GeographicLib  1.50.1
Gnomonic.hpp
Go to the documentation of this file.
1 /**
2  * \file Gnomonic.hpp
3  * \brief Header for GeographicLib::Gnomonic class
4  *
5  * Copyright (c) Charles Karney (2010-2019) <charles@karney.com> and licensed
6  * under the MIT/X11 License. For more information, see
7  * https://geographiclib.sourceforge.io/
8  **********************************************************************/
9 
10 #if !defined(GEOGRAPHICLIB_GNOMONIC_HPP)
11 #define GEOGRAPHICLIB_GNOMONIC_HPP 1
12 
16 
17 namespace GeographicLib {
18 
19  /**
20  * \brief %Gnomonic projection
21  *
22  * %Gnomonic projection centered at an arbitrary position \e C on the
23  * ellipsoid. This projection is derived in Section 8 of
24  * - C. F. F. Karney,
25  * <a href="https://doi.org/10.1007/s00190-012-0578-z">
26  * Algorithms for geodesics</a>,
27  * J. Geodesy <b>87</b>, 43--55 (2013);
28  * DOI: <a href="https://doi.org/10.1007/s00190-012-0578-z">
29  * 10.1007/s00190-012-0578-z</a>;
30  * addenda:
31  * <a href="https://geographiclib.sourceforge.io/geod-addenda.html">
32  * geod-addenda.html</a>.
33  * .
34  * The projection of \e P is defined as follows: compute the geodesic line
35  * from \e C to \e P; compute the reduced length \e m12, geodesic scale \e
36  * M12, and &rho; = <i>m12</i>/\e M12; finally \e x = &rho; sin \e azi1; \e
37  * y = &rho; cos \e azi1, where \e azi1 is the azimuth of the geodesic at \e
38  * C. The Gnomonic::Forward and Gnomonic::Reverse methods also return the
39  * azimuth \e azi of the geodesic at \e P and reciprocal scale \e rk in the
40  * azimuthal direction. The scale in the radial direction if
41  * 1/<i>rk</i><sup>2</sup>.
42  *
43  * For a sphere, &rho; is reduces to \e a tan(<i>s12</i>/<i>a</i>), where \e
44  * s12 is the length of the geodesic from \e C to \e P, and the gnomonic
45  * projection has the property that all geodesics appear as straight lines.
46  * For an ellipsoid, this property holds only for geodesics interesting the
47  * centers. However geodesic segments close to the center are approximately
48  * straight.
49  *
50  * Consider a geodesic segment of length \e l. Let \e T be the point on the
51  * geodesic (extended if necessary) closest to \e C the center of the
52  * projection and \e t be the distance \e CT. To lowest order, the maximum
53  * deviation (as a true distance) of the corresponding gnomonic line segment
54  * (i.e., with the same end points) from the geodesic is<br>
55  * <br>
56  * (<i>K</i>(<i>T</i>) - <i>K</i>(<i>C</i>))
57  * <i>l</i><sup>2</sup> \e t / 32.<br>
58  * <br>
59  * where \e K is the Gaussian curvature.
60  *
61  * This result applies for any surface. For an ellipsoid of revolution,
62  * consider all geodesics whose end points are within a distance \e r of \e
63  * C. For a given \e r, the deviation is maximum when the latitude of \e C
64  * is 45&deg;, when endpoints are a distance \e r away, and when their
65  * azimuths from the center are &plusmn; 45&deg; or &plusmn; 135&deg;.
66  * To lowest order in \e r and the flattening \e f, the deviation is \e f
67  * (<i>r</i>/2<i>a</i>)<sup>3</sup> \e r.
68  *
69  * The conversions all take place using a Geodesic object (by default
70  * Geodesic::WGS84()). For more information on geodesics see \ref geodesic.
71  *
72  * \warning The definition of this projection for a sphere is
73  * standard. However, there is no standard for how it should be extended to
74  * an ellipsoid. The choices are:
75  * - Declare that the projection is undefined for an ellipsoid.
76  * - Project to a tangent plane from the center of the ellipsoid. This
77  * causes great ellipses to appear as straight lines in the projection;
78  * i.e., it generalizes the spherical great circle to a great ellipse.
79  * This was proposed by independently by Bowring and Williams in 1997.
80  * - Project to the conformal sphere with the constant of integration chosen
81  * so that the values of the latitude match for the center point and
82  * perform a central projection onto the plane tangent to the conformal
83  * sphere at the center point. This causes normal sections through the
84  * center point to appear as straight lines in the projection; i.e., it
85  * generalizes the spherical great circle to a normal section. This was
86  * proposed by I. G. Letoval'tsev, Generalization of the gnomonic
87  * projection for a spheroid and the principal geodetic problems involved
88  * in the alignment of surface routes, Geodesy and Aerophotography (5),
89  * 271--274 (1963).
90  * - The projection given here. This causes geodesics close to the center
91  * point to appear as straight lines in the projection; i.e., it
92  * generalizes the spherical great circle to a geodesic.
93  *
94  * Example of use:
95  * \include example-Gnomonic.cpp
96  *
97  * <a href="GeodesicProj.1.html">GeodesicProj</a> is a command-line utility
98  * providing access to the functionality of AzimuthalEquidistant, Gnomonic,
99  * and CassiniSoldner.
100  **********************************************************************/
101 
103  private:
104  typedef Math::real real;
105  real eps0_, eps_;
106  Geodesic _earth;
107  real _a, _f;
108  // numit_ increased from 10 to 20 to fix convergence failure with high
109  // precision (e.g., GEOGRAPHICLIB_DIGITS=2000) calculations. Reverse uses
110  // Newton's method which converges quadratically and so numit_ = 10 would
111  // normally be big enough. However, since the Geodesic class is based on a
112  // series it is of limited accuracy; in particular, the derivative rules
113  // used by Reverse only hold approximately. Consequently, after a few
114  // iterations, the convergence in the Reverse falls back to improvements in
115  // each step by a constant (albeit small) factor.
116  static const int numit_ = 20;
117  public:
118 
119  /**
120  * Constructor for Gnomonic.
121  *
122  * @param[in] earth the Geodesic object to use for geodesic calculations.
123  * By default this uses the WGS84 ellipsoid.
124  **********************************************************************/
125  explicit Gnomonic(const Geodesic& earth = Geodesic::WGS84());
126 
127  /**
128  * Forward projection, from geographic to gnomonic.
129  *
130  * @param[in] lat0 latitude of center point of projection (degrees).
131  * @param[in] lon0 longitude of center point of projection (degrees).
132  * @param[in] lat latitude of point (degrees).
133  * @param[in] lon longitude of point (degrees).
134  * @param[out] x easting of point (meters).
135  * @param[out] y northing of point (meters).
136  * @param[out] azi azimuth of geodesic at point (degrees).
137  * @param[out] rk reciprocal of azimuthal scale at point.
138  *
139  * \e lat0 and \e lat should be in the range [&minus;90&deg;, 90&deg;].
140  * The scale of the projection is 1/<i>rk</i><sup>2</sup> in the "radial"
141  * direction, \e azi clockwise from true north, and is 1/\e rk in the
142  * direction perpendicular to this. If the point lies "over the horizon",
143  * i.e., if \e rk &le; 0, then NaNs are returned for \e x and \e y (the
144  * correct values are returned for \e azi and \e rk). A call to Forward
145  * followed by a call to Reverse will return the original (\e lat, \e lon)
146  * (to within roundoff) provided the point in not over the horizon.
147  **********************************************************************/
148  void Forward(real lat0, real lon0, real lat, real lon,
149  real& x, real& y, real& azi, real& rk) const;
150 
151  /**
152  * Reverse projection, from gnomonic to geographic.
153  *
154  * @param[in] lat0 latitude of center point of projection (degrees).
155  * @param[in] lon0 longitude of center point of projection (degrees).
156  * @param[in] x easting of point (meters).
157  * @param[in] y northing of point (meters).
158  * @param[out] lat latitude of point (degrees).
159  * @param[out] lon longitude of point (degrees).
160  * @param[out] azi azimuth of geodesic at point (degrees).
161  * @param[out] rk reciprocal of azimuthal scale at point.
162  *
163  * \e lat0 should be in the range [&minus;90&deg;, 90&deg;]. \e lat will
164  * be in the range [&minus;90&deg;, 90&deg;] and \e lon will be in the
165  * range [&minus;180&deg;, 180&deg;]. The scale of the projection is
166  * 1/<i>rk</i><sup>2</sup> in the "radial" direction, \e azi clockwise from
167  * true north, and is 1/\e rk in the direction perpendicular to this. Even
168  * though all inputs should return a valid \e lat and \e lon, it's possible
169  * that the procedure fails to converge for very large \e x or \e y; in
170  * this case NaNs are returned for all the output arguments. A call to
171  * Reverse followed by a call to Forward will return the original (\e x, \e
172  * y) (to roundoff).
173  **********************************************************************/
174  void Reverse(real lat0, real lon0, real x, real y,
175  real& lat, real& lon, real& azi, real& rk) const;
176 
177  /**
178  * Gnomonic::Forward without returning the azimuth and scale.
179  **********************************************************************/
180  void Forward(real lat0, real lon0, real lat, real lon,
181  real& x, real& y) const {
182  real azi, rk;
183  Forward(lat0, lon0, lat, lon, x, y, azi, rk);
184  }
185 
186  /**
187  * Gnomonic::Reverse without returning the azimuth and scale.
188  **********************************************************************/
189  void Reverse(real lat0, real lon0, real x, real y,
190  real& lat, real& lon) const {
191  real azi, rk;
192  Reverse(lat0, lon0, x, y, lat, lon, azi, rk);
193  }
194 
195  /** \name Inspector functions
196  **********************************************************************/
197  ///@{
198  /**
199  * @return \e a the equatorial radius of the ellipsoid (meters). This is
200  * the value inherited from the Geodesic object used in the constructor.
201  **********************************************************************/
202  Math::real EquatorialRadius() const { return _earth.EquatorialRadius(); }
203 
204  /**
205  * @return \e f the flattening of the ellipsoid. This is the value
206  * inherited from the Geodesic object used in the constructor.
207  **********************************************************************/
208  Math::real Flattening() const { return _earth.Flattening(); }
209 
210  /**
211  * \deprecated An old name for EquatorialRadius().
212  **********************************************************************/
213  // GEOGRAPHICLIB_DEPRECATED("Use EquatorialRadius()")
214  Math::real MajorRadius() const { return EquatorialRadius(); }
215  ///@}
216 
217  };
218 
219 } // namespace GeographicLib
220 
221 #endif // GEOGRAPHICLIB_GNOMONIC_HPP
real
GeographicLib::Math::real real
Definition: GeodSolve.cpp:31
GeographicLib::Geodesic::Flattening
Math::real Flattening() const
Definition: Geodesic.hpp:949
GeographicLib::Geodesic::EquatorialRadius
Math::real EquatorialRadius() const
Definition: Geodesic.hpp:943
GeographicLib
Namespace for GeographicLib.
Definition: Accumulator.cpp:12
GeographicLib::Geodesic::WGS84
static const Geodesic & WGS84()
Definition: Geodesic.cpp:89
GEOGRAPHICLIB_EXPORT
#define GEOGRAPHICLIB_EXPORT
Definition: Constants.hpp:92
GeographicLib::Gnomonic::Flattening
Math::real Flattening() const
Definition: Gnomonic.hpp:208
GeographicLib::Gnomonic
Gnomonic projection
Definition: Gnomonic.hpp:102
GeographicLib::Math::real
double real
Definition: Math.hpp:121
GeographicLib::Gnomonic::Forward
void Forward(real lat0, real lon0, real lat, real lon, real &x, real &y) const
Definition: Gnomonic.hpp:180
GeographicLib::Gnomonic::EquatorialRadius
Math::real EquatorialRadius() const
Definition: Gnomonic.hpp:202
GeodesicLine.hpp
Header for GeographicLib::GeodesicLine class.
Constants.hpp
Header for GeographicLib::Constants class.
GeographicLib::Gnomonic::MajorRadius
Math::real MajorRadius() const
Definition: Gnomonic.hpp:214
GeographicLib::Gnomonic::Reverse
void Reverse(real lat0, real lon0, real x, real y, real &lat, real &lon) const
Definition: Gnomonic.hpp:189
GeographicLib::Geodesic
Geodesic calculations
Definition: Geodesic.hpp:172
Geodesic.hpp
Header for GeographicLib::Geodesic class.