so that the results of the NGA codes are reproduced accurately.
271  * approximations used here. Details are given in \ref gravitygeoid.
272  **********************************************************************/
273  void SphericalAnomaly(real lat, real lon, real h,
274  real& Dg01, real& xi, real& eta) const;
275  ///@}
276 
277  /** \name Compute gravity in geocentric coordinates
278  **********************************************************************/
279  ///@{
280  /**
281  * Evaluate the components of the acceleration due to gravity and the
282  * centrifugal acceleration in geocentric coordinates.
283  *
284  * @param[in] X geocentric coordinate of point (meters).
285  * @param[in] Y geocentric coordinate of point (meters).
286  * @param[in] Z geocentric coordinate of point (meters).
287  * @param[out] gX the \e X component of the acceleration
288  * (m s<sup>&minus;2</sup>).
289  * @param[out] gY the \e Y component of the acceleration
290  * (m s<sup>&minus;2</sup>).
291  * @param[out] gZ the \e Z component of the acceleration
292  * (m s<sup>&minus;2</sup>).
293  * @return \e W = \e V + &Phi; the sum of the gravitational and
294  * centrifugal potentials (m<sup>2</sup> s<sup>&minus;2</sup>).
295  *
296  * This calls NormalGravity::U for ReferenceEllipsoid().
297  **********************************************************************/
298  Math::real W(real X, real Y, real Z,
299  real& gX, real& gY, real& gZ) const;
300 
301  /**
302  * Evaluate the components of the acceleration due to gravity in geocentric
303  * coordinates.
304  *
305  * @param[in] X geocentric coordinate of point (meters).
306  * @param[in] Y geocentric coordinate of point (meters).
307  * @param[in] Z geocentric coordinate of point (meters).
308  * @param[out] GX the \e X component of the acceleration
309  * (m s<sup>&minus;2</sup>).
310  * @param[out] GY the \e Y component of the acceleration
311  * (m s<sup>&minus;2</sup>).
312  * @param[out] GZ the \e Z component of the acceleration
313  * (m s<sup>&minus;2</sup>).
314  * @return \e V = \e W - &Phi; the gravitational potential
315  * (m<sup>2</sup> s<sup>&minus;2</sup>).
316  **********************************************************************/
317  Math::real V(real X, real Y, real Z,
318  real& GX, real& GY, real& GZ) const;
319 
320  /**
321  * Evaluate the components of the gravity disturbance in geocentric
322  * coordinates.
323  *
324  * @param[in] X geocentric coordinate of point (meters).
325  * @param[in] Y geocentric coordinate of point (meters).
326  * @param[in] Z geocentric coordinate of point (meters).
327  * @param[out] deltaX the \e X component of the gravity disturbance
328  * (m s<sup>&minus;2</sup>).
329  * @param[out] deltaY the \e Y component of the gravity disturbance
330  * (m s<sup>&minus;2</sup>).
331  * @param[out] deltaZ the \e Z component of the gravity disturbance
332  * (m s<sup>&minus;2</sup>).
333  * @return \e T = \e W - \e U the disturbing potential (also called the
334  * anomalous potential) (m<sup>2</sup> s<sup>&minus;2</sup>).
335  **********************************************************************/
336  Math::real T(real X, real Y, real Z,
337  real& deltaX, real& deltaY, real& deltaZ) const
338  { return InternalT(X, Y, Z, deltaX, deltaY, deltaZ, true, true); }
339 
340  /**
341  * Evaluate disturbing potential in geocentric coordinates.
342  *
343  * @param[in] X geocentric coordinate of point (meters).
344  * @param[in] Y geocentric coordinate of point (meters).
345  * @param[in] Z geocentric coordinate of point (meters).
346  * @return \e T = \e W - \e U the disturbing potential (also called the
347  * anomalous potential) (m<sup>2</sup> s<sup>&minus;2</sup>).
348  **********************************************************************/
349  Math::real T(real X, real Y, real Z) const {
350  real dummy;
351  return InternalT(X, Y, Z, dummy, dummy, dummy, false, true);
352  }
353 
354  /**
355  * Evaluate the components of the acceleration due to normal gravity and
356  * the centrifugal acceleration in geocentric coordinates.
357  *
358  * @param[in] X geocentric coordinate of point (meters).
359  * @param[in] Y geocentric coordinate of point (meters).
360  * @param[in] Z geocentric coordinate of point (meters).
361  * @param[out] gammaX the \e X component of the normal acceleration
362  * (m s<sup>&minus;2</sup>).
363  * @param[out] gammaY the \e Y component of the normal acceleration
364  * (m s<sup>&minus;2</sup>).
365  * @param[out] gammaZ the \e Z component of the normal acceleration
366  * (m s<sup>&minus;2</sup>).
367  * @return \e U = <i>V</i><sub>0</sub> + &Phi; the sum of the
368  * normal gravitational and centrifugal potentials
369  * (m<sup>2</sup> s<sup>&minus;2</sup>).
370  *
371  * This calls NormalGravity::U for ReferenceEllipsoid().
372  **********************************************************************/
373  Math::real U(real X, real Y, real Z,
374  real& gammaX, real& gammaY, real& gammaZ) const
375  { return _earth.U(X, Y, Z, gammaX, gammaY, gammaZ); }
376 
377  /**
378  * Evaluate the centrifugal acceleration in geocentric coordinates.
379  *
380  * @param[in] X geocentric coordinate of point (meters).
381  * @param[in] Y geocentric coordinate of point (meters).
382  * @param[out] fX the \e X component of the centrifugal acceleration
383  * (m s<sup>&minus;2</sup>).
384  * @param[out] fY the \e Y component of the centrifugal acceleration
385  * (m s<sup>&minus;2</sup>).
386  * @return &Phi; the centrifugal potential (m<sup>2</sup>
387  * s<sup>&minus;2</sup>).
388  *
389  * This calls NormalGravity::Phi for ReferenceEllipsoid().
390  **********************************************************************/
391  Math::real Phi(real X, real Y, real& fX, real& fY) const
392  { return _earth.Phi(X, Y, fX, fY); }
393  ///@}
394 
395  /** \name Compute gravity on a circle of constant latitude
396  **********************************************************************/
397  ///@{
398  /**
399  * Create a GravityCircle object to allow the gravity field at many points
400  * with constant \e lat and \e h and varying \e lon to be computed
401  * efficiently.
402  *
403  * @param[in] lat latitude of the point (degrees).
404  * @param[in] h the height of the point above the ellipsoid (meters).
405  * @param[in] caps bitor'ed combination of GravityModel::mask values
406  * specifying the capabilities of the resulting GravityCircle object.
407  * @exception std::bad_alloc if the memory necessary for creating a
408  * GravityCircle can't be allocated.
409  * @return a GravityCircle object whose member functions computes the
410  * gravitational field at a particular values of \e lon.
411  *
412  * The GravityModel::mask values are
413  * - \e caps |= GravityModel::GRAVITY
414  * - \e caps |= GravityModel::DISTURBANCE
415  * - \e caps |= GravityModel::DISTURBING_POTENTIAL
416  * - \e caps |= GravityModel::SPHERICAL_ANOMALY
417  * - \e caps |= GravityModel::GEOID_HEIGHT
418  * .
419  * The default value of \e caps is GravityModel::ALL which turns on all the
420  * capabilities. If an unsupported function is invoked, it will return
421  * NaNs. Note that GravityModel::GEOID_HEIGHT will only be honored if \e h
422  * = 0.
423  *
424  * If the field at several points on a circle of latitude need to be
425  * calculated then creating a GravityCircle object and using its member
426  * functions will be substantially faster, especially for high-degree
427  * models. See \ref gravityparallel for an example of using GravityCircle
428  * (together with OpenMP) to speed up the computation of geoid heights.
429  **********************************************************************/
430  GravityCircle Circle(real lat, real h, unsigned caps = ALL) const;
431  ///@}
432 
433  /** \name Inspector functions
434  **********************************************************************/
435  ///@{
436 
437  /**
438  * @return the NormalGravity object for the reference ellipsoid.
439  **********************************************************************/
440  const NormalGravity& ReferenceEllipsoid() const { return _earth; }
441 
442  /**
443  * @return the description of the gravity model, if available, in the data
444  * file; if absent, return "NONE".
445  **********************************************************************/
446  const std::string& Description() const { return _description; }
447 
448  /**
449  * @return date of the model; if absent, return "UNKNOWN".
450  **********************************************************************/
451  const std::string& DateTime() const { return _date; }
452 
453  /**
454  * @return full file name used to load the gravity model.
455  **********************************************************************/
456  const std::string& GravityFile() const { return _filename; }
457 
458  /**
459  * @return "name" used to load the gravity model (from the first argument
460  * of the constructor, but this may be overridden by the model file).
461  **********************************************************************/
462  const std::string& GravityModelName() const { return _name; }
463 
464  /**
465  * @return directory used to load the gravity model.
466  **********************************************************************/
467  const std::string& GravityModelDirectory() const { return _dir; }
468 
469  /**
470  * @return \e a the equatorial radius of the ellipsoid (meters).
471  **********************************************************************/
472  Math::real EquatorialRadius() const { return _earth.EquatorialRadius(); }
473 
474  /**
475  * @return \e GM the mass constant of the model (m<sup>3</sup>
476  * s<sup>&minus;2</sup>); this is the product of \e G the gravitational
477  * constant and \e M the mass of the earth (usually including the mass of
478  * the earth's atmosphere).
479  **********************************************************************/
480  Math::real MassConstant() const { return _GMmodel; }
481 
482  /**
483  * @return \e GM the mass constant of the ReferenceEllipsoid()
484  * (m<sup>3</sup> s<sup>&minus;2</sup>).
485  **********************************************************************/
487  { return _earth.MassConstant(); }
488 
489  /**
490  * @return &omega; the angular velocity of the model and the
491  * ReferenceEllipsoid() (rad s<sup>&minus;1</sup>).
492  **********************************************************************/
494  { return _earth.AngularVelocity(); }
495 
496  /**
497  * @return \e f the flattening of the ellipsoid.
498  **********************************************************************/
499  Math::real Flattening() const { return _earth.Flattening(); }
500 
501  /**
502  * @return \e Nmax the maximum degree of the components of the model.
503  **********************************************************************/
504  int Degree() const { return _nmx; }
505 
506  /**
507  * @return \e Mmax the maximum order of the components of the model.
508  **********************************************************************/
509  int Order() const { return _mmx; }
510 
511  /**
512  * \deprecated An old name for EquatorialRadius().
513  **********************************************************************/
514  GEOGRAPHICLIB_DEPRECATED("Use EquatorialRadius()")
515  Math::real MajorRadius() const { return EquatorialRadius(); }
516  ///@}
517 
518  /**
519  * @return the default path for gravity model data files.
520  *
521  * This is the value of the environment variable
522  * GEOGRAPHICLIB_GRAVITY_PATH, if set; otherwise, it is
523  * $GEOGRAPHICLIB_DATA/gravity if the environment variable
524  * GEOGRAPHICLIB_DATA is set; otherwise, it is a compile-time default
525  * (/usr/local/share/GeographicLib/gravity on non-Windows systems and
526  * C:/ProgramData/GeographicLib/gravity on Windows systems).
527  **********************************************************************/
528  static std::string DefaultGravityPath();
529 
530  /**
531  * @return the default name for the gravity model.
532  *
533  * This is the value of the environment variable
534  * GEOGRAPHICLIB_GRAVITY_NAME, if set; otherwise, it is "egm96". The
535  * GravityModel class does not use this function; it is just provided as a
536  * convenience for a calling program when constructing a GravityModel
537  * object.
538  **********************************************************************/
539  static std::string DefaultGravityName();
540  };
541 
542 } // namespace GeographicLib
543 
544 #if defined(_MSC_VER)
545 # pragma warning (pop)
546 #endif
547 
548 #endif // GEOGRAPHICLIB_GRAVITYMODEL_HPP
Header for GeographicLib::Constants class.
#define GEOGRAPHICLIB_EXPORT
Definition: Constants.hpp:66
#define GEOGRAPHICLIB_DEPRECATED(msg)
Definition: Constants.hpp:81
GeographicLib::Math::real real
Definition: GeodSolve.cpp:31
Header for GeographicLib::NormalGravity class.
Header for GeographicLib::SphericalHarmonic1 class.
Header for GeographicLib::SphericalHarmonic class.
Gravity on a circle of latitude.
Model of the earth's gravity field.
Math::real ReferenceMassConstant() const
const std::string & DateTime() const
const std::string & GravityModelDirectory() const
Math::real Flattening() const
Math::real T(real X, real Y, real Z, real &deltaX, real &deltaY, real &deltaZ) const
const NormalGravity & ReferenceEllipsoid() const
Math::real MassConstant() const
const std::string & GravityFile() const
Math::real EquatorialRadius() const
const std::string & Description() const
Math::real Phi(real X, real Y, real &fX, real &fY) const
const std::string & GravityModelName() const
Math::real U(real X, real Y, real Z, real &gammaX, real &gammaY, real &gammaZ) const
Math::real T(real X, real Y, real Z) const
Math::real AngularVelocity() const
Mathematical functions needed by GeographicLib.
Definition: Math.hpp:76
The normal gravity of the earth.
Math::real EquatorialRadius() const
Math::real Flattening() const
Math::real AngularVelocity() const
Math::real Phi(real X, real Y, real &fX, real &fY) const
Math::real MassConstant() const
Math::real U(real X, real Y, real Z, real &gammaX, real &gammaY, real &gammaZ) const
Spherical harmonic series with a correction to the coefficients.
Spherical harmonic series.
Namespace for GeographicLib.
Definition: Accumulator.cpp:12