GeographicLib  2.6
Geodesic.hpp
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1 /**
2  * \file Geodesic.hpp
3  * \brief Header for GeographicLib::Geodesic class
4  *
5  * Copyright (c) Charles Karney (2009-2024) <karney@alum.mit.edu> and licensed
6  * under the MIT/X11 License. For more information, see
7  * https://geographiclib.sourceforge.io/
8  **********************************************************************/
9 
10 #if !defined(GEOGRAPHICLIB_GEODESIC_HPP)
11 #define GEOGRAPHICLIB_GEODESIC_HPP 1
12 
13 #include <GeographicLib/Constants.hpp>
14 #include <GeographicLib/GeodesicExact.hpp>
15 
16 #if !defined(GEOGRAPHICLIB_GEODESIC_ORDER)
17 /**
18  * The order of the expansions used by Geodesic.
19  * GEOGRAPHICLIB_GEODESIC_ORDER can be set to any integer in [3, 8].
20  **********************************************************************/
21 # define GEOGRAPHICLIB_GEODESIC_ORDER \
22  (GEOGRAPHICLIB_PRECISION == 2 ? 6 : \
23  (GEOGRAPHICLIB_PRECISION == 1 ? 3 : \
24  (GEOGRAPHICLIB_PRECISION == 3 ? 7 : 8)))
25 #endif
26 
27 namespace GeographicLib {
28 
29  class GeodesicLine;
30 
31  /**
32  * \brief %Geodesic calculations
33  *
34  * The shortest path between two points on an ellipsoid at (\e lat1, \e lon1)
35  * and (\e lat2, \e lon2) is called the geodesic. Its length is \e s12 and
36  * the geodesic from point 1 to point 2 has azimuths \e azi1 and \e azi2 at
37  * the two end points. (The azimuth is the heading measured clockwise from
38  * north. \e azi2 is the "forward" azimuth, i.e., the heading that takes you
39  * beyond point 2 not back to point 1.) In the figure below, latitude is
40  * labeled &phi;, longitude &lambda; (with &lambda;<sub>12</sub> =
41  * &lambda;<sub>2</sub> &minus; &lambda;<sub>1</sub>), and azimuth &alpha;.
42  *
43  * <img src="https://upload.wikimedia.org/wikipedia/commons/c/cb/Geodesic_problem_on_an_ellipsoid.svg" width=250 alt="spheroidal triangle">
44  *
45  * Given \e lat1, \e lon1, \e azi1, and \e s12, we can determine \e lat2, \e
46  * lon2, and \e azi2. This is the \e direct geodesic problem and its
47  * solution is given by the function Geodesic::Direct. (If \e s12 is
48  * sufficiently large that the geodesic wraps more than halfway around the
49  * earth, there will be another geodesic between the points with a smaller \e
50  * s12.)
51  *
52  * Given \e lat1, \e lon1, \e lat2, and \e lon2, we can determine \e azi1, \e
53  * azi2, and \e s12. This is the \e inverse geodesic problem, whose solution
54  * is given by Geodesic::Inverse. Usually, the solution to the inverse
55  * problem is unique. In cases where there are multiple solutions (all with
56  * the same \e s12, of course), all the solutions can be easily generated
57  * once a particular solution is provided.
58  *
59  * The standard way of specifying the direct problem is the specify the
60  * distance \e s12 to the second point. However it is sometimes useful
61  * instead to specify the arc length \e a12 (in degrees) on the auxiliary
62  * sphere. This is a mathematical construct used in solving the geodesic
63  * problems. The solution of the direct problem in this form is provided by
64  * Geodesic::ArcDirect. An arc length in excess of 180&deg; indicates that
65  * the geodesic is not a shortest path. In addition, the arc length between
66  * an equatorial crossing and the next extremum of latitude for a geodesic is
67  * 90&deg;.
68  *
69  * This class can also calculate several other quantities related to
70  * geodesics. These are:
71  * - <i>reduced length</i>. If we fix the first point and increase \e azi1
72  * by \e dazi1 (radians), the second point is displaced \e m12 \e dazi1 in
73  * the direction \e azi2 + 90&deg;. The quantity \e m12 is called
74  * the "reduced length" and is symmetric under interchange of the two
75  * points. On a curved surface the reduced length obeys a symmetry
76  * relation, \e m12 + \e m21 = 0. On a flat surface, we have \e m12 = \e
77  * s12. The ratio <i>s12</i>/\e m12 gives the azimuthal scale for an
78  * azimuthal equidistant projection.
79  * - <i>geodesic scale</i>. Consider a reference geodesic and a second
80  * geodesic parallel to this one at point 1 and separated by a small
81  * distance \e dt. The separation of the two geodesics at point 2 is \e
82  * M12 \e dt where \e M12 is called the "geodesic scale". \e M21 is
83  * defined similarly (with the geodesics being parallel at point 2). On a
84  * flat surface, we have \e M12 = \e M21 = 1. The quantity 1/\e M12 gives
85  * the scale of the Cassini-Soldner projection.
86  * - <i>area</i>. The area between the geodesic from point 1 to point 2 and
87  * the equation is represented by \e S12; it is the area, measured
88  * counter-clockwise, of the geodesic quadrilateral with corners
89  * (<i>lat1</i>,<i>lon1</i>), (0,<i>lon1</i>), (0,<i>lon2</i>), and
90  * (<i>lat2</i>,<i>lon2</i>). It can be used to compute the area of any
91  * geodesic polygon.
92  *
93  * Overloaded versions of Geodesic::Direct, Geodesic::ArcDirect, and
94  * Geodesic::Inverse allow these quantities to be returned. In addition
95  * there are general functions Geodesic::GenDirect, and Geodesic::GenInverse
96  * which allow an arbitrary set of results to be computed. The quantities \e
97  * m12, \e M12, \e M21 which all specify the behavior of nearby geodesics
98  * obey addition rules. If points 1, 2, and 3 all lie on a single geodesic,
99  * then the following rules hold:
100  * - \e s13 = \e s12 + \e s23
101  * - \e a13 = \e a12 + \e a23
102  * - \e S13 = \e S12 + \e S23
103  * - \e m13 = \e m12 \e M23 + \e m23 \e M21
104  * - \e M13 = \e M12 \e M23 &minus; (1 &minus; \e M12 \e M21) \e m23 / \e m12
105  * - \e M31 = \e M32 \e M21 &minus; (1 &minus; \e M23 \e M32) \e m12 / \e m23
106  *
107  * Additional functionality is provided by the GeodesicLine class, which
108  * allows a sequence of points along a geodesic to be computed.
109  *
110  * The shortest distance returned by the solution of the inverse problem is
111  * (obviously) uniquely defined. However, in a few special cases there are
112  * multiple azimuths which yield the same shortest distance. Here is a
113  * catalog of those cases:
114  * - \e lat1 = &minus;\e lat2 (with neither point at a pole). If \e azi1 =
115  * \e azi2, the geodesic is unique. Otherwise there are two geodesics and
116  * the second one is obtained by setting [\e azi1, \e azi2] &rarr; [\e
117  * azi2, \e azi1], [\e M12, \e M21] &rarr; [\e M21, \e M12], \e S12 &rarr;
118  * &minus;\e S12. (This occurs when the longitude difference is near
119  * &plusmn;180&deg; for oblate ellipsoids.)
120  * - \e lon2 = \e lon1 &plusmn; 180&deg; (with neither point at a pole). If
121  * \e azi1 = 0&deg; or &plusmn;180&deg;, the geodesic is unique. Otherwise
122  * there are two geodesics and the second one is obtained by setting [\e
123  * azi1, \e azi2] &rarr; [&minus;\e azi1, &minus;\e azi2], \e S12 &rarr;
124  * &minus;\e S12. (This occurs when \e lat2 is near &minus;\e lat1 for
125  * prolate ellipsoids.)
126  * - Points 1 and 2 at opposite poles. There are infinitely many geodesics
127  * which can be generated by setting [\e azi1, \e azi2] &rarr; [\e azi1, \e
128  * azi2] + [\e d, &minus;\e d], for arbitrary \e d. (For spheres, this
129  * prescription applies when points 1 and 2 are antipodal.)
130  * - \e s12 = 0 (coincident points). There are infinitely many geodesics
131  * which can be generated by setting [\e azi1, \e azi2] &rarr;
132  * [\e azi1, \e azi2] + [\e d, \e d], for arbitrary \e d.
133  *
134  * The calculations are accurate to better than 15 nm (15 nanometers) for the
135  * WGS84 ellipsoid. See Sec. 9 of
136  * <a href="https://arxiv.org/abs/1102.1215v1">arXiv:1102.1215v1</a> for
137  * details. With \e exact = false (the default) in the constructor,
138  * the algorithms used by this class are based on series expansions
139  * using the flattening \e f as a small parameter. These are only accurate
140  * for |<i>f</i>| &lt; 0.02; however reasonably accurate results will be
141  * obtained for |<i>f</i>| &lt; 0.2. Here is a table of the approximate
142  * maximum error (expressed as a distance) for an ellipsoid with the same
143  * equatorial radius as the WGS84 ellipsoid and different values of the
144  * flattening.<pre>
145  * |f| error
146  * 0.01 25 nm
147  * 0.02 30 nm
148  * 0.05 10 um
149  * 0.1 1.5 mm
150  * 0.2 300 mm
151  * </pre>
152  * For very eccentric ellipsoids, set \e exact to true in the constructor;
153  * this will delegate the calculations to the GeodesicExact class.
154  *
155  * The algorithms are described in
156  * - C. F. F. Karney,
157  * <a href="https://doi.org/10.1007/s00190-012-0578-z">
158  * <i>Algorithms for geodesics</i></a>,
159  * J. Geodesy <b>87</b>, 43--55 (2013);
160  * DOI: <a href="https://doi.org/10.1007/s00190-012-0578-z">
161  * 10.1007/s00190-012-0578-z</a>;
162  * addenda:
163  * <a href="https://geographiclib.sourceforge.io/geod-addenda.html">
164  * geod-addenda.html</a>.
165  * .
166  * For more information on geodesics see \ref geodesic.
167  *
168  * Example of use:
169  * \include example-Geodesic.cpp
170  *
171  * <a href="GeodSolve.1.html">GeodSolve</a> is a command-line utility
172  * providing access to the functionality of Geodesic and GeodesicLine.
173  **********************************************************************/
174 
175  class GEOGRAPHICLIB_EXPORT Geodesic {
176  private:
177  typedef Math::real real;
178  friend class GeodesicLine;
179  static const int nA1_ = GEOGRAPHICLIB_GEODESIC_ORDER;
180  static const int nC1_ = GEOGRAPHICLIB_GEODESIC_ORDER;
181  static const int nC1p_ = GEOGRAPHICLIB_GEODESIC_ORDER;
182  static const int nA2_ = GEOGRAPHICLIB_GEODESIC_ORDER;
183  static const int nC2_ = GEOGRAPHICLIB_GEODESIC_ORDER;
184  static const int nA3_ = GEOGRAPHICLIB_GEODESIC_ORDER;
185  static const int nA3x_ = nA3_;
186  static const int nC3_ = GEOGRAPHICLIB_GEODESIC_ORDER;
187  static const int nC3x_ = (nC3_ * (nC3_ - 1)) / 2;
188  static const int nC4_ = GEOGRAPHICLIB_GEODESIC_ORDER;
189  static const int nC4x_ = (nC4_ * (nC4_ + 1)) / 2;
190  // Size for temporary array
191  // nC = max(max(nC1_, nC1p_, nC2_) + 1, max(nC3_, nC4_))
192  static const int nC_ = GEOGRAPHICLIB_GEODESIC_ORDER + 1;
193  static const unsigned maxit1_ = 20;
194  unsigned maxit2_;
195  real tiny_, tol0_, tol1_, tol2_, tolb_, xthresh_;
196 
197  static constexpr unsigned CAP_NONE = 0U;
198  static constexpr unsigned CAP_C1 = 1U<<0;
199  static constexpr unsigned CAP_C1p = 1U<<1;
200  static constexpr unsigned CAP_C2 = 1U<<2;
201  static constexpr unsigned CAP_C3 = 1U<<3;
202  static constexpr unsigned CAP_C4 = 1U<<4;
203  static constexpr unsigned CAP_ALL = 0x1FU;
204  static constexpr unsigned CAP_MASK = CAP_ALL;
205  static constexpr unsigned OUT_ALL = 0x7F80U;
206  static constexpr unsigned OUT_MASK = 0xFF80U; // Includes LONG_UNROLL
207 
208  static real SinCosSeries(bool sinp,
209  real sinx, real cosx, const real c[], int n);
210  static real Astroid(real x, real y);
211 
212  real _a, _f;
213  bool _exact;
214  real _f1, _e2, _ep2, _n, _b, _c2, _etol2;
215  real _aA3x[nA3x_], _cC3x[nC3x_], _cC4x[nC4x_];
216  GeodesicExact _geodexact;
217 
218  void Lengths(real eps, real sig12,
219  real ssig1, real csig1, real dn1,
220  real ssig2, real csig2, real dn2,
221  real cbet1, real cbet2, unsigned outmask,
222  real& s12s, real& m12a, real& m0,
223  real& M12, real& M21, real Ca[]) const;
224  real InverseStart(real sbet1, real cbet1, real dn1,
225  real sbet2, real cbet2, real dn2,
226  real lam12, real slam12, real clam12,
227  real& salp1, real& calp1,
228  real& salp2, real& calp2, real& dnm,
229  real Ca[]) const;
230  real Lambda12(real sbet1, real cbet1, real dn1,
231  real sbet2, real cbet2, real dn2,
232  real salp1, real calp1, real slam120, real clam120,
233  real& salp2, real& calp2, real& sig12,
234  real& ssig1, real& csig1, real& ssig2, real& csig2,
235  real& eps, real& domg12,
236  bool diffp, real& dlam12, real Ca[]) const;
237  real GenInverse(real lat1, real lon1, real lat2, real lon2,
238  unsigned outmask, real& s12,
239  real& salp1, real& calp1, real& salp2, real& calp2,
240  real& m12, real& M12, real& M21, real& S12) const;
241 
242  // These are Maxima generated functions to provide series approximations to
243  // the integrals for the ellipsoidal geodesic.
244  static real A1m1f(real eps);
245  static void C1f(real eps, real c[]);
246  static void C1pf(real eps, real c[]);
247  static real A2m1f(real eps);
248  static void C2f(real eps, real c[]);
249 
250  void A3coeff();
251  real A3f(real eps) const;
252  void C3coeff();
253  void C3f(real eps, real c[]) const;
254  void C4coeff();
255  void C4f(real k2, real c[]) const;
256  public:
257 
258  /**
259  * Bit masks for what calculations to do. These masks do double duty.
260  * They signify to the GeodesicLine constructor and to
261  * Geodesic::Line what capabilities should be included in the GeodesicLine
262  * object. They also specify which results to return in the general
263  * routines Geodesic::GenDirect and Geodesic::GenInverse routines.
264  * GeodesicLine::mask is a duplication of this enum.
265  **********************************************************************/
266  enum mask {
267  /**
268  * No capabilities, no output.
269  * @hideinitializer
270  **********************************************************************/
271  NONE = 0U,
272  /**
273  * Calculate latitude \e lat2. (It's not necessary to include this as a
274  * capability to GeodesicLine because this is included by default.)
275  * @hideinitializer
276  **********************************************************************/
277  LATITUDE = 1U<<7 | CAP_NONE,
278  /**
279  * Calculate longitude \e lon2.
280  * @hideinitializer
281  **********************************************************************/
282  LONGITUDE = 1U<<8 | CAP_C3,
283  /**
284  * Calculate azimuths \e azi1 and \e azi2. (It's not necessary to
285  * include this as a capability to GeodesicLine because this is included
286  * by default.)
287  * @hideinitializer
288  **********************************************************************/
289  AZIMUTH = 1U<<9 | CAP_NONE,
290  /**
291  * Calculate distance \e s12.
292  * @hideinitializer
293  **********************************************************************/
294  DISTANCE = 1U<<10 | CAP_C1,
295  /**
296  * A combination of the common capabilities: Geodesic::LATITUDE,
297  * Geodesic::LONGITUDE, Geodesic::AZIMUTH, Geodesic::DISTANCE.
298  * @hideinitializer
299  **********************************************************************/
300  STANDARD = LATITUDE | LONGITUDE | AZIMUTH | DISTANCE,
301  /**
302  * Allow distance \e s12 to be used as input in the direct geodesic
303  * problem.
304  * @hideinitializer
305  **********************************************************************/
306  DISTANCE_IN = 1U<<11 | CAP_C1 | CAP_C1p,
307  /**
308  * Calculate reduced length \e m12.
309  * @hideinitializer
310  **********************************************************************/
311  REDUCEDLENGTH = 1U<<12 | CAP_C1 | CAP_C2,
312  /**
313  * Calculate geodesic scales \e M12 and \e M21.
314  * @hideinitializer
315  **********************************************************************/
316  GEODESICSCALE = 1U<<13 | CAP_C1 | CAP_C2,
317  /**
318  * Calculate area \e S12.
319  * @hideinitializer
320  **********************************************************************/
321  AREA = 1U<<14 | CAP_C4,
322  /**
323  * Unroll \e lon2 in the direct calculation.
324  * @hideinitializer
325  **********************************************************************/
326  LONG_UNROLL = 1U<<15,
327  /**
328  * All capabilities, calculate everything. (Geodesic::LONG_UNROLL is not
329  * included in this mask.)
330  * @hideinitializer
331  **********************************************************************/
332  ALL = OUT_ALL| CAP_ALL,
333  };
334 
335  /** \name Constructor
336  **********************************************************************/
337  ///@{
338  /**
339  * Constructor for an ellipsoid with
340  *
341  * @param[in] a equatorial radius (meters).
342  * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
343  * Negative \e f gives a prolate ellipsoid.
344  * @param[in] exact if true use exact formulation in terms of elliptic
345  * integrals instead of series expansions (default false).
346  * @exception GeographicErr if \e a or (1 &minus; \e f) \e a is not
347  * positive.
348  *
349  * With \e exact = true, this class delegates the calculations to the
350  * GeodesicExact and GeodesicLineExact classes which solve the geodesic
351  * problems in terms of elliptic integrals.
352  **********************************************************************/
353  Geodesic(real a, real f, bool exact = false);
354  ///@}
355 
356  /** \name Direct geodesic problem specified in terms of distance.
357  **********************************************************************/
358  ///@{
359  /**
360  * Solve the direct geodesic problem where the length of the geodesic
361  * is specified in terms of distance.
362  *
363  * @param[in] lat1 latitude of point 1 (degrees).
364  * @param[in] lon1 longitude of point 1 (degrees).
365  * @param[in] azi1 azimuth at point 1 (degrees).
366  * @param[in] s12 distance between point 1 and point 2 (meters); it can be
367  * negative.
368  * @param[out] lat2 latitude of point 2 (degrees).
369  * @param[out] lon2 longitude of point 2 (degrees).
370  * @param[out] azi2 (forward) azimuth at point 2 (degrees).
371  * @param[out] m12 reduced length of geodesic (meters).
372  * @param[out] M12 geodesic scale of point 2 relative to point 1
373  * (dimensionless).
374  * @param[out] M21 geodesic scale of point 1 relative to point 2
375  * (dimensionless).
376  * @param[out] S12 area under the geodesic (meters<sup>2</sup>).
377  * @return \e a12 arc length of between point 1 and point 2 (degrees).
378  *
379  * \e lat1 should be in the range [&minus;90&deg;, 90&deg;]. The values of
380  * \e lon2 and \e azi2 returned are in the range [&minus;180&deg;,
381  * 180&deg;].
382  *
383  * If either point is at a pole, the azimuth is defined by keeping the
384  * longitude fixed, writing \e lat = &plusmn;(90&deg; &minus; &epsilon;),
385  * and taking the limit &epsilon; &rarr; 0+. An arc length greater that
386  * 180&deg; signifies a geodesic which is not a shortest path. (For a
387  * prolate ellipsoid, an additional condition is necessary for a shortest
388  * path: the longitudinal extent must not exceed of 180&deg;.)
389  *
390  * The following functions are overloaded versions of Geodesic::Direct
391  * which omit some of the output parameters. Note, however, that the arc
392  * length is always computed and returned as the function value.
393  **********************************************************************/
394  Math::real Direct(real lat1, real lon1, real azi1, real s12,
395  real& lat2, real& lon2, real& azi2,
396  real& m12, real& M12, real& M21, real& S12)
397  const {
398  real t;
399  return GenDirect(lat1, lon1, azi1, false, s12,
400  LATITUDE | LONGITUDE | AZIMUTH |
401  REDUCEDLENGTH | GEODESICSCALE | AREA,
402  lat2, lon2, azi2, t, m12, M12, M21, S12);
403  }
404 
405  /**
406  * See the documentation for Geodesic::Direct.
407  **********************************************************************/
408  Math::real Direct(real lat1, real lon1, real azi1, real s12,
409  real& lat2, real& lon2)
410  const {
411  real t;
412  return GenDirect(lat1, lon1, azi1, false, s12,
413  LATITUDE | LONGITUDE,
414  lat2, lon2, t, t, t, t, t, t);
415  }
416 
417  /**
418  * See the documentation for Geodesic::Direct.
419  **********************************************************************/
420  Math::real Direct(real lat1, real lon1, real azi1, real s12,
421  real& lat2, real& lon2, real& azi2)
422  const {
423  real t;
424  return GenDirect(lat1, lon1, azi1, false, s12,
425  LATITUDE | LONGITUDE | AZIMUTH,
426  lat2, lon2, azi2, t, t, t, t, t);
427  }
428 
429  /**
430  * See the documentation for Geodesic::Direct.
431  **********************************************************************/
432  Math::real Direct(real lat1, real lon1, real azi1, real s12,
433  real& lat2, real& lon2, real& azi2, real& m12)
434  const {
435  real t;
436  return GenDirect(lat1, lon1, azi1, false, s12,
437  LATITUDE | LONGITUDE | AZIMUTH | REDUCEDLENGTH,
438  lat2, lon2, azi2, t, m12, t, t, t);
439  }
440 
441  /**
442  * See the documentation for Geodesic::Direct.
443  **********************************************************************/
444  Math::real Direct(real lat1, real lon1, real azi1, real s12,
445  real& lat2, real& lon2, real& azi2,
446  real& M12, real& M21)
447  const {
448  real t;
449  return GenDirect(lat1, lon1, azi1, false, s12,
450  LATITUDE | LONGITUDE | AZIMUTH | GEODESICSCALE,
451  lat2, lon2, azi2, t, t, M12, M21, t);
452  }
453 
454  /**
455  * See the documentation for Geodesic::Direct.
456  **********************************************************************/
457  Math::real Direct(real lat1, real lon1, real azi1, real s12,
458  real& lat2, real& lon2, real& azi2,
459  real& m12, real& M12, real& M21)
460  const {
461  real t;
462  return GenDirect(lat1, lon1, azi1, false, s12,
463  LATITUDE | LONGITUDE | AZIMUTH |
464  REDUCEDLENGTH | GEODESICSCALE,
465  lat2, lon2, azi2, t, m12, M12, M21, t);
466  }
467  ///@}
468 
469  /** \name Direct geodesic problem specified in terms of arc length.
470  **********************************************************************/
471  ///@{
472  /**
473  * Solve the direct geodesic problem where the length of the geodesic
474  * is specified in terms of arc length.
475  *
476  * @param[in] lat1 latitude of point 1 (degrees).
477  * @param[in] lon1 longitude of point 1 (degrees).
478  * @param[in] azi1 azimuth at point 1 (degrees).
479  * @param[in] a12 arc length between point 1 and point 2 (degrees); it can
480  * be negative.
481  * @param[out] lat2 latitude of point 2 (degrees).
482  * @param[out] lon2 longitude of point 2 (degrees).
483  * @param[out] azi2 (forward) azimuth at point 2 (degrees).
484  * @param[out] s12 distance between point 1 and point 2 (meters).
485  * @param[out] m12 reduced length of geodesic (meters).
486  * @param[out] M12 geodesic scale of point 2 relative to point 1
487  * (dimensionless).
488  * @param[out] M21 geodesic scale of point 1 relative to point 2
489  * (dimensionless).
490  * @param[out] S12 area under the geodesic (meters<sup>2</sup>).
491  *
492  * \e lat1 should be in the range [&minus;90&deg;, 90&deg;]. The values of
493  * \e lon2 and \e azi2 returned are in the range [&minus;180&deg;,
494  * 180&deg;].
495  *
496  * If either point is at a pole, the azimuth is defined by keeping the
497  * longitude fixed, writing \e lat = &plusmn;(90&deg; &minus; &epsilon;),
498  * and taking the limit &epsilon; &rarr; 0+. An arc length greater that
499  * 180&deg; signifies a geodesic which is not a shortest path. (For a
500  * prolate ellipsoid, an additional condition is necessary for a shortest
501  * path: the longitudinal extent must not exceed of 180&deg;.)
502  *
503  * The following functions are overloaded versions of Geodesic::Direct
504  * which omit some of the output parameters.
505  **********************************************************************/
506  void ArcDirect(real lat1, real lon1, real azi1, real a12,
507  real& lat2, real& lon2, real& azi2, real& s12,
508  real& m12, real& M12, real& M21, real& S12)
509  const {
510  GenDirect(lat1, lon1, azi1, true, a12,
511  LATITUDE | LONGITUDE | AZIMUTH | DISTANCE |
512  REDUCEDLENGTH | GEODESICSCALE | AREA,
513  lat2, lon2, azi2, s12, m12, M12, M21, S12);
514  }
515 
516  /**
517  * See the documentation for Geodesic::ArcDirect.
518  **********************************************************************/
519  void ArcDirect(real lat1, real lon1, real azi1, real a12,
520  real& lat2, real& lon2) const {
521  real t;
522  GenDirect(lat1, lon1, azi1, true, a12,
523  LATITUDE | LONGITUDE,
524  lat2, lon2, t, t, t, t, t, t);
525  }
526 
527  /**
528  * See the documentation for Geodesic::ArcDirect.
529  **********************************************************************/
530  void ArcDirect(real lat1, real lon1, real azi1, real a12,
531  real& lat2, real& lon2, real& azi2) const {
532  real t;
533  GenDirect(lat1, lon1, azi1, true, a12,
534  LATITUDE | LONGITUDE | AZIMUTH,
535  lat2, lon2, azi2, t, t, t, t, t);
536  }
537 
538  /**
539  * See the documentation for Geodesic::ArcDirect.
540  **********************************************************************/
541  void ArcDirect(real lat1, real lon1, real azi1, real a12,
542  real& lat2, real& lon2, real& azi2, real& s12)
543  const {
544  real t;
545  GenDirect(lat1, lon1, azi1, true, a12,
546  LATITUDE | LONGITUDE | AZIMUTH | DISTANCE,
547  lat2, lon2, azi2, s12, t, t, t, t);
548  }
549 
550  /**
551  * See the documentation for Geodesic::ArcDirect.
552  **********************************************************************/
553  void ArcDirect(real lat1, real lon1, real azi1, real a12,
554  real& lat2, real& lon2, real& azi2,
555  real& s12, real& m12) const {
556  real t;
557  GenDirect(lat1, lon1, azi1, true, a12,
558  LATITUDE | LONGITUDE | AZIMUTH | DISTANCE |
559  REDUCEDLENGTH,
560  lat2, lon2, azi2, s12, m12, t, t, t);
561  }
562 
563  /**
564  * See the documentation for Geodesic::ArcDirect.
565  **********************************************************************/
566  void ArcDirect(real lat1, real lon1, real azi1, real a12,
567  real& lat2, real& lon2, real& azi2, real& s12,
568  real& M12, real& M21) const {
569  real t;
570  GenDirect(lat1, lon1, azi1, true, a12,
571  LATITUDE | LONGITUDE | AZIMUTH | DISTANCE |
572  GEODESICSCALE,
573  lat2, lon2, azi2, s12, t, M12, M21, t);
574  }
575 
576  /**
577  * See the documentation for Geodesic::ArcDirect.
578  **********************************************************************/
579  void ArcDirect(real lat1, real lon1, real azi1, real a12,
580  real& lat2, real& lon2, real& azi2, real& s12,
581  real& m12, real& M12, real& M21) const {
582  real t;
583  GenDirect(lat1, lon1, azi1, true, a12,
584  LATITUDE | LONGITUDE | AZIMUTH | DISTANCE |
585  REDUCEDLENGTH | GEODESICSCALE,
586  lat2, lon2, azi2, s12, m12, M12, M21, t);
587  }
588  ///@}
589 
590  /** \name General version of the direct geodesic solution.
591  **********************************************************************/
592  ///@{
593 
594  /**
595  * The general direct geodesic problem. Geodesic::Direct and
596  * Geodesic::ArcDirect are defined in terms of this function.
597  *
598  * @param[in] lat1 latitude of point 1 (degrees).
599  * @param[in] lon1 longitude of point 1 (degrees).
600  * @param[in] azi1 azimuth at point 1 (degrees).
601  * @param[in] arcmode boolean flag determining the meaning of the \e
602  * s12_a12.
603  * @param[in] s12_a12 if \e arcmode is false, this is the distance between
604  * point 1 and point 2 (meters); otherwise it is the arc length between
605  * point 1 and point 2 (degrees); it can be negative.
606  * @param[in] outmask a bitor'ed combination of Geodesic::mask values
607  * specifying which of the following parameters should be set.
608  * @param[out] lat2 latitude of point 2 (degrees).
609  * @param[out] lon2 longitude of point 2 (degrees).
610  * @param[out] azi2 (forward) azimuth at point 2 (degrees).
611  * @param[out] s12 distance between point 1 and point 2 (meters).
612  * @param[out] m12 reduced length of geodesic (meters).
613  * @param[out] M12 geodesic scale of point 2 relative to point 1
614  * (dimensionless).
615  * @param[out] M21 geodesic scale of point 1 relative to point 2
616  * (dimensionless).
617  * @param[out] S12 area under the geodesic (meters<sup>2</sup>).
618  * @return \e a12 arc length of between point 1 and point 2 (degrees).
619  *
620  * The Geodesic::mask values possible for \e outmask are
621  * - \e outmask |= Geodesic::LATITUDE for the latitude \e lat2;
622  * - \e outmask |= Geodesic::LONGITUDE for the latitude \e lon2;
623  * - \e outmask |= Geodesic::AZIMUTH for the latitude \e azi2;
624  * - \e outmask |= Geodesic::DISTANCE for the distance \e s12;
625  * - \e outmask |= Geodesic::REDUCEDLENGTH for the reduced length \e
626  * m12;
627  * - \e outmask |= Geodesic::GEODESICSCALE for the geodesic scales \e
628  * M12 and \e M21;
629  * - \e outmask |= Geodesic::AREA for the area \e S12;
630  * - \e outmask |= Geodesic::ALL for all of the above;
631  * - \e outmask |= Geodesic::LONG_UNROLL to unroll \e lon2 instead of
632  * wrapping it into the range [&minus;180&deg;, 180&deg;].
633  * .
634  * The function value \e a12 is always computed and returned and this
635  * equals \e s12_a12 is \e arcmode is true. If \e outmask includes
636  * Geodesic::DISTANCE and \e arcmode is false, then \e s12 = \e s12_a12.
637  * It is not necessary to include Geodesic::DISTANCE_IN in \e outmask; this
638  * is automatically included is \e arcmode is false.
639  *
640  * With the Geodesic::LONG_UNROLL bit set, the quantity \e lon2 &minus; \e
641  * lon1 indicates how many times and in what sense the geodesic encircles
642  * the ellipsoid.
643  **********************************************************************/
644  Math::real GenDirect(real lat1, real lon1, real azi1,
645  bool arcmode, real s12_a12, unsigned outmask,
646  real& lat2, real& lon2, real& azi2,
647  real& s12, real& m12, real& M12, real& M21,
648  real& S12) const;
649  ///@}
650 
651  /** \name Inverse geodesic problem.
652  **********************************************************************/
653  ///@{
654  /**
655  * Solve the inverse geodesic problem.
656  *
657  * @param[in] lat1 latitude of point 1 (degrees).
658  * @param[in] lon1 longitude of point 1 (degrees).
659  * @param[in] lat2 latitude of point 2 (degrees).
660  * @param[in] lon2 longitude of point 2 (degrees).
661  * @param[out] s12 distance between point 1 and point 2 (meters).
662  * @param[out] azi1 azimuth at point 1 (degrees).
663  * @param[out] azi2 (forward) azimuth at point 2 (degrees).
664  * @param[out] m12 reduced length of geodesic (meters).
665  * @param[out] M12 geodesic scale of point 2 relative to point 1
666  * (dimensionless).
667  * @param[out] M21 geodesic scale of point 1 relative to point 2
668  * (dimensionless).
669  * @param[out] S12 area under the geodesic (meters<sup>2</sup>).
670  * @return \e a12 arc length of between point 1 and point 2 (degrees).
671  *
672  * \e lat1 and \e lat2 should be in the range [&minus;90&deg;, 90&deg;].
673  * The values of \e azi1 and \e azi2 returned are in the range
674  * [&minus;180&deg;, 180&deg;].
675  *
676  * If either point is at a pole, the azimuth is defined by keeping the
677  * longitude fixed, writing \e lat = &plusmn;(90&deg; &minus; &epsilon;),
678  * and taking the limit &epsilon; &rarr; 0+.
679  *
680  * The solution to the inverse problem is found using Newton's method. If
681  * this fails to converge (this is very unlikely in geodetic applications
682  * but does occur for very eccentric ellipsoids), then the bisection method
683  * is used to refine the solution.
684  *
685  * The following functions are overloaded versions of Geodesic::Inverse
686  * which omit some of the output parameters. Note, however, that the arc
687  * length is always computed and returned as the function value.
688  **********************************************************************/
689  Math::real Inverse(real lat1, real lon1, real lat2, real lon2,
690  real& s12, real& azi1, real& azi2, real& m12,
691  real& M12, real& M21, real& S12) const {
692  return GenInverse(lat1, lon1, lat2, lon2,
693  DISTANCE | AZIMUTH |
694  REDUCEDLENGTH | GEODESICSCALE | AREA,
695  s12, azi1, azi2, m12, M12, M21, S12);
696  }
697 
698  /**
699  * See the documentation for Geodesic::Inverse.
700  **********************************************************************/
701  Math::real Inverse(real lat1, real lon1, real lat2, real lon2,
702  real& s12) const {
703  real t;
704  return GenInverse(lat1, lon1, lat2, lon2,
705  DISTANCE,
706  s12, t, t, t, t, t, t);
707  }
708 
709  /**
710  * See the documentation for Geodesic::Inverse.
711  **********************************************************************/
712  Math::real Inverse(real lat1, real lon1, real lat2, real lon2,
713  real& azi1, real& azi2) const {
714  real t;
715  return GenInverse(lat1, lon1, lat2, lon2,
716  AZIMUTH,
717  t, azi1, azi2, t, t, t, t);
718  }
719 
720  /**
721  * See the documentation for Geodesic::Inverse.
722  **********************************************************************/
723  Math::real Inverse(real lat1, real lon1, real lat2, real lon2,
724  real& s12, real& azi1, real& azi2)
725  const {
726  real t;
727  return GenInverse(lat1, lon1, lat2, lon2,
728  DISTANCE | AZIMUTH,
729  s12, azi1, azi2, t, t, t, t);
730  }
731 
732  /**
733  * See the documentation for Geodesic::Inverse.
734  **********************************************************************/
735  Math::real Inverse(real lat1, real lon1, real lat2, real lon2,
736  real& s12, real& azi1, real& azi2, real& m12)
737  const {
738  real t;
739  return GenInverse(lat1, lon1, lat2, lon2,
740  DISTANCE | AZIMUTH | REDUCEDLENGTH,
741  s12, azi1, azi2, m12, t, t, t);
742  }
743 
744  /**
745  * See the documentation for Geodesic::Inverse.
746  **********************************************************************/
747  Math::real Inverse(real lat1, real lon1, real lat2, real lon2,
748  real& s12, real& azi1, real& azi2,
749  real& M12, real& M21) const {
750  real t;
751  return GenInverse(lat1, lon1, lat2, lon2,
752  DISTANCE | AZIMUTH | GEODESICSCALE,
753  s12, azi1, azi2, t, M12, M21, t);
754  }
755 
756  /**
757  * See the documentation for Geodesic::Inverse.
758  **********************************************************************/
759  Math::real Inverse(real lat1, real lon1, real lat2, real lon2,
760  real& s12, real& azi1, real& azi2, real& m12,
761  real& M12, real& M21) const {
762  real t;
763  return GenInverse(lat1, lon1, lat2, lon2,
764  DISTANCE | AZIMUTH |
765  REDUCEDLENGTH | GEODESICSCALE,
766  s12, azi1, azi2, m12, M12, M21, t);
767  }
768  ///@}
769 
770  /** \name General version of inverse geodesic solution.
771  **********************************************************************/
772  ///@{
773  /**
774  * The general inverse geodesic calculation. Geodesic::Inverse is defined
775  * in terms of this function.
776  *
777  * @param[in] lat1 latitude of point 1 (degrees).
778  * @param[in] lon1 longitude of point 1 (degrees).
779  * @param[in] lat2 latitude of point 2 (degrees).
780  * @param[in] lon2 longitude of point 2 (degrees).
781  * @param[in] outmask a bitor'ed combination of Geodesic::mask values
782  * specifying which of the following parameters should be set.
783  * @param[out] s12 distance between point 1 and point 2 (meters).
784  * @param[out] azi1 azimuth at point 1 (degrees).
785  * @param[out] azi2 (forward) azimuth at point 2 (degrees).
786  * @param[out] m12 reduced length of geodesic (meters).
787  * @param[out] M12 geodesic scale of point 2 relative to point 1
788  * (dimensionless).
789  * @param[out] M21 geodesic scale of point 1 relative to point 2
790  * (dimensionless).
791  * @param[out] S12 area under the geodesic (meters<sup>2</sup>).
792  * @return \e a12 arc length of between point 1 and point 2 (degrees).
793  *
794  * The Geodesic::mask values possible for \e outmask are
795  * - \e outmask |= Geodesic::DISTANCE for the distance \e s12;
796  * - \e outmask |= Geodesic::AZIMUTH for the latitude \e azi2;
797  * - \e outmask |= Geodesic::REDUCEDLENGTH for the reduced length \e
798  * m12;
799  * - \e outmask |= Geodesic::GEODESICSCALE for the geodesic scales \e
800  * M12 and \e M21;
801  * - \e outmask |= Geodesic::AREA for the area \e S12;
802  * - \e outmask |= Geodesic::ALL for all of the above.
803  * .
804  * The arc length is always computed and returned as the function value.
805  **********************************************************************/
806  Math::real GenInverse(real lat1, real lon1, real lat2, real lon2,
807  unsigned outmask,
808  real& s12, real& azi1, real& azi2,
809  real& m12, real& M12, real& M21, real& S12) const;
810  ///@}
811 
812  /** \name Interface to GeodesicLine.
813  **********************************************************************/
814  ///@{
815 
816  /**
817  * Typedef for the class for computing multiple points on a geodesic.
818  **********************************************************************/
819  typedef GeodesicLine LineClass;
820 
821  /**
822  * Set up to compute several points on a single geodesic.
823  *
824  * @param[in] lat1 latitude of point 1 (degrees).
825  * @param[in] lon1 longitude of point 1 (degrees).
826  * @param[in] azi1 azimuth at point 1 (degrees).
827  * @param[in] caps bitor'ed combination of Geodesic::mask values
828  * specifying the capabilities the GeodesicLine object should possess,
829  * i.e., which quantities can be returned in calls to
830  * GeodesicLine::Position.
831  * @return a GeodesicLine object.
832  *
833  * \e lat1 should be in the range [&minus;90&deg;, 90&deg;].
834  *
835  * The Geodesic::mask values are
836  * - \e caps |= Geodesic::LATITUDE for the latitude \e lat2; this is
837  * added automatically;
838  * - \e caps |= Geodesic::LONGITUDE for the latitude \e lon2;
839  * - \e caps |= Geodesic::AZIMUTH for the azimuth \e azi2; this is
840  * added automatically;
841  * - \e caps |= Geodesic::DISTANCE for the distance \e s12;
842  * - \e caps |= Geodesic::REDUCEDLENGTH for the reduced length \e m12;
843  * - \e caps |= Geodesic::GEODESICSCALE for the geodesic scales \e M12
844  * and \e M21;
845  * - \e caps |= Geodesic::AREA for the area \e S12;
846  * - \e caps |= Geodesic::DISTANCE_IN permits the length of the
847  * geodesic to be given in terms of \e s12; without this capability the
848  * length can only be specified in terms of arc length;
849  * - \e caps |= Geodesic::ALL for all of the above.
850  * .
851  * The default value of \e caps is Geodesic::ALL.
852  *
853  * If the point is at a pole, the azimuth is defined by keeping \e lon1
854  * fixed, writing \e lat1 = &plusmn;(90 &minus; &epsilon;), and taking the
855  * limit &epsilon; &rarr; 0+.
856  **********************************************************************/
857  GeodesicLine Line(real lat1, real lon1, real azi1, unsigned caps = ALL)
858  const;
859 
860  /**
861  * Define a GeodesicLine in terms of the inverse geodesic problem.
862  *
863  * @param[in] lat1 latitude of point 1 (degrees).
864  * @param[in] lon1 longitude of point 1 (degrees).
865  * @param[in] lat2 latitude of point 2 (degrees).
866  * @param[in] lon2 longitude of point 2 (degrees).
867  * @param[in] caps bitor'ed combination of Geodesic::mask values
868  * specifying the capabilities the GeodesicLine object should possess,
869  * i.e., which quantities can be returned in calls to
870  * GeodesicLine::Position.
871  * @return a GeodesicLine object.
872  *
873  * This function sets point 3 of the GeodesicLine to correspond to point 2
874  * of the inverse geodesic problem.
875  *
876  * \e lat1 and \e lat2 should be in the range [&minus;90&deg;, 90&deg;].
877  **********************************************************************/
878  GeodesicLine InverseLine(real lat1, real lon1, real lat2, real lon2,
879  unsigned caps = ALL) const;
880 
881  /**
882  * Define a GeodesicLine in terms of the direct geodesic problem specified
883  * in terms of distance.
884  *
885  * @param[in] lat1 latitude of point 1 (degrees).
886  * @param[in] lon1 longitude of point 1 (degrees).
887  * @param[in] azi1 azimuth at point 1 (degrees).
888  * @param[in] s12 distance between point 1 and point 2 (meters); it can be
889  * negative.
890  * @param[in] caps bitor'ed combination of Geodesic::mask values
891  * specifying the capabilities the GeodesicLine object should possess,
892  * i.e., which quantities can be returned in calls to
893  * GeodesicLine::Position.
894  * @return a GeodesicLine object.
895  *
896  * This function sets point 3 of the GeodesicLine to correspond to point 2
897  * of the direct geodesic problem.
898  *
899  * \e lat1 should be in the range [&minus;90&deg;, 90&deg;].
900  **********************************************************************/
901  GeodesicLine DirectLine(real lat1, real lon1, real azi1, real s12,
902  unsigned caps = ALL) const;
903 
904  /**
905  * Define a GeodesicLine in terms of the direct geodesic problem specified
906  * in terms of arc length.
907  *
908  * @param[in] lat1 latitude of point 1 (degrees).
909  * @param[in] lon1 longitude of point 1 (degrees).
910  * @param[in] azi1 azimuth at point 1 (degrees).
911  * @param[in] a12 arc length between point 1 and point 2 (degrees); it can
912  * be negative.
913  * @param[in] caps bitor'ed combination of Geodesic::mask values
914  * specifying the capabilities the GeodesicLine object should possess,
915  * i.e., which quantities can be returned in calls to
916  * GeodesicLine::Position.
917  * @return a GeodesicLine object.
918  *
919  * This function sets point 3 of the GeodesicLine to correspond to point 2
920  * of the direct geodesic problem.
921  *
922  * \e lat1 should be in the range [&minus;90&deg;, 90&deg;].
923  **********************************************************************/
924  GeodesicLine ArcDirectLine(real lat1, real lon1, real azi1, real a12,
925  unsigned caps = ALL) const;
926 
927  /**
928  * Define a GeodesicLine in terms of the direct geodesic problem specified
929  * in terms of either distance or arc length.
930  *
931  * @param[in] lat1 latitude of point 1 (degrees).
932  * @param[in] lon1 longitude of point 1 (degrees).
933  * @param[in] azi1 azimuth at point 1 (degrees).
934  * @param[in] arcmode boolean flag determining the meaning of the \e
935  * s12_a12.
936  * @param[in] s12_a12 if \e arcmode is false, this is the distance between
937  * point 1 and point 2 (meters); otherwise it is the arc length between
938  * point 1 and point 2 (degrees); it can be negative.
939  * @param[in] caps bitor'ed combination of Geodesic::mask values
940  * specifying the capabilities the GeodesicLine object should possess,
941  * i.e., which quantities can be returned in calls to
942  * GeodesicLine::Position.
943  * @return a GeodesicLine object.
944  *
945  * This function sets point 3 of the GeodesicLine to correspond to point 2
946  * of the direct geodesic problem.
947  *
948  * \e lat1 should be in the range [&minus;90&deg;, 90&deg;].
949  **********************************************************************/
950  GeodesicLine GenDirectLine(real lat1, real lon1, real azi1,
951  bool arcmode, real s12_a12,
952  unsigned caps = ALL) const;
953  ///@}
954 
955  /** \name Inspector functions.
956  **********************************************************************/
957  ///@{
958 
959  /**
960  * @return \e a the equatorial radius of the ellipsoid (meters). This is
961  * the value used in the constructor.
962  **********************************************************************/
963  Math::real EquatorialRadius() const { return _a; }
964 
965  /**
966  * @return \e f the flattening of the ellipsoid. This is the
967  * value used in the constructor.
968  **********************************************************************/
969  Math::real Flattening() const { return _f; }
970 
971  /**
972  * @return \e exact whether the exact formulation is used. This is the
973  * value used in the constructor.
974  **********************************************************************/
975  bool Exact() const { return _exact; }
976 
977  /**
978  * @return total area of ellipsoid in meters<sup>2</sup>. The area of a
979  * polygon encircling a pole can be found by adding
980  * Geodesic::EllipsoidArea()/2 to the sum of \e S12 for each side of the
981  * polygon.
982  **********************************************************************/
983  Math::real EllipsoidArea() const
984  { return 4 * Math::pi() * _c2; }
985  ///@}
986 
987  /**
988  * A global instantiation of Geodesic with the parameters for the WGS84
989  * ellipsoid.
990  **********************************************************************/
991  static const Geodesic& WGS84();
992 
993  };
994 
995 } // namespace GeographicLib
996 
997 #endif // GEOGRAPHICLIB_GEODESIC_HPP
GeographicLib::Geodesic::ArcDirect
void ArcDirect(real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2, real &s12, real &m12, real &M12, real &M21) const
Definition: Geodesic.hpp:579
GeographicLib::Geodesic::LineClass
GeodesicLine LineClass
Definition: Geodesic.hpp:819
GeographicLib::Geodesic::Direct
Math::real Direct(real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2, real &m12, real &M12, real &M21) const
Definition: Geodesic.hpp:457
GeographicLib::Math::pi
static T pi()
Definition: Math.hpp:187
GEOGRAPHICLIB_EXPORT
#define GEOGRAPHICLIB_EXPORT
Definition: Constants.hpp:59
GeographicLib::Geodesic::EquatorialRadius
Math::real EquatorialRadius() const
Definition: Geodesic.hpp:963
GeographicLib::Geodesic::Inverse
Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real &azi1, real &azi2) const
Definition: Geodesic.hpp:712
GeographicLib::Geodesic::ArcDirect
void ArcDirect(real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2) const
Definition: Geodesic.hpp:519
GeographicLib::Geodesic::Direct
Math::real Direct(real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2) const
Definition: Geodesic.hpp:408
GeographicLib::Geodesic::ArcDirect
void ArcDirect(real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2, real &s12, real &m12, real &M12, real &M21, real &S12) const
Definition: Geodesic.hpp:506
GeographicLib::Geodesic::ArcDirect
void ArcDirect(real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2, real &s12, real &M12, real &M21) const
Definition: Geodesic.hpp:566
GeographicLib::Geodesic::Inverse
Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2) const
Definition: Geodesic.hpp:723
GeographicLib::Geodesic::Inverse
Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2, real &m12, real &M12, real &M21) const
Definition: Geodesic.hpp:759
GeographicLib::Geodesic::Inverse
Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2, real &M12, real &M21) const
Definition: Geodesic.hpp:747
GeographicLib
Namespace for GeographicLib.
Definition: Accumulator.cpp:12
GeographicLib::Geodesic::ArcDirect
void ArcDirect(real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2, real &s12) const
Definition: Geodesic.hpp:541
real
GeographicLib::Math::real real
Definition: Geod3Solve.cpp:25
GeographicLib::Geodesic::ArcDirect
void ArcDirect(real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2) const
Definition: Geodesic.hpp:530
GeographicLib::Geodesic::Flattening
Math::real Flattening() const
Definition: Geodesic.hpp:969
GeographicLib::Geodesic::Direct
Math::real Direct(real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2, real &m12, real &M12, real &M21, real &S12) const
Definition: Geodesic.hpp:394
GeographicLib::GeodesicExact
Exact geodesic calculations.
Definition: GeodesicExact.hpp:88
GeographicLib::Geodesic::Inverse
Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real &s12) const
Definition: Geodesic.hpp:701
GeodesicExact.hpp
Header for GeographicLib::GeodesicExact class.
GeographicLib::Geodesic::Direct
Math::real Direct(real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2, real &m12) const
Definition: Geodesic.hpp:432
Constants.hpp
Header for GeographicLib::Constants class.
GeographicLib::Geodesic::EllipsoidArea
Math::real EllipsoidArea() const
Definition: Geodesic.hpp:983
GeographicLib::Geodesic::Inverse
Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2, real &m12) const
Definition: Geodesic.hpp:735
GeographicLib::Geodesic::Direct
Math::real Direct(real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2, real &M12, real &M21) const
Definition: Geodesic.hpp:444
GEOGRAPHICLIB_GEODESIC_ORDER
#define GEOGRAPHICLIB_GEODESIC_ORDER
Definition: Geodesic.hpp:21
GeographicLib::Geodesic::ArcDirect
void ArcDirect(real lat1, real lon1, real azi1, real a12, real &lat2, real &lon2, real &azi2, real &s12, real &m12) const
Definition: Geodesic.hpp:553
GeographicLib::Geodesic
Geodesic calculations
Definition: Geodesic.hpp:175
GeographicLib::Geodesic::Direct
Math::real Direct(real lat1, real lon1, real azi1, real s12, real &lat2, real &lon2, real &azi2) const
Definition: Geodesic.hpp:420
GeographicLib::Geodesic::Inverse
Math::real Inverse(real lat1, real lon1, real lat2, real lon2, real &s12, real &azi1, real &azi2, real &m12, real &M12, real &M21, real &S12) const
Definition: Geodesic.hpp:689