GeographicLib  1.47
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.
66  * @exception GeographicErr if \e a or (1 &minus; \e f) \e a is not
67  * positive.
68  **********************************************************************/
69  Ellipsoid(real a, real f);
70  ///@}
71 
72  /** \name %Ellipsoid dimensions.
73  **********************************************************************/
74  ///@{
75 
76  /**
77  * @return \e a the equatorial radius of the ellipsoid (meters). This is
78  * the value used in the constructor.
79  **********************************************************************/
80  Math::real MajorRadius() const { return _a; }
81 
82  /**
83  * @return \e b the polar semi-axis (meters).
84  **********************************************************************/
85  Math::real MinorRadius() const { return _b; }
86 
87  /**
88  * @return \e L the distance between the equator and a pole along a
89  * meridian (meters). For a sphere \e L = (&pi;/2) \e a. The radius
90  * of a sphere with the same meridian length is \e L / (&pi;/2).
91  **********************************************************************/
92  Math::real QuarterMeridian() const;
93 
94  /**
95  * @return \e A the total area of the ellipsoid (meters<sup>2</sup>). For
96  * a sphere \e A = 4&pi; <i>a</i><sup>2</sup>. The radius of a sphere
97  * with the same area is sqrt(\e A / (4&pi;)).
98  **********************************************************************/
99  Math::real Area() const;
100 
101  /**
102  * @return \e V the total volume of the ellipsoid (meters<sup>3</sup>).
103  * For a sphere \e V = (4&pi; / 3) <i>a</i><sup>3</sup>. The radius of
104  * a sphere with the same volume is cbrt(\e V / (4&pi;/3)).
105  **********************************************************************/
107  { return (4 * Math::pi()) * Math::sq(_a) * _b / 3; }
108  ///@}
109 
110  /** \name %Ellipsoid shape
111  **********************************************************************/
112  ///@{
113 
114  /**
115  * @return \e f = (\e a &minus; \e b) / \e a, the flattening of the
116  * ellipsoid. This is the value used in the constructor. This is zero,
117  * positive, or negative for a sphere, oblate ellipsoid, or prolate
118  * ellipsoid.
119  **********************************************************************/
120  Math::real Flattening() const { return _f; }
121 
122  /**
123  * @return \e f ' = (\e a &minus; \e b) / \e b, the second flattening of
124  * the ellipsoid. This is zero, positive, or negative for a sphere,
125  * oblate ellipsoid, or prolate ellipsoid.
126  **********************************************************************/
127  Math::real SecondFlattening() const { return _f / (1 - _f); }
128 
129  /**
130  * @return \e n = (\e a &minus; \e b) / (\e a + \e b), the third flattening
131  * of the ellipsoid. This is zero, positive, or negative for a sphere,
132  * oblate ellipsoid, or prolate ellipsoid.
133  **********************************************************************/
134  Math::real ThirdFlattening() const { return _n; }
135 
136  /**
137  * @return <i>e</i><sup>2</sup> = (<i>a</i><sup>2</sup> &minus;
138  * <i>b</i><sup>2</sup>) / <i>a</i><sup>2</sup>, the eccentricity squared
139  * of the ellipsoid. This is zero, positive, or negative for a sphere,
140  * oblate ellipsoid, or prolate ellipsoid.
141  **********************************************************************/
142  Math::real EccentricitySq() const { return _e2; }
143 
144  /**
145  * @return <i>e'</i> <sup>2</sup> = (<i>a</i><sup>2</sup> &minus;
146  * <i>b</i><sup>2</sup>) / <i>b</i><sup>2</sup>, the second eccentricity
147  * squared of the ellipsoid. This is zero, positive, or negative for a
148  * sphere, oblate ellipsoid, or prolate ellipsoid.
149  **********************************************************************/
150  Math::real SecondEccentricitySq() const { return _e12; }
151 
152  /**
153  * @return <i>e''</i> <sup>2</sup> = (<i>a</i><sup>2</sup> &minus;
154  * <i>b</i><sup>2</sup>) / (<i>a</i><sup>2</sup> + <i>b</i><sup>2</sup>),
155  * the third eccentricity squared of the ellipsoid. This is zero,
156  * positive, or negative for a sphere, oblate ellipsoid, or prolate
157  * ellipsoid.
158  **********************************************************************/
159  Math::real ThirdEccentricitySq() const { return _e2 / (2 - _e2); }
160  ///@}
161 
162  /** \name Latitude conversion.
163  **********************************************************************/
164  ///@{
165 
166  /**
167  * @param[in] phi the geographic latitude (degrees).
168  * @return &beta; the parametric latitude (degrees).
169  *
170  * The geographic latitude, &phi;, is the angle beween the equatorial
171  * plane and a vector normal to the surface of the ellipsoid.
172  *
173  * The parametric latitude (also called the reduced latitude), &beta;,
174  * allows the cartesian coordinated of a meridian to be expressed
175  * conveniently in parametric form as
176  * - \e R = \e a cos &beta;
177  * - \e Z = \e b sin &beta;
178  * .
179  * where \e a and \e b are the equatorial radius and the polar semi-axis.
180  * For a sphere &beta; = &phi;.
181  *
182  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the
183  * result is undefined if this condition does not hold. The returned value
184  * &beta; lies in [&minus;90&deg;, 90&deg;].
185  **********************************************************************/
186  Math::real ParametricLatitude(real phi) const;
187 
188  /**
189  * @param[in] beta the parametric latitude (degrees).
190  * @return &phi; the geographic latitude (degrees).
191  *
192  * &beta; must lie in the range [&minus;90&deg;, 90&deg;]; the
193  * result is undefined if this condition does not hold. The returned value
194  * &phi; lies in [&minus;90&deg;, 90&deg;].
195  **********************************************************************/
196  Math::real InverseParametricLatitude(real beta) const;
197 
198  /**
199  * @param[in] phi the geographic latitude (degrees).
200  * @return &theta; the geocentric latitude (degrees).
201  *
202  * The geocentric latitude, &theta;, is the angle beween the equatorial
203  * plane and a line between the center of the ellipsoid and a point on the
204  * ellipsoid. For a sphere &theta; = &phi;.
205  *
206  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the
207  * result is undefined if this condition does not hold. The returned value
208  * &theta; lies in [&minus;90&deg;, 90&deg;].
209  **********************************************************************/
210  Math::real GeocentricLatitude(real phi) const;
211 
212  /**
213  * @param[in] theta the geocentric latitude (degrees).
214  * @return &phi; the geographic latitude (degrees).
215  *
216  * &theta; must lie in the range [&minus;90&deg;, 90&deg;]; the
217  * result is undefined if this condition does not hold. The returned value
218  * &phi; lies in [&minus;90&deg;, 90&deg;].
219  **********************************************************************/
220  Math::real InverseGeocentricLatitude(real theta) const;
221 
222  /**
223  * @param[in] phi the geographic latitude (degrees).
224  * @return &mu; the rectifying latitude (degrees).
225  *
226  * The rectifying latitude, &mu;, has the property that the distance along
227  * a meridian of the ellipsoid between two points with rectifying latitudes
228  * &mu;<sub>1</sub> and &mu;<sub>2</sub> is equal to
229  * (&mu;<sub>2</sub> - &mu;<sub>1</sub>) \e L / 90&deg;,
230  * where \e L = QuarterMeridian(). For a sphere &mu; = &phi;.
231  *
232  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the
233  * result is undefined if this condition does not hold. The returned value
234  * &mu; lies in [&minus;90&deg;, 90&deg;].
235  **********************************************************************/
236  Math::real RectifyingLatitude(real phi) const;
237 
238  /**
239  * @param[in] mu the rectifying latitude (degrees).
240  * @return &phi; the geographic latitude (degrees).
241  *
242  * &mu; must lie in the range [&minus;90&deg;, 90&deg;]; the
243  * result is undefined if this condition does not hold. The returned value
244  * &phi; lies in [&minus;90&deg;, 90&deg;].
245  **********************************************************************/
246  Math::real InverseRectifyingLatitude(real mu) const;
247 
248  /**
249  * @param[in] phi the geographic latitude (degrees).
250  * @return &xi; the authalic latitude (degrees).
251  *
252  * The authalic latitude, &xi;, has the property that the area of the
253  * ellipsoid between two circles with authalic latitudes
254  * &xi;<sub>1</sub> and &xi;<sub>2</sub> is equal to (sin
255  * &xi;<sub>2</sub> - sin &xi;<sub>1</sub>) \e A / 2, where \e A
256  * = Area(). For a sphere &xi; = &phi;.
257  *
258  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the
259  * result is undefined if this condition does not hold. The returned value
260  * &xi; lies in [&minus;90&deg;, 90&deg;].
261  **********************************************************************/
262  Math::real AuthalicLatitude(real phi) const;
263 
264  /**
265  * @param[in] xi the authalic latitude (degrees).
266  * @return &phi; the geographic latitude (degrees).
267  *
268  * &xi; must lie in the range [&minus;90&deg;, 90&deg;]; the
269  * result is undefined if this condition does not hold. The returned value
270  * &phi; lies in [&minus;90&deg;, 90&deg;].
271  **********************************************************************/
272  Math::real InverseAuthalicLatitude(real xi) const;
273 
274  /**
275  * @param[in] phi the geographic latitude (degrees).
276  * @return &chi; the conformal latitude (degrees).
277  *
278  * The conformal latitude, &chi;, gives the mapping of the ellipsoid to a
279  * sphere which which is conformal (angles are preserved) and in which the
280  * equator of the ellipsoid maps to the equator of the sphere. For a
281  * sphere &chi; = &phi;.
282  *
283  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the
284  * result is undefined if this condition does not hold. The returned value
285  * &chi; lies in [&minus;90&deg;, 90&deg;].
286  **********************************************************************/
287  Math::real ConformalLatitude(real phi) const;
288 
289  /**
290  * @param[in] chi the conformal latitude (degrees).
291  * @return &phi; the geographic latitude (degrees).
292  *
293  * &chi; must lie in the range [&minus;90&deg;, 90&deg;]; the
294  * result is undefined if this condition does not hold. The returned value
295  * &phi; lies in [&minus;90&deg;, 90&deg;].
296  **********************************************************************/
297  Math::real InverseConformalLatitude(real chi) const;
298 
299  /**
300  * @param[in] phi the geographic latitude (degrees).
301  * @return &psi; the isometric latitude (degrees).
302  *
303  * The isometric latitude gives the mapping of the ellipsoid to a plane
304  * which which is conformal (angles are preserved) and in which the equator
305  * of the ellipsoid maps to a straight line of constant scale; this mapping
306  * defines the Mercator projection. For a sphere &psi; =
307  * sinh<sup>&minus;1</sup> tan &phi;.
308  *
309  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the result is
310  * undefined if this condition does not hold. The value returned for &phi;
311  * = &plusmn;90&deg; is some (positive or negative) large but finite value,
312  * such that InverseIsometricLatitude returns the original value of &phi;.
313  **********************************************************************/
314  Math::real IsometricLatitude(real phi) const;
315 
316  /**
317  * @param[in] psi the isometric latitude (degrees).
318  * @return &phi; the geographic latitude (degrees).
319  *
320  * The returned value &phi; lies in [&minus;90&deg;, 90&deg;]. For a
321  * sphere &phi; = tan<sup>&minus;1</sup> sinh &psi;.
322  **********************************************************************/
323  Math::real InverseIsometricLatitude(real psi) const;
324  ///@}
325 
326  /** \name Other quantities.
327  **********************************************************************/
328  ///@{
329 
330  /**
331  * @param[in] phi the geographic latitude (degrees).
332  * @return \e R = \e a cos &beta; the radius of a circle of latitude
333  * &phi; (meters). \e R (&pi;/180&deg;) gives meters per degree
334  * longitude measured along a circle of latitude.
335  *
336  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the
337  * result is undefined if this condition does not hold.
338  **********************************************************************/
339  Math::real CircleRadius(real phi) const;
340 
341  /**
342  * @param[in] phi the geographic latitude (degrees).
343  * @return \e Z = \e b sin &beta; the distance of a circle of latitude
344  * &phi; from the equator measured parallel to the ellipsoid axis
345  * (meters).
346  *
347  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the
348  * result is undefined if this condition does not hold.
349  **********************************************************************/
350  Math::real CircleHeight(real phi) const;
351 
352  /**
353  * @param[in] phi the geographic latitude (degrees).
354  * @return \e s the distance along a meridian
355  * between the equator and a point of latitude &phi; (meters). \e s is
356  * given by \e s = &mu; \e L / 90&deg;, where \e L =
357  * QuarterMeridian()).
358  *
359  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the
360  * result is undefined if this condition does not hold.
361  **********************************************************************/
362  Math::real MeridianDistance(real phi) const;
363 
364  /**
365  * @param[in] phi the geographic latitude (degrees).
366  * @return &rho; the meridional radius of curvature of the ellipsoid at
367  * latitude &phi; (meters); this is the curvature of the meridian. \e
368  * rho is given by &rho; = (180&deg;/&pi;) d\e s / d&phi;,
369  * where \e s = MeridianDistance(); thus &rho; (&pi;/180&deg;)
370  * gives meters per degree latitude measured along a meridian.
371  *
372  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the
373  * result is undefined if this condition does not hold.
374  **********************************************************************/
375  Math::real MeridionalCurvatureRadius(real phi) const;
376 
377  /**
378  * @param[in] phi the geographic latitude (degrees).
379  * @return &nu; the transverse radius of curvature of the ellipsoid at
380  * latitude &phi; (meters); this is the curvature of a curve on the
381  * ellipsoid which also lies in a plane perpendicular to the ellipsoid
382  * and to the meridian. &nu; is related to \e R = CircleRadius() by \e
383  * R = &nu; cos &phi;.
384  *
385  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the
386  * result is undefined if this condition does not hold.
387  **********************************************************************/
388  Math::real TransverseCurvatureRadius(real phi) const;
389 
390  /**
391  * @param[in] phi the geographic latitude (degrees).
392  * @param[in] azi the angle between the meridian and the normal section
393  * (degrees).
394  * @return the radius of curvature of the ellipsoid in the normal
395  * section at latitude &phi; inclined at an angle \e azi to the
396  * meridian (meters).
397  *
398  * &phi; must lie in the range [&minus;90&deg;, 90&deg;]; the result is
399  * undefined this condition does not hold.
400  **********************************************************************/
401  Math::real NormalCurvatureRadius(real phi, real azi) const;
402  ///@}
403 
404  /** \name Eccentricity conversions.
405  **********************************************************************/
406  ///@{
407 
408  /**
409  * @param[in] fp = \e f ' = (\e a &minus; \e b) / \e b, the second
410  * flattening.
411  * @return \e f = (\e a &minus; \e b) / \e a, the flattening.
412  *
413  * \e f ' should lie in (&minus;1, &infin;).
414  * The returned value \e f lies in (&minus;&infin;, 1).
415  **********************************************************************/
417  { return fp / (1 + fp); }
418 
419  /**
420  * @param[in] f = (\e a &minus; \e b) / \e a, the flattening.
421  * @return \e f ' = (\e a &minus; \e b) / \e b, the second flattening.
422  *
423  * \e f should lie in (&minus;&infin;, 1).
424  * The returned value \e f ' lies in (&minus;1, &infin;).
425  **********************************************************************/
427  { return f / (1 - f); }
428 
429  /**
430  * @param[in] n = (\e a &minus; \e b) / (\e a + \e b), the third
431  * flattening.
432  * @return \e f = (\e a &minus; \e b) / \e a, the flattening.
433  *
434  * \e n should lie in (&minus;1, 1).
435  * The returned value \e f lies in (&minus;&infin;, 1).
436  **********************************************************************/
438  { return 2 * n / (1 + n); }
439 
440  /**
441  * @param[in] f = (\e a &minus; \e b) / \e a, the flattening.
442  * @return \e n = (\e a &minus; \e b) / (\e a + \e b), the third
443  * flattening.
444  *
445  * \e f should lie in (&minus;&infin;, 1).
446  * The returned value \e n lies in (&minus;1, 1).
447  **********************************************************************/
449  { return f / (2 - f); }
450 
451  /**
452  * @param[in] e2 = <i>e</i><sup>2</sup> = (<i>a</i><sup>2</sup> &minus;
453  * <i>b</i><sup>2</sup>) / <i>a</i><sup>2</sup>, the eccentricity
454  * squared.
455  * @return \e f = (\e a &minus; \e b) / \e a, the flattening.
456  *
457  * <i>e</i><sup>2</sup> should lie in (&minus;&infin;, 1).
458  * The returned value \e f lies in (&minus;&infin;, 1).
459  **********************************************************************/
461  { using std::sqrt; return e2 / (sqrt(1 - e2) + 1); }
462 
463  /**
464  * @param[in] f = (\e a &minus; \e b) / \e a, the flattening.
465  * @return <i>e</i><sup>2</sup> = (<i>a</i><sup>2</sup> &minus;
466  * <i>b</i><sup>2</sup>) / <i>a</i><sup>2</sup>, the eccentricity
467  * squared.
468  *
469  * \e f should lie in (&minus;&infin;, 1).
470  * The returned value <i>e</i><sup>2</sup> lies in (&minus;&infin;, 1).
471  **********************************************************************/
473  { return f * (2 - f); }
474 
475  /**
476  * @param[in] ep2 = <i>e'</i> <sup>2</sup> = (<i>a</i><sup>2</sup> &minus;
477  * <i>b</i><sup>2</sup>) / <i>b</i><sup>2</sup>, the second eccentricity
478  * squared.
479  * @return \e f = (\e a &minus; \e b) / \e a, the flattening.
480  *
481  * <i>e'</i> <sup>2</sup> should lie in (&minus;1, &infin;).
482  * The returned value \e f lies in (&minus;&infin;, 1).
483  **********************************************************************/
485  { using std::sqrt; return ep2 / (sqrt(1 + ep2) + 1 + ep2); }
486 
487  /**
488  * @param[in] f = (\e a &minus; \e b) / \e a, the flattening.
489  * @return <i>e'</i> <sup>2</sup> = (<i>a</i><sup>2</sup> &minus;
490  * <i>b</i><sup>2</sup>) / <i>b</i><sup>2</sup>, the second eccentricity
491  * squared.
492  *
493  * \e f should lie in (&minus;&infin;, 1).
494  * The returned value <i>e'</i> <sup>2</sup> lies in (&minus;1, &infin;).
495  **********************************************************************/
497  { return f * (2 - f) / Math::sq(1 - f); }
498 
499  /**
500  * @param[in] epp2 = <i>e''</i> <sup>2</sup> = (<i>a</i><sup>2</sup>
501  * &minus; <i>b</i><sup>2</sup>) / (<i>a</i><sup>2</sup> +
502  * <i>b</i><sup>2</sup>), the third eccentricity squared.
503  * @return \e f = (\e a &minus; \e b) / \e a, the flattening.
504  *
505  * <i>e''</i> <sup>2</sup> should lie in (&minus;1, 1).
506  * The returned value \e f lies in (&minus;&infin;, 1).
507  **********************************************************************/
509  { return 2 * epp2 / (sqrt((1 - epp2) * (1 + epp2)) + 1 + epp2); }
510 
511  /**
512  * @param[in] f = (\e a &minus; \e b) / \e a, the flattening.
513  * @return <i>e''</i> <sup>2</sup> = (<i>a</i><sup>2</sup> &minus;
514  * <i>b</i><sup>2</sup>) / (<i>a</i><sup>2</sup> + <i>b</i><sup>2</sup>),
515  * the third eccentricity squared.
516  *
517  * \e f should lie in (&minus;&infin;, 1).
518  * The returned value <i>e''</i> <sup>2</sup> lies in (&minus;1, 1).
519  **********************************************************************/
521  { return f * (2 - f) / (1 + Math::sq(1 - f)); }
522 
523  ///@}
524 
525  /**
526  * A global instantiation of Ellipsoid with the parameters for the WGS84
527  * ellipsoid.
528  **********************************************************************/
529  static const Ellipsoid& WGS84();
530  };
531 
532 } // namespace GeographicLib
533 
534 #endif // GEOGRAPHICLIB_ELLIPSOID_HPP
static T pi()
Definition: Math.hpp:202
#define GEOGRAPHICLIB_EXPORT
Definition: Constants.hpp:90
Math::real EccentricitySq() const
Definition: Ellipsoid.hpp:142
GeographicLib::Math::real real
Definition: GeodSolve.cpp:31
static Math::real SecondFlatteningToFlattening(real fp)
Definition: Ellipsoid.hpp:416
Math::real Volume() const
Definition: Ellipsoid.hpp:106
Math::real SecondEccentricitySq() const
Definition: Ellipsoid.hpp:150
Transverse Mercator projection.
Elliptic integrals and functions.
Header for GeographicLib::TransverseMercator class.
static Math::real FlatteningToEccentricitySq(real f)
Definition: Ellipsoid.hpp:472
Math::real Flattening() const
Definition: Ellipsoid.hpp:120
static Math::real ThirdEccentricitySqToFlattening(real epp2)
Definition: Ellipsoid.hpp:508
Header for GeographicLib::AlbersEqualArea class.
Albers equal area conic projection.
static Math::real ThirdFlatteningToFlattening(real n)
Definition: Ellipsoid.hpp:437
static T sq(T x)
Definition: Math.hpp:232
Math::real SecondFlattening() const
Definition: Ellipsoid.hpp:127
Namespace for GeographicLib.
Definition: Accumulator.cpp:12
static Math::real FlatteningToThirdEccentricitySq(real f)
Definition: Ellipsoid.hpp:520
Header for GeographicLib::EllipticFunction class.
static Math::real SecondEccentricitySqToFlattening(real ep2)
Definition: Ellipsoid.hpp:484
Properties of an ellipsoid.
Definition: Ellipsoid.hpp:39
static Math::real FlatteningToThirdFlattening(real f)
Definition: Ellipsoid.hpp:448
Math::real MajorRadius() const
Definition: Ellipsoid.hpp:80
Math::real ThirdFlattening() const
Definition: Ellipsoid.hpp:134
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:460
static Math::real FlatteningToSecondFlattening(real f)
Definition: Ellipsoid.hpp:426
Find a sequence of points on a single rhumb line.
Definition: Rhumb.hpp:436
static Math::real FlatteningToSecondEccentricitySq(real f)
Definition: Ellipsoid.hpp:496
Math::real ThirdEccentricitySq() const
Definition: Ellipsoid.hpp:159
Math::real MinorRadius() const
Definition: Ellipsoid.hpp:85