GeographicLib  1.43
Ellipsoid.hpp
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1 /**
2  * \file Ellipsoid.hpp
3  * \brief Header for GeographicLib::Ellipsoid class
4  *
5  * Copyright (c) Charles Karney (2012-2015) <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_ELLIPSOID_HPP)
11 #define GEOGRAPHICLIB_ELLIPSOID_HPP 1
12 
17 
18 namespace GeographicLib {
19 
20  /**
21  * \brief Properties of an ellipsoid
22  *
23  * This class returns various properties of the ellipsoid and converts
24  * between various types of latitudes. The latitude conversions are also
25  * possible using the various projections supported by %GeographicLib; but
26  * Ellipsoid provides more direct access (sometimes using private functions
27  * of the projection classes). Ellipsoid::RectifyingLatitude,
28  * Ellipsoid::InverseRectifyingLatitude, and Ellipsoid::MeridianDistance
29  * provide functionality which can be provided by the Geodesic class.
30  * However Geodesic uses a series approximation (valid for abs \e f < 1/150),
31  * whereas Ellipsoid computes these quantities using EllipticFunction which
32  * provides accurate results even when \e f is large. Use of this class
33  * should be limited to &minus;3 < \e f < 3/4 (i.e., 1/4 < b/a < 4).
34  *
35  * Example of use:
36  * \include example-Ellipsoid.cpp
37  **********************************************************************/
38 
40  private:
41  typedef Math::real real;
42  static const int numit_ = 10;
43  real stol_;
44  real _a, _f, _f1, _f12, _e2, _es, _e12, _n, _b;
46  EllipticFunction _ell;
47  AlbersEqualArea _au;
48 
49  // These are the alpha and beta coefficients in the Krueger series from
50  // TransverseMercator. Thy are used by RhumbSolve to compute
51  // (psi2-psi1)/(mu2-mu1).
52  const Math::real* ConformalToRectifyingCoeffs() const { return _tm._alp; }
53  const Math::real* RectifyingToConformalCoeffs() const { return _tm._bet; }
54  friend class Rhumb; friend class RhumbLine;
55  public:
56  /** \name Constructor
57  **********************************************************************/
58  ///@{
59 
60  /**
61  * Constructor for a ellipsoid with
62  *
63  * @param[in] a equatorial radius (meters).
64  * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
65  * Negative \e f gives a prolate ellipsoid. If \e f &gt; 1, set
66  * flattening to 1/\e f.
67  * @exception GeographicErr if \e a or (1 &minus; \e f) \e a is not
68  * positive.
69  **********************************************************************/
70  Ellipsoid(real a, real f);
71  ///@}
72 
73  /** \name %Ellipsoid dimensions.
74  **********************************************************************/
75  ///@{
76 
77  /**
78  * @return \e a the equatorial radius of the ellipsoid (meters). This is
79  * the value used in the constructor.
80  **********************************************************************/
81  Math::real MajorRadius() const { return _a; }
82 
83  /**
84  * @return \e b the polar semi-axis (meters).
85  **********************************************************************/
86  Math::real MinorRadius() const { return _b; }
87 
88  /**
89  * @return \e L the distance between the equator and a pole along a
90  * meridian (meters). For a sphere \e L = (&pi;/2) \e a. The radius
91  * of a sphere with the same meridian length is \e L / (&pi;/2).
92  **********************************************************************/
93  Math::real QuarterMeridian() const;
94 
95  /**
96  * @return \e A the total area of the ellipsoid (meters<sup>2</sup>). For
97  * a sphere \e A = 4&pi; <i>a</i><sup>2</sup>. The radius of a sphere
98  * with the same area is sqrt(\e A / (4&pi;)).
99  **********************************************************************/
100  Math::real Area() const;
101 
102  /**
103  * @return \e V the total volume of the ellipsoid (meters<sup>3</sup>).
104  * For a sphere \e V = (4&pi; / 3) <i>a</i><sup>3</sup>. The radius of
105  * a sphere with the same volume is cbrt(\e V / (4&pi;/3)).
106  **********************************************************************/
108  { return (4 * Math::pi()) * Math::sq(_a) * _b / 3; }
109  ///@}
110 
111  /** \name %Ellipsoid shape
112  **********************************************************************/
113  ///@{
114 
115  /**
116  * @return \e f = (\e a &minus; \e b) / \e a, the flattening of the
117  * ellipsoid. This is the value used in the constructor. This is zero,
118  * positive, or negative for a sphere, oblate ellipsoid, or prolate
119  * ellipsoid.
120  **********************************************************************/
121  Math::real Flattening() const { return _f; }
122 
123  /**
124  * @return \e f ' = (\e a &minus; \e b) / \e b, the second flattening of
125  * the ellipsoid. This is zero, positive, or negative for a sphere,
126  * oblate ellipsoid, or prolate ellipsoid.
127  **********************************************************************/
128  Math::real SecondFlattening() const { return _f / (1 - _f); }
129 
130  /**
131  * @return \e n = (\e a &minus; \e b) / (\e a + \e b), the third flattening
132  * of the ellipsoid. This is zero, positive, or negative for a sphere,
133  * oblate ellipsoid, or prolate ellipsoid.
134  **********************************************************************/
135  Math::real ThirdFlattening() const { return _n; }
136 
137  /**
138  * @return <i>e</i><sup>2</sup> = (<i>a</i><sup>2</sup> &minus;
139  * <i>b</i><sup>2</sup>) / <i>a</i><sup>2</sup>, the eccentricity squared
140  * of the ellipsoid. This is zero, positive, or negative for a sphere,
141  * oblate ellipsoid, or prolate ellipsoid.
142  **********************************************************************/
143  Math::real EccentricitySq() const { return _e2; }
144 
145  /**
146  * @return <i>e'</i> <sup>2</sup> = (<i>a</i><sup>2</sup> &minus;
147  * <i>b</i><sup>2</sup>) / <i>b</i><sup>2</sup>, the second eccentricity
148  * squared of the ellipsoid. This is zero, positive, or negative for a
149  * sphere, oblate ellipsoid, or prolate ellipsoid.
150  **********************************************************************/
151  Math::real SecondEccentricitySq() const { return _e12; }
152 
153  /**
154  * @return <i>e''</i> <sup>2</sup> = (<i>a</i><sup>2</sup> &minus;
155  * <i>b</i><sup>2</sup>) / (<i>a</i><sup>2</sup> + <i>b</i><sup>2</sup>),
156  * the third eccentricity squared of the ellipsoid. This is zero,
157  * positive, or negative for a sphere, oblate ellipsoid, or prolate
158  * ellipsoid.
159  **********************************************************************/
160  Math::real ThirdEccentricitySq() const { return _e2 / (2 - _e2); }
161  ///@}
162 
163  /** \name Latitude conversion.
164  **********************************************************************/
165  ///@{
166 
167  /**
168  * @param[in] phi the geographic latitude (degrees).
169  * @return &beta; the parametric latitude (degrees).
170  *
171  * The geographic latitude, &phi;, is the angle beween the equatorial
172  * plane and a vector normal to the surface of the ellipsoid.
173  *
174  * The parametric latitude (also called the reduced latitude), &beta;,
175  * allows the cartesian coordinated of a meridian to be expressed
176  * conveniently in parametric form as
177  * - \e R = \e a cos &beta;
178  * - \e Z = \e b sin &beta;
179  * .
180  * where \e a and \e b are the equatorial radius and the polar semi-axis.
181  * For a sphere &beta; = &phi;.
182  *
183  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the
184  * result is undefined if this condition does not hold. The returned value
185  * &beta; lies in [&minus;90&deg;, 90&deg;].
186  **********************************************************************/
187  Math::real ParametricLatitude(real phi) const;
188 
189  /**
190  * @param[in] beta the parametric latitude (degrees).
191  * @return &phi; the geographic latitude (degrees).
192  *
193  * &beta; must lie in the range [&minus;90&deg;, 90&deg;]; the
194  * result is undefined if this condition does not hold. The returned value
195  * &phi; lies in [&minus;90&deg;, 90&deg;].
196  **********************************************************************/
197  Math::real InverseParametricLatitude(real beta) const;
198 
199  /**
200  * @param[in] phi the geographic latitude (degrees).
201  * @return &theta; the geocentric latitude (degrees).
202  *
203  * The geocentric latitude, &theta;, is the angle beween the equatorial
204  * plane and a line between the center of the ellipsoid and a point on the
205  * ellipsoid. For a sphere &theta; = &phi;.
206  *
207  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the
208  * result is undefined if this condition does not hold. The returned value
209  * &theta; lies in [&minus;90&deg;, 90&deg;].
210  **********************************************************************/
211  Math::real GeocentricLatitude(real phi) const;
212 
213  /**
214  * @param[in] theta the geocentric latitude (degrees).
215  * @return &phi; the geographic latitude (degrees).
216  *
217  * &theta; must lie in the range [&minus;90&deg;, 90&deg;]; the
218  * result is undefined if this condition does not hold. The returned value
219  * &phi; lies in [&minus;90&deg;, 90&deg;].
220  **********************************************************************/
221  Math::real InverseGeocentricLatitude(real theta) const;
222 
223  /**
224  * @param[in] phi the geographic latitude (degrees).
225  * @return &mu; the rectifying latitude (degrees).
226  *
227  * The rectifying latitude, &mu;, has the property that the distance along
228  * a meridian of the ellipsoid between two points with rectifying latitudes
229  * &mu;<sub>1</sub> and &mu;<sub>2</sub> is equal to
230  * (&mu;<sub>2</sub> - &mu;<sub>1</sub>) \e L / 90&deg;,
231  * where \e L = QuarterMeridian(). For a sphere &mu; = &phi;.
232  *
233  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the
234  * result is undefined if this condition does not hold. The returned value
235  * &mu; lies in [&minus;90&deg;, 90&deg;].
236  **********************************************************************/
237  Math::real RectifyingLatitude(real phi) const;
238 
239  /**
240  * @param[in] mu the rectifying latitude (degrees).
241  * @return &phi; the geographic latitude (degrees).
242  *
243  * &mu; must lie in the range [&minus;90&deg;, 90&deg;]; the
244  * result is undefined if this condition does not hold. The returned value
245  * &phi; lies in [&minus;90&deg;, 90&deg;].
246  **********************************************************************/
247  Math::real InverseRectifyingLatitude(real mu) const;
248 
249  /**
250  * @param[in] phi the geographic latitude (degrees).
251  * @return &xi; the authalic latitude (degrees).
252  *
253  * The authalic latitude, &xi;, has the property that the area of the
254  * ellipsoid between two circles with authalic latitudes
255  * &xi;<sub>1</sub> and &xi;<sub>2</sub> is equal to (sin
256  * &xi;<sub>2</sub> - sin &xi;<sub>1</sub>) \e A / 2, where \e A
257  * = Area(). For a sphere &xi; = &phi;.
258  *
259  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the
260  * result is undefined if this condition does not hold. The returned value
261  * &xi; lies in [&minus;90&deg;, 90&deg;].
262  **********************************************************************/
263  Math::real AuthalicLatitude(real phi) const;
264 
265  /**
266  * @param[in] xi the authalic latitude (degrees).
267  * @return &phi; the geographic latitude (degrees).
268  *
269  * &xi; must lie in the range [&minus;90&deg;, 90&deg;]; the
270  * result is undefined if this condition does not hold. The returned value
271  * &phi; lies in [&minus;90&deg;, 90&deg;].
272  **********************************************************************/
273  Math::real InverseAuthalicLatitude(real xi) const;
274 
275  /**
276  * @param[in] phi the geographic latitude (degrees).
277  * @return &chi; the conformal latitude (degrees).
278  *
279  * The conformal latitude, &chi;, gives the mapping of the ellipsoid to a
280  * sphere which which is conformal (angles are preserved) and in which the
281  * equator of the ellipsoid maps to the equator of the sphere. For a
282  * sphere &chi; = &phi;.
283  *
284  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the
285  * result is undefined if this condition does not hold. The returned value
286  * &chi; lies in [&minus;90&deg;, 90&deg;].
287  **********************************************************************/
288  Math::real ConformalLatitude(real phi) const;
289 
290  /**
291  * @param[in] chi the conformal latitude (degrees).
292  * @return &phi; the geographic latitude (degrees).
293  *
294  * &chi; must lie in the range [&minus;90&deg;, 90&deg;]; the
295  * result is undefined if this condition does not hold. The returned value
296  * &phi; lies in [&minus;90&deg;, 90&deg;].
297  **********************************************************************/
298  Math::real InverseConformalLatitude(real chi) const;
299 
300  /**
301  * @param[in] phi the geographic latitude (degrees).
302  * @return &psi; the isometric latitude (degrees).
303  *
304  * The isometric latitude gives the mapping of the ellipsoid to a plane
305  * which which is conformal (angles are preserved) and in which the equator
306  * of the ellipsoid maps to a straight line of constant scale; this mapping
307  * defines the Mercator projection. For a sphere &psi; =
308  * sinh<sup>&minus;1</sup> tan &phi;.
309  *
310  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the result is
311  * undefined if this condition does not hold. The value returned for &phi;
312  * = &plusmn;90&deg; is some (positive or negative) large but finite value,
313  * such that InverseIsometricLatitude returns the original value of &phi;.
314  **********************************************************************/
315  Math::real IsometricLatitude(real phi) const;
316 
317  /**
318  * @param[in] psi the isometric latitude (degrees).
319  * @return &phi; the geographic latitude (degrees).
320  *
321  * The returned value &phi; lies in [&minus;90&deg;, 90&deg;]. For a
322  * sphere &phi; = tan<sup>&minus;1</sup> sinh &psi;.
323  **********************************************************************/
324  Math::real InverseIsometricLatitude(real psi) const;
325  ///@}
326 
327  /** \name Other quantities.
328  **********************************************************************/
329  ///@{
330 
331  /**
332  * @param[in] phi the geographic latitude (degrees).
333  * @return \e R = \e a cos &beta; the radius of a circle of latitude
334  * &phi; (meters). \e R (&pi;/180&deg;) gives meters per degree
335  * longitude measured along a circle of latitude.
336  *
337  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the
338  * result is undefined if this condition does not hold.
339  **********************************************************************/
340  Math::real CircleRadius(real phi) const;
341 
342  /**
343  * @param[in] phi the geographic latitude (degrees).
344  * @return \e Z = \e b sin &beta; the distance of a circle of latitude
345  * &phi; from the equator measured parallel to the ellipsoid axis
346  * (meters).
347  *
348  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the
349  * result is undefined if this condition does not hold.
350  **********************************************************************/
351  Math::real CircleHeight(real phi) const;
352 
353  /**
354  * @param[in] phi the geographic latitude (degrees).
355  * @return \e s the distance along a meridian
356  * between the equator and a point of latitude &phi; (meters). \e s is
357  * given by \e s = &mu; \e L / 90&deg;, where \e L =
358  * QuarterMeridian()).
359  *
360  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the
361  * result is undefined if this condition does not hold.
362  **********************************************************************/
363  Math::real MeridianDistance(real phi) const;
364 
365  /**
366  * @param[in] phi the geographic latitude (degrees).
367  * @return &rho; the meridional radius of curvature of the ellipsoid at
368  * latitude &phi; (meters); this is the curvature of the meridian. \e
369  * rho is given by &rho; = (180&deg;/&pi;) d\e s / d&phi;,
370  * where \e s = MeridianDistance(); thus &rho; (&pi;/180&deg;)
371  * gives meters per degree latitude measured along a meridian.
372  *
373  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the
374  * result is undefined if this condition does not hold.
375  **********************************************************************/
376  Math::real MeridionalCurvatureRadius(real phi) const;
377 
378  /**
379  * @param[in] phi the geographic latitude (degrees).
380  * @return &nu; the transverse radius of curvature of the ellipsoid at
381  * latitude &phi; (meters); this is the curvature of a curve on the
382  * ellipsoid which also lies in a plane perpendicular to the ellipsoid
383  * and to the meridian. &nu; is related to \e R = CircleRadius() by \e
384  * R = &nu; cos &phi;.
385  *
386  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the
387  * result is undefined if this condition does not hold.
388  **********************************************************************/
389  Math::real TransverseCurvatureRadius(real phi) const;
390 
391  /**
392  * @param[in] phi the geographic latitude (degrees).
393  * @param[in] azi the angle between the meridian and the normal section
394  * (degrees).
395  * @return the radius of curvature of the ellipsoid in the normal
396  * section at latitude &phi; inclined at an angle \e azi to the
397  * meridian (meters).
398  *
399  * &phi; must lie in the range [&minus;90&deg;, 90&deg;] and \e
400  * azi must lie in the range [&minus;540&deg;, 540&deg;); the
401  * result is undefined if either of conditions does not hold.
402  **********************************************************************/
403  Math::real NormalCurvatureRadius(real phi, real azi) const;
404  ///@}
405 
406  /** \name Eccentricity conversions.
407  **********************************************************************/
408  ///@{
409 
410  /**
411  * @param[in] fp = \e f ' = (\e a &minus; \e b) / \e b, the second
412  * flattening.
413  * @return \e f = (\e a &minus; \e b) / \e a, the flattening.
414  *
415  * \e f ' should lie in (&minus;1, &infin;).
416  * The returned value \e f lies in (&minus;&infin;, 1).
417  **********************************************************************/
419  { return fp / (1 + fp); }
420 
421  /**
422  * @param[in] f = (\e a &minus; \e b) / \e a, the flattening.
423  * @return \e f ' = (\e a &minus; \e b) / \e b, the second flattening.
424  *
425  * \e f should lie in (&minus;&infin;, 1).
426  * The returned value \e f ' lies in (&minus;1, &infin;).
427  **********************************************************************/
429  { return f / (1 - f); }
430 
431  /**
432  * @param[in] n = (\e a &minus; \e b) / (\e a + \e b), the third
433  * flattening.
434  * @return \e f = (\e a &minus; \e b) / \e a, the flattening.
435  *
436  * \e n should lie in (&minus;1, 1).
437  * The returned value \e f lies in (&minus;&infin;, 1).
438  **********************************************************************/
440  { return 2 * n / (1 + n); }
441 
442  /**
443  * @param[in] f = (\e a &minus; \e b) / \e a, the flattening.
444  * @return \e n = (\e a &minus; \e b) / (\e a + \e b), the third
445  * flattening.
446  *
447  * \e f should lie in (&minus;&infin;, 1).
448  * The returned value \e n lies in (&minus;1, 1).
449  **********************************************************************/
451  { return f / (2 - f); }
452 
453  /**
454  * @param[in] e2 = <i>e</i><sup>2</sup> = (<i>a</i><sup>2</sup> &minus;
455  * <i>b</i><sup>2</sup>) / <i>a</i><sup>2</sup>, the eccentricity
456  * squared.
457  * @return \e f = (\e a &minus; \e b) / \e a, the flattening.
458  *
459  * <i>e</i><sup>2</sup> should lie in (&minus;&infin;, 1).
460  * The returned value \e f lies in (&minus;&infin;, 1).
461  **********************************************************************/
463  { using std::sqrt; return e2 / (sqrt(1 - e2) + 1); }
464 
465  /**
466  * @param[in] f = (\e a &minus; \e b) / \e a, the flattening.
467  * @return <i>e</i><sup>2</sup> = (<i>a</i><sup>2</sup> &minus;
468  * <i>b</i><sup>2</sup>) / <i>a</i><sup>2</sup>, the eccentricity
469  * squared.
470  *
471  * \e f should lie in (&minus;&infin;, 1).
472  * The returned value <i>e</i><sup>2</sup> lies in (&minus;&infin;, 1).
473  **********************************************************************/
475  { return f * (2 - f); }
476 
477  /**
478  * @param[in] ep2 = <i>e'</i> <sup>2</sup> = (<i>a</i><sup>2</sup> &minus;
479  * <i>b</i><sup>2</sup>) / <i>b</i><sup>2</sup>, the second eccentricity
480  * squared.
481  * @return \e f = (\e a &minus; \e b) / \e a, the flattening.
482  *
483  * <i>e'</i> <sup>2</sup> should lie in (&minus;1, &infin;).
484  * The returned value \e f lies in (&minus;&infin;, 1).
485  **********************************************************************/
487  { using std::sqrt; return ep2 / (sqrt(1 + ep2) + 1 + ep2); }
488 
489  /**
490  * @param[in] f = (\e a &minus; \e b) / \e a, the flattening.
491  * @return <i>e'</i> <sup>2</sup> = (<i>a</i><sup>2</sup> &minus;
492  * <i>b</i><sup>2</sup>) / <i>b</i><sup>2</sup>, the second eccentricity
493  * squared.
494  *
495  * \e f should lie in (&minus;&infin;, 1).
496  * The returned value <i>e'</i> <sup>2</sup> lies in (&minus;1, &infin;).
497  **********************************************************************/
499  { return f * (2 - f) / Math::sq(1 - f); }
500 
501  /**
502  * @param[in] epp2 = <i>e''</i> <sup>2</sup> = (<i>a</i><sup>2</sup>
503  * &minus; <i>b</i><sup>2</sup>) / (<i>a</i><sup>2</sup> +
504  * <i>b</i><sup>2</sup>), the third eccentricity squared.
505  * @return \e f = (\e a &minus; \e b) / \e a, the flattening.
506  *
507  * <i>e''</i> <sup>2</sup> should lie in (&minus;1, 1).
508  * The returned value \e f lies in (&minus;&infin;, 1).
509  **********************************************************************/
511  { return 2 * epp2 / (sqrt((1 - epp2) * (1 + epp2)) + 1 + epp2); }
512 
513  /**
514  * @param[in] f = (\e a &minus; \e b) / \e a, the flattening.
515  * @return <i>e''</i> <sup>2</sup> = (<i>a</i><sup>2</sup> &minus;
516  * <i>b</i><sup>2</sup>) / (<i>a</i><sup>2</sup> + <i>b</i><sup>2</sup>),
517  * the third eccentricity squared.
518  *
519  * \e f should lie in (&minus;&infin;, 1).
520  * The returned value <i>e''</i> <sup>2</sup> lies in (&minus;1, 1).
521  **********************************************************************/
523  { return f * (2 - f) / (1 + Math::sq(1 - f)); }
524 
525  ///@}
526 
527  /**
528  * A global instantiation of Ellipsoid with the parameters for the WGS84
529  * ellipsoid.
530  **********************************************************************/
531  static const Ellipsoid& WGS84();
532  };
533 
534 } // namespace GeographicLib
535 
536 #endif // GEOGRAPHICLIB_ELLIPSOID_HPP
Math::real SecondFlattening() const
Definition: Ellipsoid.hpp:128
static T pi()
Definition: Math.hpp:214
#define GEOGRAPHICLIB_EXPORT
Definition: Constants.hpp:90
Math::real SecondEccentricitySq() const
Definition: Ellipsoid.hpp:151
GeographicLib::Math::real real
Definition: GeodSolve.cpp:32
static Math::real SecondFlatteningToFlattening(real fp)
Definition: Ellipsoid.hpp:418
Transverse Mercator projection.
Elliptic integrals and functions.
Header for GeographicLib::TransverseMercator class.
static Math::real FlatteningToEccentricitySq(real f)
Definition: Ellipsoid.hpp:474
static Math::real ThirdEccentricitySqToFlattening(real epp2)
Definition: Ellipsoid.hpp:510
Header for GeographicLib::AlbersEqualArea class.
Albers equal area conic projection.
Math::real EccentricitySq() const
Definition: Ellipsoid.hpp:143
static Math::real ThirdFlatteningToFlattening(real n)
Definition: Ellipsoid.hpp:439
static T sq(T x)
Definition: Math.hpp:244
Math::real MajorRadius() const
Definition: Ellipsoid.hpp:81
Math::real MinorRadius() const
Definition: Ellipsoid.hpp:86
Namespace for GeographicLib.
Definition: Accumulator.cpp:12
static Math::real FlatteningToThirdEccentricitySq(real f)
Definition: Ellipsoid.hpp:522
Header for GeographicLib::EllipticFunction class.
static Math::real SecondEccentricitySqToFlattening(real ep2)
Definition: Ellipsoid.hpp:486
Properties of an ellipsoid.
Definition: Ellipsoid.hpp:39
static Math::real FlatteningToThirdFlattening(real f)
Definition: Ellipsoid.hpp:450
Math::real Volume() const
Definition: Ellipsoid.hpp:107
Header for GeographicLib::Constants class.
Solve of the direct and inverse rhumb problems.
Definition: Rhumb.hpp:66
static Math::real EccentricitySqToFlattening(real e2)
Definition: Ellipsoid.hpp:462
static Math::real FlatteningToSecondFlattening(real f)
Definition: Ellipsoid.hpp:428
Find a sequence of points on a single rhumb line.
Definition: Rhumb.hpp:447
Math::real ThirdFlattening() const
Definition: Ellipsoid.hpp:135
static Math::real FlatteningToSecondEccentricitySq(real f)
Definition: Ellipsoid.hpp:498
Math::real ThirdEccentricitySq() const
Definition: Ellipsoid.hpp:160
Math::real Flattening() const
Definition: Ellipsoid.hpp:121