GeographicLib  1.42
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  * Do not wrap the \e lon2 in the direct calculation.
222  * @hideinitializer
223  **********************************************************************/
224  LONG_NOWRAP = 1U<<15,
225  /**
226  * Calculate everything. (LONG_NOWRAP 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. If \e f &gt; 1, set
238  * flattening to 1/\e f.
239  * @param[in] exact if true (the default) use an addition theorem for
240  * elliptic integrals to compute divided differences; otherwise use
241  * series expansion (accurate for |<i>f</i>| < 0.01).
242  * @exception GeographicErr if \e a or (1 &minus; \e f) \e a is not
243  * positive.
244  *
245  * See \ref rhumb, for a detailed description of the \e exact parameter.
246  **********************************************************************/
247  Rhumb(real a, real f, bool exact = true);
248 
249  /**
250  * Solve the direct rhumb problem returning also the area.
251  *
252  * @param[in] lat1 latitude of point 1 (degrees).
253  * @param[in] lon1 longitude of point 1 (degrees).
254  * @param[in] azi12 azimuth of the rhumb line (degrees).
255  * @param[in] s12 distance between point 1 and point 2 (meters); it can be
256  * negative.
257  * @param[out] lat2 latitude of point 2 (degrees).
258  * @param[out] lon2 longitude of point 2 (degrees).
259  * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
260  *
261  * \e lat1 should be in the range [&minus;90&deg;, 90&deg;]; \e lon1 and \e
262  * azi12 should be in the range [&minus;540&deg;, 540&deg;). The value of
263  * \e lon2 returned is in the range [&minus;180&deg;, 180&deg;).
264  *
265  * If point 1 is a pole, the cosine of its latitude is taken to be
266  * 1/&epsilon;<sup>2</sup> (where &epsilon; is 2<sup>-52</sup>). This
267  * position, which is extremely close to the actual pole, allows the
268  * calculation to be carried out in finite terms. If \e s12 is large
269  * enough that the rhumb line crosses a pole, the longitude of point 2
270  * is indeterminate (a NaN is returned for \e lon2 and \e S12).
271  **********************************************************************/
272  void Direct(real lat1, real lon1, real azi12, real s12,
273  real& lat2, real& lon2, real& S12) const {
274  GenDirect(lat1, lon1, azi12, s12,
275  LATITUDE | LONGITUDE | AREA, lat2, lon2, S12);
276  }
277 
278  /**
279  * Solve the direct rhumb problem without the area.
280  **********************************************************************/
281  void Direct(real lat1, real lon1, real azi12, real s12,
282  real& lat2, real& lon2) const {
283  real t;
284  GenDirect(lat1, lon1, azi12, s12, LATITUDE | LONGITUDE, lat2, lon2, t);
285  }
286 
287  /**
288  * The general direct rhumb problem. Rhumb::Direct is defined in terms
289  * of this function.
290  *
291  * @param[in] lat1 latitude of point 1 (degrees).
292  * @param[in] lon1 longitude of point 1 (degrees).
293  * @param[in] azi12 azimuth of the rhumb line (degrees).
294  * @param[in] s12 distance between point 1 and point 2 (meters); it can be
295  * negative.
296  * @param[in] outmask a bitor'ed combination of Rhumb::mask values
297  * specifying which of the following parameters should be set.
298  * @param[out] lat2 latitude of point 2 (degrees).
299  * @param[out] lon2 longitude of point 2 (degrees).
300  * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
301  *
302  * The Rhumb::mask values possible for \e outmask are
303  * - \e outmask |= Rhumb::LATITUDE for the latitude \e lat2;
304  * - \e outmask |= Rhumb::LONGITUDE for the latitude \e lon2;
305  * - \e outmask |= Rhumb::AREA for the area \e S12;
306  * - \e outmask |= Rhumb::ALL for all of the above;
307  * - \e outmask |= Rhumb::LONG_NOWRAP stops the returned value of \e
308  * lon2 being wrapped into the range [&minus;180&deg;, 180&deg;).
309  * .
310  * With the LONG_NOWRAP bit set, the quantity \e lon2 &minus; \e lon1
311  * indicates how many times the rhumb line wrapped around the ellipsoid.
312  * Because \e lon2 might be outside the normal allowed range for
313  * longitudes, [&minus;540&deg;, 540&deg;), be sure to normalize it with
314  * Math::AngNormalize2 before using it in other GeographicLib calls.
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;]; \e lon1 and \e lon2 should be in the range [&minus;540&deg;,
334  * 540&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;]; \e lon1 and \e
390  * azi12 should be in the range [&minus;540&deg;, 540&deg;).
391  *
392  * If point 1 is a pole, the cosine of its latitude is taken to be
393  * 1/&epsilon;<sup>2</sup> (where &epsilon; is 2<sup>-52</sup>). This
394  * position, which is extremely close to the actual pole, allows the
395  * calculation to be carried out in finite terms.
396  **********************************************************************/
397  RhumbLine Line(real lat1, real lon1, real azi12) const;
398 
399  /** \name Inspector functions.
400  **********************************************************************/
401  ///@{
402 
403  /**
404  * @return \e a the equatorial radius of the ellipsoid (meters). This is
405  * the value used in the constructor.
406  **********************************************************************/
407  Math::real MajorRadius() const { return _ell.MajorRadius(); }
408 
409  /**
410  * @return \e f the flattening of the ellipsoid. This is the
411  * value used in the constructor.
412  **********************************************************************/
413  Math::real Flattening() const { return _ell.Flattening(); }
414 
415  Math::real EllipsoidArea() const { return _ell.Area(); }
416 
417  /**
418  * A global instantiation of Rhumb with the parameters for the WGS84
419  * ellipsoid.
420  **********************************************************************/
421  static const Rhumb& WGS84();
422  };
423 
424  /**
425  * \brief Find a sequence of points on a single rhumb line.
426  *
427  * RhumbLine facilitates the determination of a series of points on a single
428  * rhumb line. The starting point (\e lat1, \e lon1) and the azimuth \e
429  * azi12 are specified in the call to Rhumb::Line which returns a RhumbLine
430  * object. RhumbLine.Position returns the location of point 2 (and,
431  * optionally, the corresponding area, \e S12) a distance \e s12 along the
432  * rhumb line.
433  *
434  * There is no public constructor for this class. (Use Rhumb::Line to create
435  * an instance.) The Rhumb object used to create a RhumbLine must stay in
436  * scope as long as the RhumbLine.
437  *
438  * Example of use:
439  * \include example-RhumbLine.cpp
440  **********************************************************************/
441 
443  private:
444  typedef Math::real real;
445  friend class Rhumb;
446  const Rhumb& _rh;
447  bool _exact;
448  real _lat1, _lon1, _azi12, _salp, _calp, _mu1, _psi1, _r1;
449  RhumbLine& operator=(const RhumbLine&); // copy assignment not allowed
450  RhumbLine(const Rhumb& rh, real lat1, real lon1, real azi12,
451  bool exact);
452  public:
453 
454  enum mask {
455  /**
456  * No output.
457  * @hideinitializer
458  **********************************************************************/
459  NONE = Rhumb::NONE,
460  /**
461  * Calculate latitude \e lat2.
462  * @hideinitializer
463  **********************************************************************/
464  LATITUDE = Rhumb::LATITUDE,
465  /**
466  * Calculate longitude \e lon2.
467  * @hideinitializer
468  **********************************************************************/
469  LONGITUDE = Rhumb::LONGITUDE,
470  /**
471  * Calculate azimuth \e azi12.
472  * @hideinitializer
473  **********************************************************************/
474  AZIMUTH = Rhumb::AZIMUTH,
475  /**
476  * Calculate distance \e s12.
477  * @hideinitializer
478  **********************************************************************/
479  DISTANCE = Rhumb::DISTANCE,
480  /**
481  * Calculate area \e S12.
482  * @hideinitializer
483  **********************************************************************/
484  AREA = Rhumb::AREA,
485  /**
486  * Do wrap the \e lon2 in the direct calculation.
487  * @hideinitializer
488  **********************************************************************/
489  LONG_NOWRAP = Rhumb::LONG_NOWRAP,
490  /**
491  * Calculate everything. (LONG_NOWRAP is not included in this mask.)
492  * @hideinitializer
493  **********************************************************************/
494  ALL = Rhumb::ALL,
495  };
496 
497  /**
498  * Compute the position of point 2 which is a distance \e s12 (meters) from
499  * point 1. The area is also computed.
500  *
501  * @param[in] s12 distance between point 1 and point 2 (meters); it can be
502  * negative.
503  * @param[out] lat2 latitude of point 2 (degrees).
504  * @param[out] lon2 longitude of point 2 (degrees).
505  * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
506  *
507  * The value of \e lon2 returned is in the range [&minus;180&deg;,
508  * 180&deg;).
509  *
510  * If \e s12 is large enough that the rhumb line crosses a pole, the
511  * longitude of point 2 is indeterminate (a NaN is returned for \e lon2 and
512  * \e S12).
513  **********************************************************************/
514  void Position(real s12, real& lat2, real& lon2, real& S12) const {
515  GenPosition(s12, LATITUDE | LONGITUDE | AREA, lat2, lon2, S12);
516  }
517 
518  /**
519  * Compute the position of point 2 which is a distance \e s12 (meters) from
520  * point 1. The area is not computed.
521  **********************************************************************/
522  void Position(real s12, real& lat2, real& lon2) const {
523  real t;
524  GenPosition(s12, LATITUDE | LONGITUDE, lat2, lon2, t);
525  }
526 
527  /**
528  * The general position routine. RhumbLine::Position is defined in term so
529  * this function.
530  *
531  * @param[in] s12 distance between point 1 and point 2 (meters); it can be
532  * negative.
533  * @param[in] outmask a bitor'ed combination of Rhumb::mask values
534  * specifying which of the following parameters should be set.
535  * @param[out] lat2 latitude of point 2 (degrees).
536  * @param[out] lon2 longitude of point 2 (degrees).
537  * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
538  *
539  * The Rhumb::mask values possible for \e outmask are
540  * - \e outmask |= Rhumb::LATITUDE for the latitude \e lat2;
541  * - \e outmask |= Rhumb::LONGITUDE for the latitude \e lon2;
542  * - \e outmask |= Rhumb::AREA for the area \e S12;
543  * - \e outmask |= Rhumb::ALL for all of the above;
544  * - \e outmask |= Rhumb::LONG_NOWRAP stops the returned value of \e
545  * lon2 being wrapped into the range [&minus;180&deg;, 180&deg;).
546  * .
547  * With the LONG_NOWRAP bit set, the quantity \e lon2 &minus; \e lon1
548  * indicates how many times the rhumb line wrapped around the ellipsoid.
549  * Because \e lon2 might be outside the normal allowed range for
550  * longitudes, [&minus;540&deg;, 540&deg;), be sure to normalize it with
551  * Math::AngNormalize2 before using it in other GeographicLib calls.
552  *
553  * If \e s12 is large enough that the rhumb line crosses a pole, the
554  * longitude of point 2 is indeterminate (a NaN is returned for \e lon2 and
555  * \e S12).
556  **********************************************************************/
557  void GenPosition(real s12, unsigned outmask,
558  real& lat2, real& lon2, real& S12) const;
559 
560  /** \name Inspector functions
561  **********************************************************************/
562  ///@{
563 
564  /**
565  * @return \e lat1 the latitude of point 1 (degrees).
566  **********************************************************************/
567  Math::real Latitude() const { return _lat1; }
568 
569  /**
570  * @return \e lon1 the longitude of point 1 (degrees).
571  **********************************************************************/
572  Math::real Longitude() const { return _lon1; }
573 
574  /**
575  * @return \e azi12 the azimuth of the rhumb line (degrees).
576  **********************************************************************/
577  Math::real Azimuth() const { return _azi12; }
578 
579  /**
580  * @return \e a the equatorial radius of the ellipsoid (meters). This is
581  * the value inherited from the Rhumb object used in the constructor.
582  **********************************************************************/
583  Math::real MajorRadius() const { return _rh.MajorRadius(); }
584 
585  /**
586  * @return \e f the flattening of the ellipsoid. This is the value
587  * inherited from the Rhumb object used in the constructor.
588  **********************************************************************/
589  Math::real Flattening() const { return _rh.Flattening(); }
590  };
591 
592 } // namespace GeographicLib
593 
594 #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:522
#define GEOGRAPHICLIB_RHUMBAREA_ORDER
Definition: Rhumb.hpp:21
static T atanh(T x)
Definition: Math.hpp:340
Math::real Latitude() const
Definition: Rhumb.hpp:567
Math::real MajorRadius() const
Definition: Rhumb.hpp:407
static T asinh(T x)
Definition: Math.hpp:323
Math::real EllipsoidArea() const
Definition: Rhumb.hpp:415
static T hypot(T x, T y)
Definition: Math.hpp:255
Math::real MajorRadius() const
Definition: Ellipsoid.hpp:81
Math::real Azimuth() const
Definition: Rhumb.hpp:577
Math::real Longitude() const
Definition: Rhumb.hpp:572
Namespace for GeographicLib.
Definition: Accumulator.cpp:12
#define GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER
Header for GeographicLib::Ellipsoid class.
Math::real Flattening() const
Definition: Rhumb.hpp:413
Properties of an ellipsoid.
Definition: Ellipsoid.hpp:39
Math::real MajorRadius() const
Definition: Rhumb.hpp:583
static T tand(T x)
Definition: Math.hpp:500
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:442
Math::real Flattening() const
Definition: Rhumb.hpp:589
void Direct(real lat1, real lon1, real azi12, real s12, real &lat2, real &lon2) const
Definition: Rhumb.hpp:281
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:272
void Position(real s12, real &lat2, real &lon2, real &S12) const
Definition: Rhumb.hpp:514