222 lines
10 KiB
C++
222 lines
10 KiB
C++
/**
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* \file Gnomonic.hpp
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* \brief Header for GeographicLib::Gnomonic class
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*
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* Copyright (c) Charles Karney (2010-2020) <charles@karney.com> and licensed
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* under the MIT/X11 License. For more information, see
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* https://geographiclib.sourceforge.io/
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**********************************************************************/
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#if !defined(GEOGRAPHICLIB_GNOMONIC_HPP)
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#define GEOGRAPHICLIB_GNOMONIC_HPP 1
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#include <GeographicLib/Geodesic.hpp>
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#include <GeographicLib/GeodesicLine.hpp>
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#include <GeographicLib/Constants.hpp>
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namespace GeographicLib {
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/**
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* \brief %Gnomonic projection
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*
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* %Gnomonic projection centered at an arbitrary position \e C on the
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* ellipsoid. This projection is derived in Section 8 of
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* - C. F. F. Karney,
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* <a href="https://doi.org/10.1007/s00190-012-0578-z">
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* Algorithms for geodesics</a>,
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* J. Geodesy <b>87</b>, 43--55 (2013);
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* DOI: <a href="https://doi.org/10.1007/s00190-012-0578-z">
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* 10.1007/s00190-012-0578-z</a>;
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* addenda:
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* <a href="https://geographiclib.sourceforge.io/geod-addenda.html">
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* geod-addenda.html</a>.
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* .
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* The projection of \e P is defined as follows: compute the geodesic line
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* from \e C to \e P; compute the reduced length \e m12, geodesic scale \e
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* M12, and ρ = <i>m12</i>/\e M12; finally \e x = ρ sin \e azi1; \e
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* y = ρ cos \e azi1, where \e azi1 is the azimuth of the geodesic at \e
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* C. The Gnomonic::Forward and Gnomonic::Reverse methods also return the
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* azimuth \e azi of the geodesic at \e P and reciprocal scale \e rk in the
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* azimuthal direction. The scale in the radial direction if
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* 1/<i>rk</i><sup>2</sup>.
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*
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* For a sphere, ρ is reduces to \e a tan(<i>s12</i>/<i>a</i>), where \e
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* s12 is the length of the geodesic from \e C to \e P, and the gnomonic
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* projection has the property that all geodesics appear as straight lines.
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* For an ellipsoid, this property holds only for geodesics interesting the
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* centers. However geodesic segments close to the center are approximately
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* straight.
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*
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* Consider a geodesic segment of length \e l. Let \e T be the point on the
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* geodesic (extended if necessary) closest to \e C the center of the
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* projection and \e t be the distance \e CT. To lowest order, the maximum
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* deviation (as a true distance) of the corresponding gnomonic line segment
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* (i.e., with the same end points) from the geodesic is<br>
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* <br>
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* (<i>K</i>(<i>T</i>) - <i>K</i>(<i>C</i>))
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* <i>l</i><sup>2</sup> \e t / 32.<br>
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* <br>
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* where \e K is the Gaussian curvature.
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*
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* This result applies for any surface. For an ellipsoid of revolution,
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* consider all geodesics whose end points are within a distance \e r of \e
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* C. For a given \e r, the deviation is maximum when the latitude of \e C
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* is 45°, when endpoints are a distance \e r away, and when their
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* azimuths from the center are ± 45° or ± 135°.
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* To lowest order in \e r and the flattening \e f, the deviation is \e f
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* (<i>r</i>/2<i>a</i>)<sup>3</sup> \e r.
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*
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* The conversions all take place using a Geodesic object (by default
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* Geodesic::WGS84()). For more information on geodesics see \ref geodesic.
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*
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* \warning The definition of this projection for a sphere is
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* standard. However, there is no standard for how it should be extended to
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* an ellipsoid. The choices are:
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* - Declare that the projection is undefined for an ellipsoid.
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* - Project to a tangent plane from the center of the ellipsoid. This
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* causes great ellipses to appear as straight lines in the projection;
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* i.e., it generalizes the spherical great circle to a great ellipse.
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* This was proposed by independently by Bowring and Williams in 1997.
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* - Project to the conformal sphere with the constant of integration chosen
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* so that the values of the latitude match for the center point and
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* perform a central projection onto the plane tangent to the conformal
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* sphere at the center point. This causes normal sections through the
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* center point to appear as straight lines in the projection; i.e., it
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* generalizes the spherical great circle to a normal section. This was
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* proposed by I. G. Letoval'tsev, Generalization of the gnomonic
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* projection for a spheroid and the principal geodetic problems involved
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* in the alignment of surface routes, Geodesy and Aerophotography (5),
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* 271--274 (1963).
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* - The projection given here. This causes geodesics close to the center
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* point to appear as straight lines in the projection; i.e., it
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* generalizes the spherical great circle to a geodesic.
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*
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* Example of use:
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* \include example-Gnomonic.cpp
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*
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* <a href="GeodesicProj.1.html">GeodesicProj</a> is a command-line utility
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* providing access to the functionality of AzimuthalEquidistant, Gnomonic,
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* and CassiniSoldner.
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**********************************************************************/
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class GEOGRAPHICLIB_EXPORT Gnomonic {
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private:
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typedef Math::real real;
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real eps0_, eps_;
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Geodesic _earth;
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real _a, _f;
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// numit_ increased from 10 to 20 to fix convergence failure with high
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// precision (e.g., GEOGRAPHICLIB_DIGITS=2000) calculations. Reverse uses
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// Newton's method which converges quadratically and so numit_ = 10 would
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// normally be big enough. However, since the Geodesic class is based on a
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// series it is of limited accuracy; in particular, the derivative rules
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// used by Reverse only hold approximately. Consequently, after a few
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// iterations, the convergence in the Reverse falls back to improvements in
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// each step by a constant (albeit small) factor.
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static const int numit_ = 20;
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public:
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/**
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* Constructor for Gnomonic.
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*
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* @param[in] earth the Geodesic object to use for geodesic calculations.
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* By default this uses the WGS84 ellipsoid.
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**********************************************************************/
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explicit Gnomonic(const Geodesic& earth = Geodesic::WGS84());
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/**
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* Forward projection, from geographic to gnomonic.
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*
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* @param[in] lat0 latitude of center point of projection (degrees).
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* @param[in] lon0 longitude of center point of projection (degrees).
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* @param[in] lat latitude of point (degrees).
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* @param[in] lon longitude of point (degrees).
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* @param[out] x easting of point (meters).
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* @param[out] y northing of point (meters).
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* @param[out] azi azimuth of geodesic at point (degrees).
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* @param[out] rk reciprocal of azimuthal scale at point.
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*
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* \e lat0 and \e lat should be in the range [−90°, 90°].
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* The scale of the projection is 1/<i>rk</i><sup>2</sup> in the "radial"
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* direction, \e azi clockwise from true north, and is 1/\e rk in the
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* direction perpendicular to this. If the point lies "over the horizon",
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* i.e., if \e rk ≤ 0, then NaNs are returned for \e x and \e y (the
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* correct values are returned for \e azi and \e rk). A call to Forward
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* followed by a call to Reverse will return the original (\e lat, \e lon)
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* (to within roundoff) provided the point in not over the horizon.
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**********************************************************************/
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void Forward(real lat0, real lon0, real lat, real lon,
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real& x, real& y, real& azi, real& rk) const;
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/**
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* Reverse projection, from gnomonic to geographic.
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*
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* @param[in] lat0 latitude of center point of projection (degrees).
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* @param[in] lon0 longitude of center point of projection (degrees).
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* @param[in] x easting of point (meters).
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* @param[in] y northing of point (meters).
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* @param[out] lat latitude of point (degrees).
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* @param[out] lon longitude of point (degrees).
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* @param[out] azi azimuth of geodesic at point (degrees).
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* @param[out] rk reciprocal of azimuthal scale at point.
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*
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* \e lat0 should be in the range [−90°, 90°]. \e lat will
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* be in the range [−90°, 90°] and \e lon will be in the
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* range [−180°, 180°]. The scale of the projection is
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* 1/<i>rk</i><sup>2</sup> in the "radial" direction, \e azi clockwise from
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* true north, and is 1/\e rk in the direction perpendicular to this. Even
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* though all inputs should return a valid \e lat and \e lon, it's possible
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* that the procedure fails to converge for very large \e x or \e y; in
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* this case NaNs are returned for all the output arguments. A call to
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* Reverse followed by a call to Forward will return the original (\e x, \e
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* y) (to roundoff).
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**********************************************************************/
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void Reverse(real lat0, real lon0, real x, real y,
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real& lat, real& lon, real& azi, real& rk) const;
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/**
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* Gnomonic::Forward without returning the azimuth and scale.
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**********************************************************************/
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void Forward(real lat0, real lon0, real lat, real lon,
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real& x, real& y) const {
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real azi, rk;
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Forward(lat0, lon0, lat, lon, x, y, azi, rk);
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}
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/**
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* Gnomonic::Reverse without returning the azimuth and scale.
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**********************************************************************/
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void Reverse(real lat0, real lon0, real x, real y,
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real& lat, real& lon) const {
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real azi, rk;
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Reverse(lat0, lon0, x, y, lat, lon, azi, rk);
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}
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/** \name Inspector functions
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**********************************************************************/
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///@{
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/**
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* @return \e a the equatorial radius of the ellipsoid (meters). This is
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* the value inherited from the Geodesic object used in the constructor.
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**********************************************************************/
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Math::real EquatorialRadius() const { return _earth.EquatorialRadius(); }
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/**
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* @return \e f the flattening of the ellipsoid. This is the value
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* inherited from the Geodesic object used in the constructor.
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**********************************************************************/
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Math::real Flattening() const { return _earth.Flattening(); }
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/**
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* \deprecated An old name for EquatorialRadius().
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**********************************************************************/
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GEOGRAPHICLIB_DEPRECATED("Use EquatorialRadius()")
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Math::real MajorRadius() const { return EquatorialRadius(); }
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///@}
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};
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} // namespace GeographicLib
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#endif // GEOGRAPHICLIB_GNOMONIC_HPP
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