/usr/include/GeographicLib/Rhumb.hpp is in libgeographic-dev 1.37-3.
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* \file Rhumb.hpp
* \brief Header for GeographicLib::Rhumb and GeographicLib::RhumbLine classes
*
* Copyright (c) Charles Karney (2012) <charles@karney.com> and licensed under
* the MIT/X11 License. For more information, see
* http://geographiclib.sourceforge.net/
**********************************************************************/
#if !defined(GEOGRAPHICLIB_RHUMB_HPP)
#define GEOGRAPHICLIB_RHUMB_HPP 1
#include <GeographicLib/Constants.hpp>
#include <GeographicLib/Ellipsoid.hpp>
namespace GeographicLib {
class RhumbLine;
/**
* \brief Solve of the direct and inverse rhumb problems.
*
* The path of constant azimuth between two points on a ellipsoid at (\e
* lat1, \e lon1) and (\e lat2, \e lon2) is called the rhumb line (also
* called the loxodrome). Its length is \e s12 and its azimuth is \e azi12
* and \e azi2. (The azimuth is the heading measured clockwise from north.)
*
* Given \e lat1, \e lon1, \e azi12, and \e s12, we can determine \e lat2,
* and \e lon2. This is the \e direct rhumb problem and its solution is
* given by the function Rhumb::Direct.
*
* Given \e lat1, \e lon1, \e lat2, and \e lon2, we can determine \e azi12
* and \e s12. This is the \e inverse rhumb problem, whose solution is
* given by Rhumb::Inverse. This finds the shortest such rhumb line, i.e.,
* the one that wraps no more than half way around the earth .
*
* Note that rhumb lines may be appreciably longer (up to 50%) than the
* corresponding Geodesic. For example the distance between London Heathrow
* and Tokyo Narita via the rhumb line is 11400 km which is 18% longer than
* the geodesic distance 9600 km.
*
* For more information on rhumb lines see \ref rhumb.
*
* Example of use:
* \include example-Rhumb.cpp
**********************************************************************/
class GEOGRAPHICLIB_EXPORT Rhumb {
private:
typedef Math::real real;
friend class RhumbLine;
Ellipsoid _ell;
bool _exact;
static const int tm_maxord = GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER;
static inline real overflow() {
// Overflow value s.t. atan(overflow_) = pi/2
static const real
overflow = 1 / Math::sq(std::numeric_limits<real>::epsilon());
return overflow;
}
static inline real tano(real x) {
using std::abs; using std::tan;
return
2 * abs(x) == Math::pi() ? (x < 0 ? - overflow() : overflow()) :
tan(x);
}
static inline real gd(real x)
{ using std::atan; using std::sinh; return atan(sinh(x)); }
// Use divided differences to determine (mu2 - mu1) / (psi2 - psi1)
// accurately
//
// Definition: Df(x,y,d) = (f(x) - f(y)) / (x - y)
// See:
// W. M. Kahan and R. J. Fateman,
// Symbolic computation of divided differences,
// SIGSAM Bull. 33(3), 7-28 (1999)
// http://dx.doi.org/10.1145/334714.334716
// http://www.cs.berkeley.edu/~fateman/papers/divdiff.pdf
static inline real Dtan(real x, real y) {
real d = x - y, tx = tano(x), ty = tano(y), txy = tx * ty;
return d ? (2 * txy > -1 ? (1 + txy) * tano(d) : tx - ty) / d :
1 + txy;
}
static inline real Datan(real x, real y) {
using std::atan;
real d = x - y, xy = x * y;
return d ? (2 * xy > -1 ? atan( d / (1 + xy) ) : atan(x) - atan(y)) / d :
1 / (1 + xy);
}
static inline real Dsin(real x, real y) {
using std::sin; using std::cos;
real d = (x - y)/2;
return cos((x + y)/2) * (d ? sin(d) / d : 1);
}
static inline real Dsinh(real x, real y) {
using std::sinh; using std::cosh;
real d = (x - y)/2;
return cosh((x + y)/2) * (d ? sinh(d) / d : 1);
}
static inline real Dasinh(real x, real y) {
real d = x - y,
hx = Math::hypot(real(1), x), hy = Math::hypot(real(1), y);
return d ? Math::asinh(x*y > 0 ? d * (x + y) / (x*hy + y*hx) :
x*hy - y*hx) / d :
1 / hx;
}
static inline real Dgd(real x, real y) {
using std::sinh;
return Datan(sinh(x), sinh(y)) * Dsinh(x, y);
}
static inline real Dgdinv(real x, real y) {
return Dasinh(tano(x), tano(y)) * Dtan(x, y);
}
// Copied from LambertConformalConic...
// e * atanh(e * x) = log( ((1 + e*x)/(1 - e*x))^(e/2) ) if f >= 0
// - sqrt(-e2) * atan( sqrt(-e2) * x) if f < 0
inline real eatanhe(real x) const {
using std::atan;
return _ell._f >= 0 ? _ell._e * Math::atanh(_ell._e * x) :
- _ell._e * atan(_ell._e * x);
}
// Copied from LambertConformalConic...
// Deatanhe(x,y) = eatanhe((x-y)/(1-e^2*x*y))/(x-y)
inline real Deatanhe(real x, real y) const {
real t = x - y, d = 1 - _ell._e2 * x * y;
return t ? eatanhe(t / d) / t : _ell._e2 / d;
}
// (E(x) - E(y)) / (x - y) -- E = incomplete elliptic integral of 2nd kind
real DE(real x, real y) const;
// (mux - muy) / (phix - phiy) using elliptic integrals
real DRectifying(real latx, real laty) const;
// (psix - psiy) / (phix - phiy)
real DIsometric(real latx, real laty) const;
// (sum(c[j]*sin(2*j*x),j=1..n) - sum(c[j]*sin(2*j*x),j=1..n)) / (x - y)
static real SinSeries(real x, real y, const real c[], int n);
// (mux - muy) / (chix - chiy) using Krueger's series
real DConformalToRectifying(real chix, real chiy) const;
// (chix - chiy) / (mux - muy) using Krueger's series
real DRectifyingToConformal(real mux, real muy) const;
// (mux - muy) / (psix - psiy)
real DIsometricToRectifying(real psix, real psiy) const;
// (psix - psiy) / (mux - muy)
real DRectifyingToIsometric(real mux, real muy) const;
public:
/**
* Constructor for a ellipsoid with
*
* @param[in] a equatorial radius (meters).
* @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
* Negative \e f gives a prolate ellipsoid. If \e f > 1, set
* flattening to 1/\e f.
* @param[in] exact if true (the default) use an addition theorem for
* elliptic integrals to compute divided differences; otherwise use
* series expansion (accurate for |<i>f</i>| < 0.01).
* @exception GeographicErr if \e a or (1 − \e f) \e a is not
* positive.
*
* See \ref rhumb, for a detailed description of the \e exact parameter.
**********************************************************************/
Rhumb(real a, real f, bool exact = true) : _ell(a, f), _exact(exact) {}
/**
* Solve the direct rhumb problem.
*
* @param[in] lat1 latitude of point 1 (degrees).
* @param[in] lon1 longitude of point 1 (degrees).
* @param[in] azi12 azimuth of the rhumb line (degrees).
* @param[in] s12 distance between point 1 and point 2 (meters); it can be
* negative.
* @param[out] lat2 latitude of point 2 (degrees).
* @param[out] lon2 longitude of point 2 (degrees).
*
* \e lat1 should be in the range [−90°, 90°]; \e lon1 and \e
* azi1 should be in the range [−540°, 540°). The values of
* \e lon2 and \e azi2 returned are in the range [−180°,
* 180°).
*
* If point 1 is a pole, the cosine of its latitude is taken to be
* 1/ε<sup>2</sup> (where ε is 2<sup>-52</sup>). This
* position, which is extremely close to the actual pole, allows the
* calculation to be carried out in finite terms. If \e s12 is large
* enough that the rhumb line crosses a pole, the longitude of point 2
* is indeterminate (a NaN is returned for \e lon2).
**********************************************************************/
void Direct(real lat1, real lon1, real azi12, real s12,
real& lat2, real& lon2) const;
/**
* Solve the inverse rhumb problem.
*
* @param[in] lat1 latitude of point 1 (degrees).
* @param[in] lon1 longitude of point 1 (degrees).
* @param[in] lat2 latitude of point 2 (degrees).
* @param[in] lon2 longitude of point 2 (degrees).
* @param[out] s12 rhumb distance between point 1 and point 2 (meters).
* @param[out] azi12 azimuth of the rhumb line (degrees).
*
* The shortest rhumb line is found. \e lat1 and \e lat2 should be in the
* range [−90°, 90°]; \e lon1 and \e lon2 should be in the
* range [−540°, 540°). The value of \e azi12 returned is in
* the range [−180°, 180°).
*
* If either point is a pole, the cosine of its latitude is taken to be
* 1/ε<sup>2</sup> (where ε is 2<sup>-52</sup>). This
* position, which is extremely close to the actual pole, allows the
* calculation to be carried out in finite terms.
**********************************************************************/
void Inverse(real lat1, real lon1, real lat2, real lon2,
real& s12, real& azi12) const;
/**
* Set up to compute several points on a single rhumb line.
*
* @param[in] lat1 latitude of point 1 (degrees).
* @param[in] lon1 longitude of point 1 (degrees).
* @param[in] azi12 azimuth of the rhumb line (degrees).
* @return a RhumbLine object.
*
* \e lat1 should be in the range [−90°, 90°]; \e lon1 and \e
* azi12 should be in the range [−540°, 540°).
*
* If point 1 is a pole, the cosine of its latitude is taken to be
* 1/ε<sup>2</sup> (where ε is 2<sup>-52</sup>). This
* position, which is extremely close to the actual pole, allows the
* calculation to be carried out in finite terms.
**********************************************************************/
RhumbLine Line(real lat1, real lon1, real azi12) const;
/** \name Inspector functions.
**********************************************************************/
///@{
/**
* @return \e a the equatorial radius of the ellipsoid (meters). This is
* the value used in the constructor.
**********************************************************************/
Math::real MajorRadius() const { return _ell.MajorRadius(); }
/**
* @return \e f the flattening of the ellipsoid. This is the
* value used in the constructor.
**********************************************************************/
Math::real Flattening() const { return _ell.Flattening(); }
/**
* A global instantiation of Rhumb with the parameters for the WGS84
* ellipsoid.
**********************************************************************/
static const Rhumb& WGS84();
};
/**
* \brief Find a sequence of points on a single rhumb line.
*
* RhumbLine facilitates the determination of a series of points on a single
* rhumb line. The starting point (\e lat1, \e lon1) and the azimuth \e
* azi12 are specified in the call to Rhumb::Line which returns a RhumbLine
* object. RhumbLine.Position returns the location of point 2 a distance \e
* s12 along the rhumb line.
* There is no public constructor for this class. (Use Rhumb::Line to create
* an instance.) The Rhumb object used to create a RhumbLine must stay in
* scope as long as the RhumbLine.
*
* Example of use:
* \include example-RhumbLine.cpp
**********************************************************************/
class GEOGRAPHICLIB_EXPORT RhumbLine {
private:
typedef Math::real real;
friend class Rhumb;
const Rhumb& _rh;
bool _exact;
real _lat1, _lon1, _azi12, _salp, _calp, _mu1, _psi1, _r1;
RhumbLine& operator=(const RhumbLine&); // copy assignment not allowed
RhumbLine(const Rhumb& rh, real lat1, real lon1, real azi12,
bool exact);
public:
/**
* Compute the position of point 2 which is a distance \e s12 (meters) from
* point 1.
*
* @param[in] s12 distance between point 1 and point 2 (meters); it can be
* negative.
* @param[out] lat2 latitude of point 2 (degrees).
* @param[out] lon2 longitude of point 2 (degrees).
*
* The values of \e lon2 and \e azi2 returned are in the range
* [−180°, 180°).
*
* If \e s12 is large enough that the rhumb line crosses a pole, the
* longitude of point 2 is indeterminate (a NaN is returned for \e lon2).
**********************************************************************/
void Position(real s12, real& lat2, real& lon2) const;
/** \name Inspector functions
**********************************************************************/
///@{
/**
* @return \e lat1 the latitude of point 1 (degrees).
**********************************************************************/
Math::real Latitude() const { return _lat1; }
/**
* @return \e lon1 the longitude of point 1 (degrees).
**********************************************************************/
Math::real Longitude() const { return _lon1; }
/**
* @return \e azi12 the azimuth of the rhumb line (degrees).
**********************************************************************/
Math::real Azimuth() const { return _azi12; }
/**
* @return \e a the equatorial radius of the ellipsoid (meters). This is
* the value inherited from the Rhumb object used in the constructor.
**********************************************************************/
Math::real MajorRadius() const { return _rh.MajorRadius(); }
/**
* @return \e f the flattening of the ellipsoid. This is the value
* inherited from the Rhumb object used in the constructor.
**********************************************************************/
Math::real Flattening() const { return _rh.Flattening(); }
};
} // namespace GeographicLib
#endif // GEOGRAPHICLIB_RHUMB_HPP
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