331 lines
15 KiB
C++
331 lines
15 KiB
C++
/**
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* \file LambertConformalConic.hpp
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* \brief Header for GeographicLib::LambertConformalConic 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_LAMBERTCONFORMALCONIC_HPP)
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#define GEOGRAPHICLIB_LAMBERTCONFORMALCONIC_HPP 1
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#include <GeographicLib/Constants.hpp>
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namespace GeographicLib {
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/**
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* \brief Lambert conformal conic projection
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*
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* Implementation taken from the report,
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* - J. P. Snyder,
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* <a href="http://pubs.er.usgs.gov/usgspubs/pp/pp1395"> Map Projections: A
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* Working Manual</a>, USGS Professional Paper 1395 (1987),
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* pp. 107--109.
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*
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* This is a implementation of the equations in Snyder except that divided
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* differences have been used to transform the expressions into ones which
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* may be evaluated accurately and that Newton's method is used to invert the
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* projection. In this implementation, the projection correctly becomes the
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* Mercator projection or the polar stereographic projection when the
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* standard latitude is the equator or a pole. The accuracy of the
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* projections is about 10 nm (10 nanometers).
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*
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* The ellipsoid parameters, the standard parallels, and the scale on the
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* standard parallels are set in the constructor. Internally, the case with
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* two standard parallels is converted into a single standard parallel, the
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* latitude of tangency (also the latitude of minimum scale), with a scale
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* specified on this parallel. This latitude is also used as the latitude of
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* origin which is returned by LambertConformalConic::OriginLatitude. The
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* scale on the latitude of origin is given by
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* LambertConformalConic::CentralScale. The case with two distinct standard
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* parallels where one is a pole is singular and is disallowed. The central
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* meridian (which is a trivial shift of the longitude) is specified as the
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* \e lon0 argument of the LambertConformalConic::Forward and
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* LambertConformalConic::Reverse functions.
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*
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* This class also returns the meridian convergence \e gamma and scale \e k.
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* The meridian convergence is the bearing of grid north (the \e y axis)
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* measured clockwise from true north.
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*
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* There is no provision in this
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* class for specifying a false easting or false northing or a different
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* latitude of origin. However these are can be simply included by the
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* calling function. For example the Pennsylvania South state coordinate
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* system (<a href="https://www.spatialreference.org/ref/epsg/3364/">
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* EPSG:3364</a>) is obtained by:
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* \include example-LambertConformalConic.cpp
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*
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* <a href="ConicProj.1.html">ConicProj</a> is a command-line utility
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* providing access to the functionality of LambertConformalConic and
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* AlbersEqualArea.
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**********************************************************************/
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class GEOGRAPHICLIB_EXPORT LambertConformalConic {
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private:
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typedef Math::real real;
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real eps_, epsx_, ahypover_;
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real _a, _f, _fm, _e2, _es;
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real _sign, _n, _nc, _t0nm1, _scale, _lat0, _k0;
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real _scbet0, _tchi0, _scchi0, _psi0, _nrho0, _drhomax;
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static const int numit_ = 5;
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static real hyp(real x) {
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using std::hypot;
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return hypot(real(1), x);
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}
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// Divided differences
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// Definition: Df(x,y) = (f(x)-f(y))/(x-y)
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// See:
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// W. M. Kahan and R. J. Fateman,
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// Symbolic computation of divided differences,
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// SIGSAM Bull. 33(3), 7-28 (1999)
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// https://doi.org/10.1145/334714.334716
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// http://www.cs.berkeley.edu/~fateman/papers/divdiff.pdf
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//
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// General rules
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// h(x) = f(g(x)): Dh(x,y) = Df(g(x),g(y))*Dg(x,y)
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// h(x) = f(x)*g(x):
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// Dh(x,y) = Df(x,y)*g(x) + Dg(x,y)*f(y)
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// = Df(x,y)*g(y) + Dg(x,y)*f(x)
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// = Df(x,y)*(g(x)+g(y))/2 + Dg(x,y)*(f(x)+f(y))/2
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//
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// hyp(x) = sqrt(1+x^2): Dhyp(x,y) = (x+y)/(hyp(x)+hyp(y))
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static real Dhyp(real x, real y, real hx, real hy)
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// hx = hyp(x)
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{ return (x + y) / (hx + hy); }
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// sn(x) = x/sqrt(1+x^2): Dsn(x,y) = (x+y)/((sn(x)+sn(y))*(1+x^2)*(1+y^2))
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static real Dsn(real x, real y, real sx, real sy) {
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// sx = x/hyp(x)
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real t = x * y;
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return t > 0 ? (x + y) * Math::sq( (sx * sy)/t ) / (sx + sy) :
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(x - y != 0 ? (sx - sy) / (x - y) : 1);
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}
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// Dlog1p(x,y) = log1p((x-y)/(1+y))/(x-y)
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static real Dlog1p(real x, real y) {
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using std::log1p;
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real t = x - y; if (t < 0) { t = -t; y = x; }
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return t != 0 ? log1p(t / (1 + y)) / t : 1 / (1 + x);
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}
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// Dexp(x,y) = exp((x+y)/2) * 2*sinh((x-y)/2)/(x-y)
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static real Dexp(real x, real y) {
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using std::sinh; using std::exp;
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real t = (x - y)/2;
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return (t != 0 ? sinh(t)/t : 1) * exp((x + y)/2);
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}
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// Dsinh(x,y) = 2*sinh((x-y)/2)/(x-y) * cosh((x+y)/2)
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// cosh((x+y)/2) = (c+sinh(x)*sinh(y)/c)/2
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// c=sqrt((1+cosh(x))*(1+cosh(y)))
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// cosh((x+y)/2) = sqrt( (sinh(x)*sinh(y) + cosh(x)*cosh(y) + 1)/2 )
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static real Dsinh(real x, real y, real sx, real sy, real cx, real cy)
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// sx = sinh(x), cx = cosh(x)
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{
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// real t = (x - y)/2, c = sqrt((1 + cx) * (1 + cy));
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// return (t ? sinh(t)/t : real(1)) * (c + sx * sy / c) /2;
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using std::sinh; using std::sqrt;
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real t = (x - y)/2;
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return (t != 0 ? sinh(t)/t : 1) * sqrt((sx * sy + cx * cy + 1) /2);
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}
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// Dasinh(x,y) = asinh((x-y)*(x+y)/(x*sqrt(1+y^2)+y*sqrt(1+x^2)))/(x-y)
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// = asinh((x*sqrt(1+y^2)-y*sqrt(1+x^2)))/(x-y)
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static real Dasinh(real x, real y, real hx, real hy) {
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// hx = hyp(x)
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using std::asinh;
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real t = x - y;
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return t != 0 ?
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asinh(x*y > 0 ? t * (x + y) / (x*hy + y*hx) : x*hy - y*hx) / t :
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1 / hx;
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}
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// Deatanhe(x,y) = eatanhe((x-y)/(1-e^2*x*y))/(x-y)
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real Deatanhe(real x, real y) const {
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real t = x - y, d = 1 - _e2 * x * y;
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return t != 0 ? Math::eatanhe(t / d, _es) / t : _e2 / d;
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}
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void Init(real sphi1, real cphi1, real sphi2, real cphi2, real k1);
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public:
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/**
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* Constructor with a single standard parallel.
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*
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* @param[in] a equatorial radius of ellipsoid (meters).
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* @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
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* Negative \e f gives a prolate ellipsoid.
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* @param[in] stdlat standard parallel (degrees), the circle of tangency.
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* @param[in] k0 scale on the standard parallel.
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* @exception GeographicErr if \e a, (1 − \e f) \e a, or \e k0 is
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* not positive.
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* @exception GeographicErr if \e stdlat is not in [−90°,
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* 90°].
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**********************************************************************/
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LambertConformalConic(real a, real f, real stdlat, real k0);
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/**
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* Constructor with two standard parallels.
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*
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* @param[in] a equatorial radius of ellipsoid (meters).
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* @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
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* Negative \e f gives a prolate ellipsoid.
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* @param[in] stdlat1 first standard parallel (degrees).
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* @param[in] stdlat2 second standard parallel (degrees).
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* @param[in] k1 scale on the standard parallels.
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* @exception GeographicErr if \e a, (1 − \e f) \e a, or \e k1 is
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* not positive.
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* @exception GeographicErr if \e stdlat1 or \e stdlat2 is not in
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* [−90°, 90°], or if either \e stdlat1 or \e
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* stdlat2 is a pole and \e stdlat1 is not equal \e stdlat2.
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**********************************************************************/
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LambertConformalConic(real a, real f, real stdlat1, real stdlat2, real k1);
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/**
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* Constructor with two standard parallels specified by sines and cosines.
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*
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* @param[in] a equatorial radius of ellipsoid (meters).
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* @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
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* Negative \e f gives a prolate ellipsoid.
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* @param[in] sinlat1 sine of first standard parallel.
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* @param[in] coslat1 cosine of first standard parallel.
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* @param[in] sinlat2 sine of second standard parallel.
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* @param[in] coslat2 cosine of second standard parallel.
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* @param[in] k1 scale on the standard parallels.
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* @exception GeographicErr if \e a, (1 − \e f) \e a, or \e k1 is
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* not positive.
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* @exception GeographicErr if \e stdlat1 or \e stdlat2 is not in
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* [−90°, 90°], or if either \e stdlat1 or \e
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* stdlat2 is a pole and \e stdlat1 is not equal \e stdlat2.
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*
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* This allows parallels close to the poles to be specified accurately.
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* This routine computes the latitude of origin and the scale at this
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* latitude. In the case where \e lat1 and \e lat2 are different, the
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* errors in this routines are as follows: if \e dlat = abs(\e lat2 −
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* \e lat1) ≤ 160° and max(abs(\e lat1), abs(\e lat2)) ≤ 90
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* − min(0.0002, 2.2 × 10<sup>−6</sup>(180 − \e
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* dlat), 6 × 10<sup>−8</sup> <i>dlat</i><sup>2</sup>) (in
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* degrees), then the error in the latitude of origin is less than 4.5
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* × 10<sup>−14</sup>d and the relative error in the scale is
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* less than 7 × 10<sup>−15</sup>.
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**********************************************************************/
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LambertConformalConic(real a, real f,
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real sinlat1, real coslat1,
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real sinlat2, real coslat2,
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real k1);
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/**
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* Set the scale for the projection.
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*
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* @param[in] lat (degrees).
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* @param[in] k scale at latitude \e lat (default 1).
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* @exception GeographicErr \e k is not positive.
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* @exception GeographicErr if \e lat is not in [−90°,
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* 90°].
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**********************************************************************/
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void SetScale(real lat, real k = real(1));
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/**
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* Forward projection, from geographic to Lambert conformal conic.
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*
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* @param[in] lon0 central meridian longitude (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] gamma meridian convergence at point (degrees).
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* @param[out] k scale of projection at point.
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*
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* The latitude origin is given by LambertConformalConic::LatitudeOrigin().
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* No false easting or northing is added and \e lat should be in the range
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* [−90°, 90°]. The error in the projection is less than
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* about 10 nm (10 nanometers), true distance, and the errors in the
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* meridian convergence and scale are consistent with this. The values of
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* \e x and \e y returned for points which project to infinity (i.e., one
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* or both of the poles) will be large but finite.
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**********************************************************************/
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void Forward(real lon0, real lat, real lon,
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real& x, real& y, real& gamma, real& k) const;
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/**
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* Reverse projection, from Lambert conformal conic to geographic.
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*
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* @param[in] lon0 central meridian longitude (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] gamma meridian convergence at point (degrees).
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* @param[out] k scale of projection at point.
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*
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* The latitude origin is given by LambertConformalConic::LatitudeOrigin().
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* No false easting or northing is added. The value of \e lon returned is
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* in the range [−180°, 180°]. The error in the projection
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* is less than about 10 nm (10 nanometers), true distance, and the errors
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* in the meridian convergence and scale are consistent with this.
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**********************************************************************/
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void Reverse(real lon0, real x, real y,
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real& lat, real& lon, real& gamma, real& k) const;
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/**
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* LambertConformalConic::Forward without returning the convergence and
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* scale.
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**********************************************************************/
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void Forward(real lon0, real lat, real lon,
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real& x, real& y) const {
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real gamma, k;
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Forward(lon0, lat, lon, x, y, gamma, k);
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}
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/**
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* LambertConformalConic::Reverse without returning the convergence and
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* scale.
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**********************************************************************/
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void Reverse(real lon0, real x, real y,
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real& lat, real& lon) const {
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real gamma, k;
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Reverse(lon0, x, y, lat, lon, gamma, k);
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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 used in the constructor.
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**********************************************************************/
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Math::real EquatorialRadius() const { return _a; }
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/**
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* @return \e f the flattening of the ellipsoid. This is the
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* value used in the constructor.
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**********************************************************************/
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Math::real Flattening() const { return _f; }
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/**
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* @return latitude of the origin for the projection (degrees).
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*
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* This is the latitude of minimum scale and equals the \e stdlat in the
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* 1-parallel constructor and lies between \e stdlat1 and \e stdlat2 in the
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* 2-parallel constructors.
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**********************************************************************/
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Math::real OriginLatitude() const { return _lat0; }
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/**
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* @return central scale for the projection. This is the scale on the
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* latitude of origin.
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**********************************************************************/
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Math::real CentralScale() const { return _k0; }
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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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* A global instantiation of LambertConformalConic with the WGS84
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* ellipsoid, \e stdlat = 0, and \e k0 = 1. This degenerates to the
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* Mercator projection.
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**********************************************************************/
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static const LambertConformalConic& Mercator();
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};
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} // namespace GeographicLib
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#endif // GEOGRAPHICLIB_LAMBERTCONFORMALCONIC_HPP
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