GeographicLib  1.47
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 (2014-2016) <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://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.
222  * @hideinitializer
223  **********************************************************************/
224  LONG_UNROLL = 1U<<15,
225  /**
226  * Calculate everything. (LONG_UNROLL is not included in this mask.)
227  * @hideinitializer
228  **********************************************************************/
229  ALL = 0x7F80U,
230  };
231 
232  /**
233  * Constructor for a ellipsoid with
234  *
235  * @param[in] a equatorial radius (meters).
236  * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
237  * Negative \e f gives a prolate ellipsoid.
238  * @param[in] exact if true (the default) use an addition theorem for
239  * elliptic integrals to compute divided differences; otherwise use
240  * series expansion (accurate for |<i>f</i>| < 0.01).
241  * @exception GeographicErr if \e a or (1 &minus; \e f) \e a is not
242  * positive.
243  *
244  * See \ref rhumb, for a detailed description of the \e exact parameter.
245  **********************************************************************/
246  Rhumb(real a, real f, bool exact = true);
247 
248  /**
249  * Solve the direct rhumb problem returning also the area.
250  *
251  * @param[in] lat1 latitude of point 1 (degrees).
252  * @param[in] lon1 longitude of point 1 (degrees).
253  * @param[in] azi12 azimuth of the rhumb line (degrees).
254  * @param[in] s12 distance between point 1 and point 2 (meters); it can be
255  * negative.
256  * @param[out] lat2 latitude of point 2 (degrees).
257  * @param[out] lon2 longitude of point 2 (degrees).
258  * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
259  *
260  * \e lat1 should be in the range [&minus;90&deg;, 90&deg;]. The value of
261  * \e lon2 returned is in the range [&minus;180&deg;, 180&deg;].
262  *
263  * If point 1 is a pole, the cosine of its latitude is taken to be
264  * 1/&epsilon;<sup>2</sup> (where &epsilon; is 2<sup>-52</sup>). This
265  * position, which is extremely close to the actual pole, allows the
266  * calculation to be carried out in finite terms. If \e s12 is large
267  * enough that the rhumb line crosses a pole, the longitude of point 2
268  * is indeterminate (a NaN is returned for \e lon2 and \e S12).
269  **********************************************************************/
270  void Direct(real lat1, real lon1, real azi12, real s12,
271  real& lat2, real& lon2, real& S12) const {
272  GenDirect(lat1, lon1, azi12, s12,
273  LATITUDE | LONGITUDE | AREA, lat2, lon2, S12);
274  }
275 
276  /**
277  * Solve the direct rhumb problem without the area.
278  **********************************************************************/
279  void Direct(real lat1, real lon1, real azi12, real s12,
280  real& lat2, real& lon2) const {
281  real t;
282  GenDirect(lat1, lon1, azi12, s12, LATITUDE | LONGITUDE, lat2, lon2, t);
283  }
284 
285  /**
286  * The general direct rhumb problem. Rhumb::Direct is defined in terms
287  * of this function.
288  *
289  * @param[in] lat1 latitude of point 1 (degrees).
290  * @param[in] lon1 longitude of point 1 (degrees).
291  * @param[in] azi12 azimuth of the rhumb line (degrees).
292  * @param[in] s12 distance between point 1 and point 2 (meters); it can be
293  * negative.
294  * @param[in] outmask a bitor'ed combination of Rhumb::mask values
295  * specifying which of the following parameters should be set.
296  * @param[out] lat2 latitude of point 2 (degrees).
297  * @param[out] lon2 longitude of point 2 (degrees).
298  * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
299  *
300  * The Rhumb::mask values possible for \e outmask are
301  * - \e outmask |= Rhumb::LATITUDE for the latitude \e lat2;
302  * - \e outmask |= Rhumb::LONGITUDE for the latitude \e lon2;
303  * - \e outmask |= Rhumb::AREA for the area \e S12;
304  * - \e outmask |= Rhumb::ALL for all of the above;
305  * - \e outmask |= Rhumb::LONG_UNROLL to unroll \e lon2 instead of wrapping
306  * it into the range [&minus;180&deg;, 180&deg;].
307  * .
308  * With the Rhumb::LONG_UNROLL bit set, the quantity \e lon2 &minus;
309  * \e lon1 indicates how many times and in what sense the rhumb line
310  * encircles the ellipsoid.
311  **********************************************************************/
312  void GenDirect(real lat1, real lon1, real azi12, real s12, unsigned outmask,
313  real& lat2, real& lon2, real& S12) const;
314 
315  /**
316  * Solve the inverse rhumb problem returning also the area.
317  *
318  * @param[in] lat1 latitude of point 1 (degrees).
319  * @param[in] lon1 longitude of point 1 (degrees).
320  * @param[in] lat2 latitude of point 2 (degrees).
321  * @param[in] lon2 longitude of point 2 (degrees).
322  * @param[out] s12 rhumb distance between point 1 and point 2 (meters).
323  * @param[out] azi12 azimuth of the rhumb line (degrees).
324  * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
325  *
326  * The shortest rhumb line is found. If the end points are on opposite
327  * meridians, there are two shortest rhumb lines and the east-going one is
328  * chosen. \e lat1 and \e lat2 should be in the range [&minus;90&deg;,
329  * 90&deg;]. The value of \e azi12 returned is in the range
330  * [&minus;180&deg;, 180&deg;].
331  *
332  * If either point is a pole, the cosine of its latitude is taken to be
333  * 1/&epsilon;<sup>2</sup> (where &epsilon; is 2<sup>-52</sup>). This
334  * position, which is extremely close to the actual pole, allows the
335  * calculation to be carried out in finite terms.
336  **********************************************************************/
337  void Inverse(real lat1, real lon1, real lat2, real lon2,
338  real& s12, real& azi12, real& S12) const {
339  GenInverse(lat1, lon1, lat2, lon2,
340  DISTANCE | AZIMUTH | AREA, s12, azi12, S12);
341  }
342 
343  /**
344  * Solve the inverse rhumb problem without the area.
345  **********************************************************************/
346  void Inverse(real lat1, real lon1, real lat2, real lon2,
347  real& s12, real& azi12) const {
348  real t;
349  GenInverse(lat1, lon1, lat2, lon2, DISTANCE | AZIMUTH, s12, azi12, t);
350  }
351 
352  /**
353  * The general inverse rhumb problem. Rhumb::Inverse is defined in terms
354  * of this function.
355  *
356  * @param[in] lat1 latitude of point 1 (degrees).
357  * @param[in] lon1 longitude of point 1 (degrees).
358  * @param[in] lat2 latitude of point 2 (degrees).
359  * @param[in] lon2 longitude of point 2 (degrees).
360  * @param[in] outmask a bitor'ed combination of Rhumb::mask values
361  * specifying which of the following parameters should be set.
362  * @param[out] s12 rhumb distance between point 1 and point 2 (meters).
363  * @param[out] azi12 azimuth of the rhumb line (degrees).
364  * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
365  *
366  * The Rhumb::mask values possible for \e outmask are
367  * - \e outmask |= Rhumb::DISTANCE for the latitude \e s12;
368  * - \e outmask |= Rhumb::AZIMUTH for the latitude \e azi12;
369  * - \e outmask |= Rhumb::AREA for the area \e S12;
370  * - \e outmask |= Rhumb::ALL for all of the above;
371  **********************************************************************/
372  void GenInverse(real lat1, real lon1, real lat2, real lon2,
373  unsigned outmask,
374  real& s12, real& azi12, real& S12) const;
375 
376  /**
377  * Set up to compute several points on a single rhumb line.
378  *
379  * @param[in] lat1 latitude of point 1 (degrees).
380  * @param[in] lon1 longitude of point 1 (degrees).
381  * @param[in] azi12 azimuth of the rhumb line (degrees).
382  * @return a RhumbLine object.
383  *
384  * \e lat1 should be in the range [&minus;90&deg;, 90&deg;].
385  *
386  * If point 1 is a pole, the cosine of its latitude is taken to be
387  * 1/&epsilon;<sup>2</sup> (where &epsilon; is 2<sup>-52</sup>). This
388  * position, which is extremely close to the actual pole, allows the
389  * calculation to be carried out in finite terms.
390  **********************************************************************/
391  RhumbLine Line(real lat1, real lon1, real azi12) const;
392 
393  /** \name Inspector functions.
394  **********************************************************************/
395  ///@{
396 
397  /**
398  * @return \e a the equatorial radius of the ellipsoid (meters). This is
399  * the value used in the constructor.
400  **********************************************************************/
401  Math::real MajorRadius() const { return _ell.MajorRadius(); }
402 
403  /**
404  * @return \e f the flattening of the ellipsoid. This is the
405  * value used in the constructor.
406  **********************************************************************/
407  Math::real Flattening() const { return _ell.Flattening(); }
408 
409  Math::real EllipsoidArea() const { return _ell.Area(); }
410 
411  /**
412  * A global instantiation of Rhumb with the parameters for the WGS84
413  * ellipsoid.
414  **********************************************************************/
415  static const Rhumb& WGS84();
416  };
417 
418  /**
419  * \brief Find a sequence of points on a single rhumb line.
420  *
421  * RhumbLine facilitates the determination of a series of points on a single
422  * rhumb line. The starting point (\e lat1, \e lon1) and the azimuth \e
423  * azi12 are specified in the call to Rhumb::Line which returns a RhumbLine
424  * object. RhumbLine.Position returns the location of point 2 (and,
425  * optionally, the corresponding area, \e S12) a distance \e s12 along the
426  * rhumb line.
427  *
428  * There is no public constructor for this class. (Use Rhumb::Line to create
429  * an instance.) The Rhumb object used to create a RhumbLine must stay in
430  * scope as long as the RhumbLine.
431  *
432  * Example of use:
433  * \include example-RhumbLine.cpp
434  **********************************************************************/
435 
437  private:
438  typedef Math::real real;
439  friend class Rhumb;
440  const Rhumb& _rh;
441  bool _exact;
442  real _lat1, _lon1, _azi12, _salp, _calp, _mu1, _psi1, _r1;
443  RhumbLine& operator=(const RhumbLine&); // copy assignment not allowed
444  RhumbLine(const Rhumb& rh, real lat1, real lon1, real azi12,
445  bool exact);
446  public:
447 
448  /**
449  * This is a duplication of Rhumb::mask.
450  **********************************************************************/
451  enum mask {
452  /**
453  * No output.
454  * @hideinitializer
455  **********************************************************************/
456  NONE = Rhumb::NONE,
457  /**
458  * Calculate latitude \e lat2.
459  * @hideinitializer
460  **********************************************************************/
461  LATITUDE = Rhumb::LATITUDE,
462  /**
463  * Calculate longitude \e lon2.
464  * @hideinitializer
465  **********************************************************************/
466  LONGITUDE = Rhumb::LONGITUDE,
467  /**
468  * Calculate azimuth \e azi12.
469  * @hideinitializer
470  **********************************************************************/
471  AZIMUTH = Rhumb::AZIMUTH,
472  /**
473  * Calculate distance \e s12.
474  * @hideinitializer
475  **********************************************************************/
476  DISTANCE = Rhumb::DISTANCE,
477  /**
478  * Calculate area \e S12.
479  * @hideinitializer
480  **********************************************************************/
481  AREA = Rhumb::AREA,
482  /**
483  * Unroll \e lon2 in the direct calculation.
484  * @hideinitializer
485  **********************************************************************/
486  LONG_UNROLL = Rhumb::LONG_UNROLL,
487  /**
488  * Calculate everything. (LONG_UNROLL is not included in this mask.)
489  * @hideinitializer
490  **********************************************************************/
491  ALL = Rhumb::ALL,
492  };
493 
494  /**
495  * Compute the position of point 2 which is a distance \e s12 (meters) from
496  * point 1. The area is also computed.
497  *
498  * @param[in] s12 distance between point 1 and point 2 (meters); it can be
499  * negative.
500  * @param[out] lat2 latitude of point 2 (degrees).
501  * @param[out] lon2 longitude of point 2 (degrees).
502  * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
503  *
504  * The value of \e lon2 returned is in the range [&minus;180&deg;,
505  * 180&deg;].
506  *
507  * If \e s12 is large enough that the rhumb line crosses a pole, the
508  * longitude of point 2 is indeterminate (a NaN is returned for \e lon2 and
509  * \e S12).
510  **********************************************************************/
511  void Position(real s12, real& lat2, real& lon2, real& S12) const {
512  GenPosition(s12, LATITUDE | LONGITUDE | AREA, lat2, lon2, S12);
513  }
514 
515  /**
516  * Compute the position of point 2 which is a distance \e s12 (meters) from
517  * point 1. The area is not computed.
518  **********************************************************************/
519  void Position(real s12, real& lat2, real& lon2) const {
520  real t;
521  GenPosition(s12, LATITUDE | LONGITUDE, lat2, lon2, t);
522  }
523 
524  /**
525  * The general position routine. RhumbLine::Position is defined in term so
526  * this function.
527  *
528  * @param[in] s12 distance between point 1 and point 2 (meters); it can be
529  * negative.
530  * @param[in] outmask a bitor'ed combination of RhumbLine::mask values
531  * specifying which of the following parameters should be set.
532  * @param[out] lat2 latitude of point 2 (degrees).
533  * @param[out] lon2 longitude of point 2 (degrees).
534  * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
535  *
536  * The RhumbLine::mask values possible for \e outmask are
537  * - \e outmask |= RhumbLine::LATITUDE for the latitude \e lat2;
538  * - \e outmask |= RhumbLine::LONGITUDE for the latitude \e lon2;
539  * - \e outmask |= RhumbLine::AREA for the area \e S12;
540  * - \e outmask |= RhumbLine::ALL for all of the above;
541  * - \e outmask |= RhumbLine::LONG_UNROLL to unroll \e lon2 instead of
542  * wrapping it into the range [&minus;180&deg;, 180&deg;].
543  * .
544  * With the RhumbLine::LONG_UNROLL bit set, the quantity \e lon2 &minus; \e
545  * lon1 indicates how many times and in what sense the rhumb line encircles
546  * the ellipsoid.
547  *
548  * If \e s12 is large enough that the rhumb line crosses a pole, the
549  * longitude of point 2 is indeterminate (a NaN is returned for \e lon2 and
550  * \e S12).
551  **********************************************************************/
552  void GenPosition(real s12, unsigned outmask,
553  real& lat2, real& lon2, real& S12) const;
554 
555  /** \name Inspector functions
556  **********************************************************************/
557  ///@{
558 
559  /**
560  * @return \e lat1 the latitude of point 1 (degrees).
561  **********************************************************************/
562  Math::real Latitude() const { return _lat1; }
563 
564  /**
565  * @return \e lon1 the longitude of point 1 (degrees).
566  **********************************************************************/
567  Math::real Longitude() const { return _lon1; }
568 
569  /**
570  * @return \e azi12 the azimuth of the rhumb line (degrees).
571  **********************************************************************/
572  Math::real Azimuth() const { return _azi12; }
573 
574  /**
575  * @return \e a the equatorial radius of the ellipsoid (meters). This is
576  * the value inherited from the Rhumb object used in the constructor.
577  **********************************************************************/
578  Math::real MajorRadius() const { return _rh.MajorRadius(); }
579 
580  /**
581  * @return \e f the flattening of the ellipsoid. This is the value
582  * inherited from the Rhumb object used in the constructor.
583  **********************************************************************/
584  Math::real Flattening() const { return _rh.Flattening(); }
585  };
586 
587 } // namespace GeographicLib
588 
589 #endif // GEOGRAPHICLIB_RHUMB_HPP
Math::real Flattening() const
Definition: Rhumb.hpp:407
#define GEOGRAPHICLIB_EXPORT
Definition: Constants.hpp:90
void Inverse(real lat1, real lon1, real lat2, real lon2, real &s12, real &azi12) const
Definition: Rhumb.hpp:346
GeographicLib::Math::real real
Definition: GeodSolve.cpp:31
void Position(real s12, real &lat2, real &lon2, real &S12) const
Definition: Rhumb.hpp:511
Math::real Latitude() const
Definition: Rhumb.hpp:562
Mathematical functions needed by GeographicLib.
Definition: Math.hpp:102
Math::real Area() const
Definition: Ellipsoid.cpp:40
#define GEOGRAPHICLIB_RHUMBAREA_ORDER
Definition: Rhumb.hpp:21
static T atanh(T x)
Definition: Math.hpp:328
Math::real Azimuth() const
Definition: Rhumb.hpp:572
Math::real Flattening() const
Definition: Ellipsoid.hpp:120
static T asinh(T x)
Definition: Math.hpp:311
static T hypot(T x, T y)
Definition: Math.hpp:243
Math::real Longitude() const
Definition: Rhumb.hpp:567
Namespace for GeographicLib.
Definition: Accumulator.cpp:12
#define GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER
Header for GeographicLib::Ellipsoid class.
Properties of an ellipsoid.
Definition: Ellipsoid.hpp:39
void Inverse(real lat1, real lon1, real lat2, real lon2, real &s12, real &azi12, real &S12) const
Definition: Rhumb.hpp:337
Math::real MajorRadius() const
Definition: Ellipsoid.hpp:80
void Direct(real lat1, real lon1, real azi12, real s12, real &lat2, real &lon2) const
Definition: Rhumb.hpp:279
static T tand(T x)
Definition: Math.hpp:664
Header for GeographicLib::Constants class.
Math::real MajorRadius() const
Definition: Rhumb.hpp:401
void Position(real s12, real &lat2, real &lon2) const
Definition: Rhumb.hpp:519
Solve of the direct and inverse rhumb problems.
Definition: Rhumb.hpp:66
Find a sequence of points on a single rhumb line.
Definition: Rhumb.hpp:436
Math::real EllipsoidArea() const
Definition: Rhumb.hpp:409
Math::real Flattening() const
Definition: Rhumb.hpp:584
static T eatanhe(T x, T es)
Definition: Math.cpp:21
void Direct(real lat1, real lon1, real azi12, real s12, real &lat2, real &lon2, real &S12) const
Definition: Rhumb.hpp:270
Math::real MajorRadius() const
Definition: Rhumb.hpp:578