GeographicLib  1.44
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. If \e f &gt; 1, set
242  * flattening to 1/\e f.
243  * @param[in] exact if true (the default) use an addition theorem for
244  * elliptic integrals to compute divided differences; otherwise use
245  * series expansion (accurate for |<i>f</i>| < 0.01).
246  * @exception GeographicErr if \e a or (1 &minus; \e f) \e a is not
247  * positive.
248  *
249  * See \ref rhumb, for a detailed description of the \e exact parameter.
250  **********************************************************************/
251  Rhumb(real a, real f, bool exact = true);
252 
253  /**
254  * Solve the direct rhumb problem returning also the area.
255  *
256  * @param[in] lat1 latitude of point 1 (degrees).
257  * @param[in] lon1 longitude of point 1 (degrees).
258  * @param[in] azi12 azimuth of the rhumb line (degrees).
259  * @param[in] s12 distance between point 1 and point 2 (meters); it can be
260  * negative.
261  * @param[out] lat2 latitude of point 2 (degrees).
262  * @param[out] lon2 longitude of point 2 (degrees).
263  * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
264  *
265  * \e lat1 should be in the range [&minus;90&deg;, 90&deg;]. The value of
266  * \e lon2 returned is in the range [&minus;180&deg;, 180&deg;).
267  *
268  * If point 1 is a pole, the cosine of its latitude is taken to be
269  * 1/&epsilon;<sup>2</sup> (where &epsilon; is 2<sup>-52</sup>). This
270  * position, which is extremely close to the actual pole, allows the
271  * calculation to be carried out in finite terms. If \e s12 is large
272  * enough that the rhumb line crosses a pole, the longitude of point 2
273  * is indeterminate (a NaN is returned for \e lon2 and \e S12).
274  **********************************************************************/
275  void Direct(real lat1, real lon1, real azi12, real s12,
276  real& lat2, real& lon2, real& S12) const {
277  GenDirect(lat1, lon1, azi12, s12,
278  LATITUDE | LONGITUDE | AREA, lat2, lon2, S12);
279  }
280 
281  /**
282  * Solve the direct rhumb problem without the area.
283  **********************************************************************/
284  void Direct(real lat1, real lon1, real azi12, real s12,
285  real& lat2, real& lon2) const {
286  real t;
287  GenDirect(lat1, lon1, azi12, s12, LATITUDE | LONGITUDE, lat2, lon2, t);
288  }
289 
290  /**
291  * The general direct rhumb problem. Rhumb::Direct is defined in terms
292  * of this function.
293  *
294  * @param[in] lat1 latitude of point 1 (degrees).
295  * @param[in] lon1 longitude of point 1 (degrees).
296  * @param[in] azi12 azimuth of the rhumb line (degrees).
297  * @param[in] s12 distance between point 1 and point 2 (meters); it can be
298  * negative.
299  * @param[in] outmask a bitor'ed combination of Rhumb::mask values
300  * specifying which of the following parameters should be set.
301  * @param[out] lat2 latitude of point 2 (degrees).
302  * @param[out] lon2 longitude of point 2 (degrees).
303  * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
304  *
305  * The Rhumb::mask values possible for \e outmask are
306  * - \e outmask |= Rhumb::LATITUDE for the latitude \e lat2;
307  * - \e outmask |= Rhumb::LONGITUDE for the latitude \e lon2;
308  * - \e outmask |= Rhumb::AREA for the area \e S12;
309  * - \e outmask |= Rhumb::ALL for all of the above;
310  * - \e outmask |= Rhumb::LONG_UNROLL to unroll \e lon2 instead of wrapping
311  * it into the range [&minus;180&deg;, 180&deg;).
312  * .
313  * With the Rhumb::LONG_UNROLL bit set, the quantity \e lon2 &minus;
314  * \e lon1 indicates how many times and in what sense the rhumb line
315  * encircles the ellipsoid.
316  **********************************************************************/
317  void GenDirect(real lat1, real lon1, real azi12, real s12, unsigned outmask,
318  real& lat2, real& lon2, real& S12) const;
319 
320  /**
321  * Solve the inverse rhumb problem returning also the area.
322  *
323  * @param[in] lat1 latitude of point 1 (degrees).
324  * @param[in] lon1 longitude of point 1 (degrees).
325  * @param[in] lat2 latitude of point 2 (degrees).
326  * @param[in] lon2 longitude of point 2 (degrees).
327  * @param[out] s12 rhumb distance between point 1 and point 2 (meters).
328  * @param[out] azi12 azimuth of the rhumb line (degrees).
329  * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
330  *
331  * The shortest rhumb line is found. If the end points are on opposite
332  * meridians, there are two shortest rhumb lines and the east-going one is
333  * chosen. \e lat1 and \e lat2 should be in the range [&minus;90&deg;,
334  * 90&deg;]. The value of \e azi12 returned is in the range
335  * [&minus;180&deg;, 180&deg;).
336  *
337  * If either point is a pole, the cosine of its latitude is taken to be
338  * 1/&epsilon;<sup>2</sup> (where &epsilon; is 2<sup>-52</sup>). This
339  * position, which is extremely close to the actual pole, allows the
340  * calculation to be carried out in finite terms.
341  **********************************************************************/
342  void Inverse(real lat1, real lon1, real lat2, real lon2,
343  real& s12, real& azi12, real& S12) const {
344  GenInverse(lat1, lon1, lat2, lon2,
345  DISTANCE | AZIMUTH | AREA, s12, azi12, S12);
346  }
347 
348  /**
349  * Solve the inverse rhumb problem without the area.
350  **********************************************************************/
351  void Inverse(real lat1, real lon1, real lat2, real lon2,
352  real& s12, real& azi12) const {
353  real t;
354  GenInverse(lat1, lon1, lat2, lon2, DISTANCE | AZIMUTH, s12, azi12, t);
355  }
356 
357  /**
358  * The general inverse rhumb problem. Rhumb::Inverse is defined in terms
359  * of this function.
360  *
361  * @param[in] lat1 latitude of point 1 (degrees).
362  * @param[in] lon1 longitude of point 1 (degrees).
363  * @param[in] lat2 latitude of point 2 (degrees).
364  * @param[in] lon2 longitude of point 2 (degrees).
365  * @param[in] outmask a bitor'ed combination of Rhumb::mask values
366  * specifying which of the following parameters should be set.
367  * @param[out] s12 rhumb distance between point 1 and point 2 (meters).
368  * @param[out] azi12 azimuth of the rhumb line (degrees).
369  * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
370  *
371  * The Rhumb::mask values possible for \e outmask are
372  * - \e outmask |= Rhumb::DISTANCE for the latitude \e s12;
373  * - \e outmask |= Rhumb::AZIMUTH for the latitude \e azi12;
374  * - \e outmask |= Rhumb::AREA for the area \e S12;
375  * - \e outmask |= Rhumb::ALL for all of the above;
376  **********************************************************************/
377  void GenInverse(real lat1, real lon1, real lat2, real lon2,
378  unsigned outmask,
379  real& s12, real& azi12, real& S12) const;
380 
381  /**
382  * Set up to compute several points on a single rhumb line.
383  *
384  * @param[in] lat1 latitude of point 1 (degrees).
385  * @param[in] lon1 longitude of point 1 (degrees).
386  * @param[in] azi12 azimuth of the rhumb line (degrees).
387  * @return a RhumbLine object.
388  *
389  * \e lat1 should be in the range [&minus;90&deg;, 90&deg;].
390  *
391  * If point 1 is a pole, the cosine of its latitude is taken to be
392  * 1/&epsilon;<sup>2</sup> (where &epsilon; is 2<sup>-52</sup>). This
393  * position, which is extremely close to the actual pole, allows the
394  * calculation to be carried out in finite terms.
395  **********************************************************************/
396  RhumbLine Line(real lat1, real lon1, real azi12) const;
397 
398  /** \name Inspector functions.
399  **********************************************************************/
400  ///@{
401 
402  /**
403  * @return \e a the equatorial radius of the ellipsoid (meters). This is
404  * the value used in the constructor.
405  **********************************************************************/
406  Math::real MajorRadius() const { return _ell.MajorRadius(); }
407 
408  /**
409  * @return \e f the flattening of the ellipsoid. This is the
410  * value used in the constructor.
411  **********************************************************************/
412  Math::real Flattening() const { return _ell.Flattening(); }
413 
414  Math::real EllipsoidArea() const { return _ell.Area(); }
415 
416  /**
417  * A global instantiation of Rhumb with the parameters for the WGS84
418  * ellipsoid.
419  **********************************************************************/
420  static const Rhumb& WGS84();
421  };
422 
423  /**
424  * \brief Find a sequence of points on a single rhumb line.
425  *
426  * RhumbLine facilitates the determination of a series of points on a single
427  * rhumb line. The starting point (\e lat1, \e lon1) and the azimuth \e
428  * azi12 are specified in the call to Rhumb::Line which returns a RhumbLine
429  * object. RhumbLine.Position returns the location of point 2 (and,
430  * optionally, the corresponding area, \e S12) a distance \e s12 along the
431  * rhumb line.
432  *
433  * There is no public constructor for this class. (Use Rhumb::Line to create
434  * an instance.) The Rhumb object used to create a RhumbLine must stay in
435  * scope as long as the RhumbLine.
436  *
437  * Example of use:
438  * \include example-RhumbLine.cpp
439  **********************************************************************/
440 
442  private:
443  typedef Math::real real;
444  friend class Rhumb;
445  const Rhumb& _rh;
446  bool _exact;
447  real _lat1, _lon1, _azi12, _salp, _calp, _mu1, _psi1, _r1;
448  RhumbLine& operator=(const RhumbLine&); // copy assignment not allowed
449  RhumbLine(const Rhumb& rh, real lat1, real lon1, real azi12,
450  bool exact);
451  public:
452 
453  /**
454  * This is a duplication of Rhumb::mask.
455  **********************************************************************/
456  enum mask {
457  /**
458  * No output.
459  * @hideinitializer
460  **********************************************************************/
461  NONE = Rhumb::NONE,
462  /**
463  * Calculate latitude \e lat2.
464  * @hideinitializer
465  **********************************************************************/
466  LATITUDE = Rhumb::LATITUDE,
467  /**
468  * Calculate longitude \e lon2.
469  * @hideinitializer
470  **********************************************************************/
471  LONGITUDE = Rhumb::LONGITUDE,
472  /**
473  * Calculate azimuth \e azi12.
474  * @hideinitializer
475  **********************************************************************/
476  AZIMUTH = Rhumb::AZIMUTH,
477  /**
478  * Calculate distance \e s12.
479  * @hideinitializer
480  **********************************************************************/
481  DISTANCE = Rhumb::DISTANCE,
482  /**
483  * Calculate area \e S12.
484  * @hideinitializer
485  **********************************************************************/
486  AREA = Rhumb::AREA,
487  /**
488  * Unroll \e lon2 in the direct calculation. (This flag used to be
489  * called LONG_NOWRAP.)
490  * @hideinitializer
491  **********************************************************************/
492  LONG_UNROLL = Rhumb::LONG_UNROLL,
493  /// \cond SKIP
494  LONG_NOWRAP = LONG_UNROLL,
495  /// \endcond
496  /**
497  * Calculate everything. (LONG_UNROLL is not included in this mask.)
498  * @hideinitializer
499  **********************************************************************/
500  ALL = Rhumb::ALL,
501  };
502 
503  /**
504  * Compute the position of point 2 which is a distance \e s12 (meters) from
505  * point 1. The area is also computed.
506  *
507  * @param[in] s12 distance between point 1 and point 2 (meters); it can be
508  * negative.
509  * @param[out] lat2 latitude of point 2 (degrees).
510  * @param[out] lon2 longitude of point 2 (degrees).
511  * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
512  *
513  * The value of \e lon2 returned is in the range [&minus;180&deg;,
514  * 180&deg;).
515  *
516  * If \e s12 is large enough that the rhumb line crosses a pole, the
517  * longitude of point 2 is indeterminate (a NaN is returned for \e lon2 and
518  * \e S12).
519  **********************************************************************/
520  void Position(real s12, real& lat2, real& lon2, real& S12) const {
521  GenPosition(s12, LATITUDE | LONGITUDE | AREA, lat2, lon2, S12);
522  }
523 
524  /**
525  * Compute the position of point 2 which is a distance \e s12 (meters) from
526  * point 1. The area is not computed.
527  **********************************************************************/
528  void Position(real s12, real& lat2, real& lon2) const {
529  real t;
530  GenPosition(s12, LATITUDE | LONGITUDE, lat2, lon2, t);
531  }
532 
533  /**
534  * The general position routine. RhumbLine::Position is defined in term so
535  * this function.
536  *
537  * @param[in] s12 distance between point 1 and point 2 (meters); it can be
538  * negative.
539  * @param[in] outmask a bitor'ed combination of RhumbLine::mask values
540  * specifying which of the following parameters should be set.
541  * @param[out] lat2 latitude of point 2 (degrees).
542  * @param[out] lon2 longitude of point 2 (degrees).
543  * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
544  *
545  * The RhumbLine::mask values possible for \e outmask are
546  * - \e outmask |= RhumbLine::LATITUDE for the latitude \e lat2;
547  * - \e outmask |= RhumbLine::LONGITUDE for the latitude \e lon2;
548  * - \e outmask |= RhumbLine::AREA for the area \e S12;
549  * - \e outmask |= RhumbLine::ALL for all of the above;
550  * - \e outmask |= RhumbLine::LONG_UNROLL to unroll \e lon2 instead of
551  * wrapping it into the range [&minus;180&deg;, 180&deg;).
552  * .
553  * With the RhumbLine::LONG_UNROLL bit set, the quantity \e lon2 &minus; \e
554  * lon1 indicates how many times and in what sense the rhumb line encircles
555  * the ellipsoid.
556  *
557  * If \e s12 is large enough that the rhumb line crosses a pole, the
558  * longitude of point 2 is indeterminate (a NaN is returned for \e lon2 and
559  * \e S12).
560  **********************************************************************/
561  void GenPosition(real s12, unsigned outmask,
562  real& lat2, real& lon2, real& S12) const;
563 
564  /** \name Inspector functions
565  **********************************************************************/
566  ///@{
567 
568  /**
569  * @return \e lat1 the latitude of point 1 (degrees).
570  **********************************************************************/
571  Math::real Latitude() const { return _lat1; }
572 
573  /**
574  * @return \e lon1 the longitude of point 1 (degrees).
575  **********************************************************************/
576  Math::real Longitude() const { return _lon1; }
577 
578  /**
579  * @return \e azi12 the azimuth of the rhumb line (degrees).
580  **********************************************************************/
581  Math::real Azimuth() const { return _azi12; }
582 
583  /**
584  * @return \e a the equatorial radius of the ellipsoid (meters). This is
585  * the value inherited from the Rhumb object used in the constructor.
586  **********************************************************************/
587  Math::real MajorRadius() const { return _rh.MajorRadius(); }
588 
589  /**
590  * @return \e f the flattening of the ellipsoid. This is the value
591  * inherited from the Rhumb object used in the constructor.
592  **********************************************************************/
593  Math::real Flattening() const { return _rh.Flattening(); }
594  };
595 
596 } // namespace GeographicLib
597 
598 #endif // GEOGRAPHICLIB_RHUMB_HPP
void Inverse(real lat1, real lon1, real lat2, real lon2, real &s12, real &azi12, real &S12) const
Definition: Rhumb.hpp:342
#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:528
#define GEOGRAPHICLIB_RHUMBAREA_ORDER
Definition: Rhumb.hpp:21
static T atanh(T x)
Definition: Math.hpp:342
Math::real Latitude() const
Definition: Rhumb.hpp:571
Math::real MajorRadius() const
Definition: Rhumb.hpp:406
static T asinh(T x)
Definition: Math.hpp:325
Math::real EllipsoidArea() const
Definition: Rhumb.hpp:414
static T hypot(T x, T y)
Definition: Math.hpp:257
Math::real MajorRadius() const
Definition: Ellipsoid.hpp:81
Math::real Azimuth() const
Definition: Rhumb.hpp:581
Math::real Longitude() const
Definition: Rhumb.hpp:576
Namespace for GeographicLib.
Definition: Accumulator.cpp:12
#define GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER
Header for GeographicLib::Ellipsoid class.
Math::real Flattening() const
Definition: Rhumb.hpp:412
Properties of an ellipsoid.
Definition: Ellipsoid.hpp:39
Math::real MajorRadius() const
Definition: Rhumb.hpp:587
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:351
Find a sequence of points on a single rhumb line.
Definition: Rhumb.hpp:441
Math::real Flattening() const
Definition: Rhumb.hpp:593
void Direct(real lat1, real lon1, real azi12, real s12, real &lat2, real &lon2) const
Definition: Rhumb.hpp:284
Math::real Flattening() const
Definition: Ellipsoid.hpp:121
void Direct(real lat1, real lon1, real azi12, real s12, real &lat2, real &lon2, real &S12) const
Definition: Rhumb.hpp:275
void Position(real s12, real &lat2, real &lon2, real &S12) const
Definition: Rhumb.hpp:520