GeographicLib  1.45
Rhumb.hpp
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1 /**
2  * \file Rhumb.hpp
3  * \brief Header for GeographicLib::Rhumb and GeographicLib::RhumbLine classes
4  *
5  * Copyright (c) Charles Karney (2014-2015) <charles@karney.com> and licensed
6  * under the MIT/X11 License. For more information, see
7  * http://geographiclib.sourceforge.net/
8  **********************************************************************/
9 
10 #if !defined(GEOGRAPHICLIB_RHUMB_HPP)
11 #define GEOGRAPHICLIB_RHUMB_HPP 1
12 
15 
16 #if !defined(GEOGRAPHICLIB_RHUMBAREA_ORDER)
17 /**
18  * The order of the series approximation used in rhumb area calculations.
19  * GEOGRAPHICLIB_RHUMBAREA_ORDER can be set to any integer in [4, 8].
20  **********************************************************************/
21 # define GEOGRAPHICLIB_RHUMBAREA_ORDER \
22  (GEOGRAPHICLIB_PRECISION == 2 ? 6 : \
23  (GEOGRAPHICLIB_PRECISION == 1 ? 4 : 8))
24 #endif
25 
26 namespace GeographicLib {
27 
28  class RhumbLine;
29  template <class T> class PolygonAreaT;
30 
31  /**
32  * \brief Solve of the direct and inverse rhumb problems.
33  *
34  * The path of constant azimuth between two points on a ellipsoid at (\e
35  * lat1, \e lon1) and (\e lat2, \e lon2) is called the rhumb line (also
36  * called the loxodrome). Its length is \e s12 and its azimuth is \e azi12.
37  * (The azimuth is the heading measured clockwise from north.)
38  *
39  * Given \e lat1, \e lon1, \e azi12, and \e s12, we can determine \e lat2,
40  * and \e lon2. This is the \e direct rhumb problem and its solution is
41  * given by the function Rhumb::Direct.
42  *
43  * Given \e lat1, \e lon1, \e lat2, and \e lon2, we can determine \e azi12
44  * and \e s12. This is the \e inverse rhumb problem, whose solution is given
45  * by Rhumb::Inverse. This finds the shortest such rhumb line, i.e., the one
46  * that wraps no more than half way around the earth. If the end points are
47  * on opposite meridians, there are two shortest rhumb lines and the
48  * east-going one is chosen.
49  *
50  * These routines also optionally calculate the area under the rhumb line, \e
51  * S12. This is the area, measured counter-clockwise, of the rhumb line
52  * quadrilateral with corners (<i>lat1</i>,<i>lon1</i>), (0,<i>lon1</i>),
53  * (0,<i>lon2</i>), and (<i>lat2</i>,<i>lon2</i>).
54  *
55  * Note that rhumb lines may be appreciably longer (up to 50%) than the
56  * corresponding Geodesic. For example the distance between London Heathrow
57  * and Tokyo Narita via the rhumb line is 11400 km which is 18% longer than
58  * the geodesic distance 9600 km.
59  *
60  * For more information on rhumb lines see \ref rhumb.
61  *
62  * Example of use:
63  * \include example-Rhumb.cpp
64  **********************************************************************/
65 
67  private:
68  typedef Math::real real;
69  friend class RhumbLine;
70  template <class T> friend class PolygonAreaT;
71  Ellipsoid _ell;
72  bool _exact;
73  real _c2;
74  static const int tm_maxord = GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER;
75  static const int maxpow_ = GEOGRAPHICLIB_RHUMBAREA_ORDER;
76  // _R[0] unused
77  real _R[maxpow_ + 1];
78  static inline real gd(real x)
79  { using std::atan; using std::sinh; return atan(sinh(x)); }
80 
81  // Use divided differences to determine (mu2 - mu1) / (psi2 - psi1)
82  // accurately
83  //
84  // Definition: Df(x,y,d) = (f(x) - f(y)) / (x - y)
85  // See:
86  // W. M. Kahan and R. J. Fateman,
87  // Symbolic computation of divided differences,
88  // SIGSAM Bull. 33(3), 7-28 (1999)
89  // https://dx.doi.org/10.1145/334714.334716
90  // http://www.cs.berkeley.edu/~fateman/papers/divdiff.pdf
91 
92  static inline real Dlog(real x, real y) {
93  real t = x - y;
94  return t ? 2 * Math::atanh(t / (x + y)) / t : 1 / x;
95  }
96  // N.B., x and y are in degrees
97  static inline real Dtan(real x, real y) {
98  real d = x - y, tx = Math::tand(x), ty = Math::tand(y), txy = tx * ty;
99  return d ?
100  (2 * txy > -1 ? (1 + txy) * Math::tand(d) : tx - ty) /
101  (d * Math::degree()) :
102  1 + txy;
103  }
104  static inline real Datan(real x, real y) {
105  using std::atan;
106  real d = x - y, xy = x * y;
107  return d ? (2 * xy > -1 ? atan( d / (1 + xy) ) : atan(x) - atan(y)) / d :
108  1 / (1 + xy);
109  }
110  static inline real Dsin(real x, real y) {
111  using std::sin; using std::cos;
112  real d = (x - y) / 2;
113  return cos((x + y)/2) * (d ? sin(d) / d : 1);
114  }
115  static inline real Dsinh(real x, real y) {
116  using std::sinh; using std::cosh;
117  real d = (x - y) / 2;
118  return cosh((x + y) / 2) * (d ? sinh(d) / d : 1);
119  }
120  static inline real Dcosh(real x, real y) {
121  using std::sinh;
122  real d = (x - y) / 2;
123  return sinh((x + y) / 2) * (d ? sinh(d) / d : 1);
124  }
125  static inline real Dasinh(real x, real y) {
126  real d = x - y,
127  hx = Math::hypot(real(1), x), hy = Math::hypot(real(1), y);
128  return d ? Math::asinh(x*y > 0 ? d * (x + y) / (x*hy + y*hx) :
129  x*hy - y*hx) / d :
130  1 / hx;
131  }
132  static inline real Dgd(real x, real y) {
133  using std::sinh;
134  return Datan(sinh(x), sinh(y)) * Dsinh(x, y);
135  }
136  // N.B., x and y are the tangents of the angles
137  static inline real Dgdinv(real x, real y)
138  { return Dasinh(x, y) / Datan(x, y); }
139  // Copied from LambertConformalConic...
140  // Deatanhe(x,y) = eatanhe((x-y)/(1-e^2*x*y))/(x-y)
141  inline real Deatanhe(real x, real y) const {
142  real t = x - y, d = 1 - _ell._e2 * x * y;
143  return t ? Math::eatanhe(t / d, _ell._es) / t : _ell._e2 / d;
144  }
145  // (E(x) - E(y)) / (x - y) -- E = incomplete elliptic integral of 2nd kind
146  real DE(real x, real y) const;
147  // (mux - muy) / (phix - phiy) using elliptic integrals
148  real DRectifying(real latx, real laty) const;
149  // (psix - psiy) / (phix - phiy)
150  real DIsometric(real latx, real laty) const;
151 
152  // (sum(c[j]*sin(2*j*x),j=1..n) - sum(c[j]*sin(2*j*x),j=1..n)) / (x - y)
153  static real SinCosSeries(bool sinp,
154  real x, real y, const real c[], int n);
155  // (mux - muy) / (chix - chiy) using Krueger's series
156  real DConformalToRectifying(real chix, real chiy) const;
157  // (chix - chiy) / (mux - muy) using Krueger's series
158  real DRectifyingToConformal(real mux, real muy) const;
159 
160  // (mux - muy) / (psix - psiy)
161  // N.B., psix and psiy are in degrees
162  real DIsometricToRectifying(real psix, real psiy) const;
163  // (psix - psiy) / (mux - muy)
164  real DRectifyingToIsometric(real mux, real muy) const;
165 
166  real MeanSinXi(real psi1, real psi2) const;
167 
168  // The following two functions (with lots of ignored arguments) mimic the
169  // interface to the corresponding Geodesic function. These are needed by
170  // PolygonAreaT.
171  void GenDirect(real lat1, real lon1, real azi12,
172  bool, real s12, unsigned outmask,
173  real& lat2, real& lon2, real&, real&, real&, real&, real&,
174  real& S12) const {
175  GenDirect(lat1, lon1, azi12, s12, outmask, lat2, lon2, S12);
176  }
177  void GenInverse(real lat1, real lon1, real lat2, real lon2,
178  unsigned outmask, real& s12, real& azi12,
179  real&, real& , real& , real& , real& S12) const {
180  GenInverse(lat1, lon1, lat2, lon2, outmask, s12, azi12, S12);
181  }
182  public:
183 
184  /**
185  * Bit masks for what calculations to do. They specify which results to
186  * return in the general routines Rhumb::GenDirect and Rhumb::GenInverse
187  * routines. RhumbLine::mask is a duplication of this enum.
188  **********************************************************************/
189  enum mask {
190  /**
191  * No output.
192  * @hideinitializer
193  **********************************************************************/
194  NONE = 0U,
195  /**
196  * Calculate latitude \e lat2.
197  * @hideinitializer
198  **********************************************************************/
199  LATITUDE = 1U<<7,
200  /**
201  * Calculate longitude \e lon2.
202  * @hideinitializer
203  **********************************************************************/
204  LONGITUDE = 1U<<8,
205  /**
206  * Calculate azimuth \e azi12.
207  * @hideinitializer
208  **********************************************************************/
209  AZIMUTH = 1U<<9,
210  /**
211  * Calculate distance \e s12.
212  * @hideinitializer
213  **********************************************************************/
214  DISTANCE = 1U<<10,
215  /**
216  * Calculate area \e S12.
217  * @hideinitializer
218  **********************************************************************/
219  AREA = 1U<<14,
220  /**
221  * Unroll \e lon2 in the direct calculation. (This flag used to be
222  * called LONG_NOWRAP.)
223  * @hideinitializer
224  **********************************************************************/
225  LONG_UNROLL = 1U<<15,
226  /// \cond SKIP
227  LONG_NOWRAP = LONG_UNROLL,
228  /// \endcond
229  /**
230  * Calculate everything. (LONG_UNROLL is not included in this mask.)
231  * @hideinitializer
232  **********************************************************************/
233  ALL = 0x7F80U,
234  };
235 
236  /**
237  * Constructor for a ellipsoid with
238  *
239  * @param[in] a equatorial radius (meters).
240  * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
241  * Negative \e f gives a prolate ellipsoid.
242  * @param[in] exact if true (the default) use an addition theorem for
243  * elliptic integrals to compute divided differences; otherwise use
244  * series expansion (accurate for |<i>f</i>| < 0.01).
245  * @exception GeographicErr if \e a or (1 &minus; \e f) \e a is not
246  * positive.
247  *
248  * See \ref rhumb, for a detailed description of the \e exact parameter.
249  **********************************************************************/
250  Rhumb(real a, real f, bool exact = true);
251 
252  /**
253  * Solve the direct rhumb problem returning also the area.
254  *
255  * @param[in] lat1 latitude of point 1 (degrees).
256  * @param[in] lon1 longitude of point 1 (degrees).
257  * @param[in] azi12 azimuth of the rhumb line (degrees).
258  * @param[in] s12 distance between point 1 and point 2 (meters); it can be
259  * negative.
260  * @param[out] lat2 latitude of point 2 (degrees).
261  * @param[out] lon2 longitude of point 2 (degrees).
262  * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
263  *
264  * \e lat1 should be in the range [&minus;90&deg;, 90&deg;]. The value of
265  * \e lon2 returned is in the range [&minus;180&deg;, 180&deg;).
266  *
267  * If point 1 is a pole, the cosine of its latitude is taken to be
268  * 1/&epsilon;<sup>2</sup> (where &epsilon; is 2<sup>-52</sup>). This
269  * position, which is extremely close to the actual pole, allows the
270  * calculation to be carried out in finite terms. If \e s12 is large
271  * enough that the rhumb line crosses a pole, the longitude of point 2
272  * is indeterminate (a NaN is returned for \e lon2 and \e S12).
273  **********************************************************************/
274  void Direct(real lat1, real lon1, real azi12, real s12,
275  real& lat2, real& lon2, real& S12) const {
276  GenDirect(lat1, lon1, azi12, s12,
277  LATITUDE | LONGITUDE | AREA, lat2, lon2, S12);
278  }
279 
280  /**
281  * Solve the direct rhumb problem without the area.
282  **********************************************************************/
283  void Direct(real lat1, real lon1, real azi12, real s12,
284  real& lat2, real& lon2) const {
285  real t;
286  GenDirect(lat1, lon1, azi12, s12, LATITUDE | LONGITUDE, lat2, lon2, t);
287  }
288 
289  /**
290  * The general direct rhumb problem. Rhumb::Direct is defined in terms
291  * of this function.
292  *
293  * @param[in] lat1 latitude of point 1 (degrees).
294  * @param[in] lon1 longitude of point 1 (degrees).
295  * @param[in] azi12 azimuth of the rhumb line (degrees).
296  * @param[in] s12 distance between point 1 and point 2 (meters); it can be
297  * negative.
298  * @param[in] outmask a bitor'ed combination of Rhumb::mask values
299  * specifying which of the following parameters should be set.
300  * @param[out] lat2 latitude of point 2 (degrees).
301  * @param[out] lon2 longitude of point 2 (degrees).
302  * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
303  *
304  * The Rhumb::mask values possible for \e outmask are
305  * - \e outmask |= Rhumb::LATITUDE for the latitude \e lat2;
306  * - \e outmask |= Rhumb::LONGITUDE for the latitude \e lon2;
307  * - \e outmask |= Rhumb::AREA for the area \e S12;
308  * - \e outmask |= Rhumb::ALL for all of the above;
309  * - \e outmask |= Rhumb::LONG_UNROLL to unroll \e lon2 instead of wrapping
310  * it into the range [&minus;180&deg;, 180&deg;).
311  * .
312  * With the Rhumb::LONG_UNROLL bit set, the quantity \e lon2 &minus;
313  * \e lon1 indicates how many times and in what sense the rhumb line
314  * encircles the ellipsoid.
315  **********************************************************************/
316  void GenDirect(real lat1, real lon1, real azi12, real s12, unsigned outmask,
317  real& lat2, real& lon2, real& S12) const;
318 
319  /**
320  * Solve the inverse rhumb problem returning also the area.
321  *
322  * @param[in] lat1 latitude of point 1 (degrees).
323  * @param[in] lon1 longitude of point 1 (degrees).
324  * @param[in] lat2 latitude of point 2 (degrees).
325  * @param[in] lon2 longitude of point 2 (degrees).
326  * @param[out] s12 rhumb distance between point 1 and point 2 (meters).
327  * @param[out] azi12 azimuth of the rhumb line (degrees).
328  * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
329  *
330  * The shortest rhumb line is found. If the end points are on opposite
331  * meridians, there are two shortest rhumb lines and the east-going one is
332  * chosen. \e lat1 and \e lat2 should be in the range [&minus;90&deg;,
333  * 90&deg;]. The value of \e azi12 returned is in the range
334  * [&minus;180&deg;, 180&deg;).
335  *
336  * If either point is a pole, the cosine of its latitude is taken to be
337  * 1/&epsilon;<sup>2</sup> (where &epsilon; is 2<sup>-52</sup>). This
338  * position, which is extremely close to the actual pole, allows the
339  * calculation to be carried out in finite terms.
340  **********************************************************************/
341  void Inverse(real lat1, real lon1, real lat2, real lon2,
342  real& s12, real& azi12, real& S12) const {
343  GenInverse(lat1, lon1, lat2, lon2,
344  DISTANCE | AZIMUTH | AREA, s12, azi12, S12);
345  }
346 
347  /**
348  * Solve the inverse rhumb problem without the area.
349  **********************************************************************/
350  void Inverse(real lat1, real lon1, real lat2, real lon2,
351  real& s12, real& azi12) const {
352  real t;
353  GenInverse(lat1, lon1, lat2, lon2, DISTANCE | AZIMUTH, s12, azi12, t);
354  }
355 
356  /**
357  * The general inverse rhumb problem. Rhumb::Inverse is defined in terms
358  * of this function.
359  *
360  * @param[in] lat1 latitude of point 1 (degrees).
361  * @param[in] lon1 longitude of point 1 (degrees).
362  * @param[in] lat2 latitude of point 2 (degrees).
363  * @param[in] lon2 longitude of point 2 (degrees).
364  * @param[in] outmask a bitor'ed combination of Rhumb::mask values
365  * specifying which of the following parameters should be set.
366  * @param[out] s12 rhumb distance between point 1 and point 2 (meters).
367  * @param[out] azi12 azimuth of the rhumb line (degrees).
368  * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
369  *
370  * The Rhumb::mask values possible for \e outmask are
371  * - \e outmask |= Rhumb::DISTANCE for the latitude \e s12;
372  * - \e outmask |= Rhumb::AZIMUTH for the latitude \e azi12;
373  * - \e outmask |= Rhumb::AREA for the area \e S12;
374  * - \e outmask |= Rhumb::ALL for all of the above;
375  **********************************************************************/
376  void GenInverse(real lat1, real lon1, real lat2, real lon2,
377  unsigned outmask,
378  real& s12, real& azi12, real& S12) const;
379 
380  /**
381  * Set up to compute several points on a single rhumb line.
382  *
383  * @param[in] lat1 latitude of point 1 (degrees).
384  * @param[in] lon1 longitude of point 1 (degrees).
385  * @param[in] azi12 azimuth of the rhumb line (degrees).
386  * @return a RhumbLine object.
387  *
388  * \e lat1 should be in the range [&minus;90&deg;, 90&deg;].
389  *
390  * If point 1 is a pole, the cosine of its latitude is taken to be
391  * 1/&epsilon;<sup>2</sup> (where &epsilon; is 2<sup>-52</sup>). This
392  * position, which is extremely close to the actual pole, allows the
393  * calculation to be carried out in finite terms.
394  **********************************************************************/
395  RhumbLine Line(real lat1, real lon1, real azi12) const;
396 
397  /** \name Inspector functions.
398  **********************************************************************/
399  ///@{
400 
401  /**
402  * @return \e a the equatorial radius of the ellipsoid (meters). This is
403  * the value used in the constructor.
404  **********************************************************************/
405  Math::real MajorRadius() const { return _ell.MajorRadius(); }
406 
407  /**
408  * @return \e f the flattening of the ellipsoid. This is the
409  * value used in the constructor.
410  **********************************************************************/
411  Math::real Flattening() const { return _ell.Flattening(); }
412 
413  Math::real EllipsoidArea() const { return _ell.Area(); }
414 
415  /**
416  * A global instantiation of Rhumb with the parameters for the WGS84
417  * ellipsoid.
418  **********************************************************************/
419  static const Rhumb& WGS84();
420  };
421 
422  /**
423  * \brief Find a sequence of points on a single rhumb line.
424  *
425  * RhumbLine facilitates the determination of a series of points on a single
426  * rhumb line. The starting point (\e lat1, \e lon1) and the azimuth \e
427  * azi12 are specified in the call to Rhumb::Line which returns a RhumbLine
428  * object. RhumbLine.Position returns the location of point 2 (and,
429  * optionally, the corresponding area, \e S12) a distance \e s12 along the
430  * rhumb line.
431  *
432  * There is no public constructor for this class. (Use Rhumb::Line to create
433  * an instance.) The Rhumb object used to create a RhumbLine must stay in
434  * scope as long as the RhumbLine.
435  *
436  * Example of use:
437  * \include example-RhumbLine.cpp
438  **********************************************************************/
439 
441  private:
442  typedef Math::real real;
443  friend class Rhumb;
444  const Rhumb& _rh;
445  bool _exact;
446  real _lat1, _lon1, _azi12, _salp, _calp, _mu1, _psi1, _r1;
447  RhumbLine& operator=(const RhumbLine&); // copy assignment not allowed
448  RhumbLine(const Rhumb& rh, real lat1, real lon1, real azi12,
449  bool exact);
450  public:
451 
452  /**
453  * This is a duplication of Rhumb::mask.
454  **********************************************************************/
455  enum mask {
456  /**
457  * No output.
458  * @hideinitializer
459  **********************************************************************/
460  NONE = Rhumb::NONE,
461  /**
462  * Calculate latitude \e lat2.
463  * @hideinitializer
464  **********************************************************************/
465  LATITUDE = Rhumb::LATITUDE,
466  /**
467  * Calculate longitude \e lon2.
468  * @hideinitializer
469  **********************************************************************/
470  LONGITUDE = Rhumb::LONGITUDE,
471  /**
472  * Calculate azimuth \e azi12.
473  * @hideinitializer
474  **********************************************************************/
475  AZIMUTH = Rhumb::AZIMUTH,
476  /**
477  * Calculate distance \e s12.
478  * @hideinitializer
479  **********************************************************************/
480  DISTANCE = Rhumb::DISTANCE,
481  /**
482  * Calculate area \e S12.
483  * @hideinitializer
484  **********************************************************************/
485  AREA = Rhumb::AREA,
486  /**
487  * Unroll \e lon2 in the direct calculation. (This flag used to be
488  * called LONG_NOWRAP.)
489  * @hideinitializer
490  **********************************************************************/
491  LONG_UNROLL = Rhumb::LONG_UNROLL,
492  /// \cond SKIP
493  LONG_NOWRAP = LONG_UNROLL,
494  /// \endcond
495  /**
496  * Calculate everything. (LONG_UNROLL is not included in this mask.)
497  * @hideinitializer
498  **********************************************************************/
499  ALL = Rhumb::ALL,
500  };
501 
502  /**
503  * Compute the position of point 2 which is a distance \e s12 (meters) from
504  * point 1. The area is also computed.
505  *
506  * @param[in] s12 distance between point 1 and point 2 (meters); it can be
507  * negative.
508  * @param[out] lat2 latitude of point 2 (degrees).
509  * @param[out] lon2 longitude of point 2 (degrees).
510  * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
511  *
512  * The value of \e lon2 returned is in the range [&minus;180&deg;,
513  * 180&deg;).
514  *
515  * If \e s12 is large enough that the rhumb line crosses a pole, the
516  * longitude of point 2 is indeterminate (a NaN is returned for \e lon2 and
517  * \e S12).
518  **********************************************************************/
519  void Position(real s12, real& lat2, real& lon2, real& S12) const {
520  GenPosition(s12, LATITUDE | LONGITUDE | AREA, lat2, lon2, S12);
521  }
522 
523  /**
524  * Compute the position of point 2 which is a distance \e s12 (meters) from
525  * point 1. The area is not computed.
526  **********************************************************************/
527  void Position(real s12, real& lat2, real& lon2) const {
528  real t;
529  GenPosition(s12, LATITUDE | LONGITUDE, lat2, lon2, t);
530  }
531 
532  /**
533  * The general position routine. RhumbLine::Position is defined in term so
534  * this function.
535  *
536  * @param[in] s12 distance between point 1 and point 2 (meters); it can be
537  * negative.
538  * @param[in] outmask a bitor'ed combination of RhumbLine::mask values
539  * specifying which of the following parameters should be set.
540  * @param[out] lat2 latitude of point 2 (degrees).
541  * @param[out] lon2 longitude of point 2 (degrees).
542  * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
543  *
544  * The RhumbLine::mask values possible for \e outmask are
545  * - \e outmask |= RhumbLine::LATITUDE for the latitude \e lat2;
546  * - \e outmask |= RhumbLine::LONGITUDE for the latitude \e lon2;
547  * - \e outmask |= RhumbLine::AREA for the area \e S12;
548  * - \e outmask |= RhumbLine::ALL for all of the above;
549  * - \e outmask |= RhumbLine::LONG_UNROLL to unroll \e lon2 instead of
550  * wrapping it into the range [&minus;180&deg;, 180&deg;).
551  * .
552  * With the RhumbLine::LONG_UNROLL bit set, the quantity \e lon2 &minus; \e
553  * lon1 indicates how many times and in what sense the rhumb line encircles
554  * the ellipsoid.
555  *
556  * If \e s12 is large enough that the rhumb line crosses a pole, the
557  * longitude of point 2 is indeterminate (a NaN is returned for \e lon2 and
558  * \e S12).
559  **********************************************************************/
560  void GenPosition(real s12, unsigned outmask,
561  real& lat2, real& lon2, real& S12) const;
562 
563  /** \name Inspector functions
564  **********************************************************************/
565  ///@{
566 
567  /**
568  * @return \e lat1 the latitude of point 1 (degrees).
569  **********************************************************************/
570  Math::real Latitude() const { return _lat1; }
571 
572  /**
573  * @return \e lon1 the longitude of point 1 (degrees).
574  **********************************************************************/
575  Math::real Longitude() const { return _lon1; }
576 
577  /**
578  * @return \e azi12 the azimuth of the rhumb line (degrees).
579  **********************************************************************/
580  Math::real Azimuth() const { return _azi12; }
581 
582  /**
583  * @return \e a the equatorial radius of the ellipsoid (meters). This is
584  * the value inherited from the Rhumb object used in the constructor.
585  **********************************************************************/
586  Math::real MajorRadius() const { return _rh.MajorRadius(); }
587 
588  /**
589  * @return \e f the flattening of the ellipsoid. This is the value
590  * inherited from the Rhumb object used in the constructor.
591  **********************************************************************/
592  Math::real Flattening() const { return _rh.Flattening(); }
593  };
594 
595 } // namespace GeographicLib
596 
597 #endif // GEOGRAPHICLIB_RHUMB_HPP
void Inverse(real lat1, real lon1, real lat2, real lon2, real &s12, real &azi12, real &S12) const
Definition: Rhumb.hpp:341
#define GEOGRAPHICLIB_EXPORT
Definition: Constants.hpp:90
GeographicLib::Math::real real
Definition: GeodSolve.cpp:32
static T eatanhe(T x, T es)
void Position(real s12, real &lat2, real &lon2) const
Definition: Rhumb.hpp:527
#define GEOGRAPHICLIB_RHUMBAREA_ORDER
Definition: Rhumb.hpp:21
static T atanh(T x)
Definition: Math.hpp:342
Math::real Latitude() const
Definition: Rhumb.hpp:570
Math::real MajorRadius() const
Definition: Rhumb.hpp:405
static T asinh(T x)
Definition: Math.hpp:325
Math::real EllipsoidArea() const
Definition: Rhumb.hpp:413
static T hypot(T x, T y)
Definition: Math.hpp:257
Math::real MajorRadius() const
Definition: Ellipsoid.hpp:80
Math::real Azimuth() const
Definition: Rhumb.hpp:580
Math::real Longitude() const
Definition: Rhumb.hpp:575
Namespace for GeographicLib.
Definition: Accumulator.cpp:12
#define GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER
Header for GeographicLib::Ellipsoid class.
Math::real Flattening() const
Definition: Rhumb.hpp:411
Properties of an ellipsoid.
Definition: Ellipsoid.hpp:39
Math::real MajorRadius() const
Definition: Rhumb.hpp:586
static T tand(T x)
Definition: Math.hpp:656
Math::real Area() const
Definition: Ellipsoid.cpp:40
Header for GeographicLib::Constants class.
Solve of the direct and inverse rhumb problems.
Definition: Rhumb.hpp:66
void Inverse(real lat1, real lon1, real lat2, real lon2, real &s12, real &azi12) const
Definition: Rhumb.hpp:350
Find a sequence of points on a single rhumb line.
Definition: Rhumb.hpp:440
Math::real Flattening() const
Definition: Rhumb.hpp:592
void Direct(real lat1, real lon1, real azi12, real s12, real &lat2, real &lon2) const
Definition: Rhumb.hpp:283
Math::real Flattening() const
Definition: Ellipsoid.hpp:120
void Direct(real lat1, real lon1, real azi12, real s12, real &lat2, real &lon2, real &S12) const
Definition: Rhumb.hpp:274
void Position(real s12, real &lat2, real &lon2, real &S12) const
Definition: Rhumb.hpp:519