/** * \file LambertConformalConic.hpp * \brief Header for GeographicLib::LambertConformalConic class * * Copyright (c) Charles Karney (2010-2020) and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ **********************************************************************/ #if !defined(GEOGRAPHICLIB_LAMBERTCONFORMALCONIC_HPP) #define GEOGRAPHICLIB_LAMBERTCONFORMALCONIC_HPP 1 #include namespace GeographicLib { /** * \brief Lambert conformal conic projection * * Implementation taken from the report, * - J. P. Snyder, * Map Projections: A * Working Manual, USGS Professional Paper 1395 (1987), * pp. 107--109. * * This is a implementation of the equations in Snyder except that divided * differences have been used to transform the expressions into ones which * may be evaluated accurately and that Newton's method is used to invert the * projection. In this implementation, the projection correctly becomes the * Mercator projection or the polar stereographic projection when the * standard latitude is the equator or a pole. The accuracy of the * projections is about 10 nm (10 nanometers). * * The ellipsoid parameters, the standard parallels, and the scale on the * standard parallels are set in the constructor. Internally, the case with * two standard parallels is converted into a single standard parallel, the * latitude of tangency (also the latitude of minimum scale), with a scale * specified on this parallel. This latitude is also used as the latitude of * origin which is returned by LambertConformalConic::OriginLatitude. The * scale on the latitude of origin is given by * LambertConformalConic::CentralScale. The case with two distinct standard * parallels where one is a pole is singular and is disallowed. The central * meridian (which is a trivial shift of the longitude) is specified as the * \e lon0 argument of the LambertConformalConic::Forward and * LambertConformalConic::Reverse functions. * * This class also returns the meridian convergence \e gamma and scale \e k. * The meridian convergence is the bearing of grid north (the \e y axis) * measured clockwise from true north. * * There is no provision in this * class for specifying a false easting or false northing or a different * latitude of origin. However these are can be simply included by the * calling function. For example the Pennsylvania South state coordinate * system ( * EPSG:3364) is obtained by: * \include example-LambertConformalConic.cpp * * ConicProj is a command-line utility * providing access to the functionality of LambertConformalConic and * AlbersEqualArea. **********************************************************************/ class GEOGRAPHICLIB_EXPORT LambertConformalConic { private: typedef Math::real real; real eps_, epsx_, ahypover_; real _a, _f, _fm, _e2, _es; real _sign, _n, _nc, _t0nm1, _scale, _lat0, _k0; real _scbet0, _tchi0, _scchi0, _psi0, _nrho0, _drhomax; static const int numit_ = 5; static real hyp(real x) { using std::hypot; return hypot(real(1), x); } // Divided differences // Definition: Df(x,y) = (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) // https://doi.org/10.1145/334714.334716 // http://www.cs.berkeley.edu/~fateman/papers/divdiff.pdf // // General rules // h(x) = f(g(x)): Dh(x,y) = Df(g(x),g(y))*Dg(x,y) // h(x) = f(x)*g(x): // Dh(x,y) = Df(x,y)*g(x) + Dg(x,y)*f(y) // = Df(x,y)*g(y) + Dg(x,y)*f(x) // = Df(x,y)*(g(x)+g(y))/2 + Dg(x,y)*(f(x)+f(y))/2 // // hyp(x) = sqrt(1+x^2): Dhyp(x,y) = (x+y)/(hyp(x)+hyp(y)) static real Dhyp(real x, real y, real hx, real hy) // hx = hyp(x) { return (x + y) / (hx + hy); } // sn(x) = x/sqrt(1+x^2): Dsn(x,y) = (x+y)/((sn(x)+sn(y))*(1+x^2)*(1+y^2)) static real Dsn(real x, real y, real sx, real sy) { // sx = x/hyp(x) real t = x * y; return t > 0 ? (x + y) * Math::sq( (sx * sy)/t ) / (sx + sy) : (x - y != 0 ? (sx - sy) / (x - y) : 1); } // Dlog1p(x,y) = log1p((x-y)/(1+y))/(x-y) static real Dlog1p(real x, real y) { using std::log1p; real t = x - y; if (t < 0) { t = -t; y = x; } return t != 0 ? log1p(t / (1 + y)) / t : 1 / (1 + x); } // Dexp(x,y) = exp((x+y)/2) * 2*sinh((x-y)/2)/(x-y) static real Dexp(real x, real y) { using std::sinh; using std::exp; real t = (x - y)/2; return (t != 0 ? sinh(t)/t : 1) * exp((x + y)/2); } // Dsinh(x,y) = 2*sinh((x-y)/2)/(x-y) * cosh((x+y)/2) // cosh((x+y)/2) = (c+sinh(x)*sinh(y)/c)/2 // c=sqrt((1+cosh(x))*(1+cosh(y))) // cosh((x+y)/2) = sqrt( (sinh(x)*sinh(y) + cosh(x)*cosh(y) + 1)/2 ) static real Dsinh(real x, real y, real sx, real sy, real cx, real cy) // sx = sinh(x), cx = cosh(x) { // real t = (x - y)/2, c = sqrt((1 + cx) * (1 + cy)); // return (t ? sinh(t)/t : real(1)) * (c + sx * sy / c) /2; using std::sinh; using std::sqrt; real t = (x - y)/2; return (t != 0 ? sinh(t)/t : 1) * sqrt((sx * sy + cx * cy + 1) /2); } // Dasinh(x,y) = asinh((x-y)*(x+y)/(x*sqrt(1+y^2)+y*sqrt(1+x^2)))/(x-y) // = asinh((x*sqrt(1+y^2)-y*sqrt(1+x^2)))/(x-y) static real Dasinh(real x, real y, real hx, real hy) { // hx = hyp(x) using std::asinh; real t = x - y; return t != 0 ? asinh(x*y > 0 ? t * (x + y) / (x*hy + y*hx) : x*hy - y*hx) / t : 1 / hx; } // Deatanhe(x,y) = eatanhe((x-y)/(1-e^2*x*y))/(x-y) real Deatanhe(real x, real y) const { real t = x - y, d = 1 - _e2 * x * y; return t != 0 ? Math::eatanhe(t / d, _es) / t : _e2 / d; } void Init(real sphi1, real cphi1, real sphi2, real cphi2, real k1); public: /** * Constructor with a single standard parallel. * * @param[in] a equatorial radius of ellipsoid (meters). * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere. * Negative \e f gives a prolate ellipsoid. * @param[in] stdlat standard parallel (degrees), the circle of tangency. * @param[in] k0 scale on the standard parallel. * @exception GeographicErr if \e a, (1 − \e f) \e a, or \e k0 is * not positive. * @exception GeographicErr if \e stdlat is not in [−90°, * 90°]. **********************************************************************/ LambertConformalConic(real a, real f, real stdlat, real k0); /** * Constructor with two standard parallels. * * @param[in] a equatorial radius of ellipsoid (meters). * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere. * Negative \e f gives a prolate ellipsoid. * @param[in] stdlat1 first standard parallel (degrees). * @param[in] stdlat2 second standard parallel (degrees). * @param[in] k1 scale on the standard parallels. * @exception GeographicErr if \e a, (1 − \e f) \e a, or \e k1 is * not positive. * @exception GeographicErr if \e stdlat1 or \e stdlat2 is not in * [−90°, 90°], or if either \e stdlat1 or \e * stdlat2 is a pole and \e stdlat1 is not equal \e stdlat2. **********************************************************************/ LambertConformalConic(real a, real f, real stdlat1, real stdlat2, real k1); /** * Constructor with two standard parallels specified by sines and cosines. * * @param[in] a equatorial radius of ellipsoid (meters). * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere. * Negative \e f gives a prolate ellipsoid. * @param[in] sinlat1 sine of first standard parallel. * @param[in] coslat1 cosine of first standard parallel. * @param[in] sinlat2 sine of second standard parallel. * @param[in] coslat2 cosine of second standard parallel. * @param[in] k1 scale on the standard parallels. * @exception GeographicErr if \e a, (1 − \e f) \e a, or \e k1 is * not positive. * @exception GeographicErr if \e stdlat1 or \e stdlat2 is not in * [−90°, 90°], or if either \e stdlat1 or \e * stdlat2 is a pole and \e stdlat1 is not equal \e stdlat2. * * This allows parallels close to the poles to be specified accurately. * This routine computes the latitude of origin and the scale at this * latitude. In the case where \e lat1 and \e lat2 are different, the * errors in this routines are as follows: if \e dlat = abs(\e lat2 − * \e lat1) ≤ 160° and max(abs(\e lat1), abs(\e lat2)) ≤ 90 * − min(0.0002, 2.2 × 10−6(180 − \e * dlat), 6 × 10−8 dlat2) (in * degrees), then the error in the latitude of origin is less than 4.5 * × 10−14d and the relative error in the scale is * less than 7 × 10−15. **********************************************************************/ LambertConformalConic(real a, real f, real sinlat1, real coslat1, real sinlat2, real coslat2, real k1); /** * Set the scale for the projection. * * @param[in] lat (degrees). * @param[in] k scale at latitude \e lat (default 1). * @exception GeographicErr \e k is not positive. * @exception GeographicErr if \e lat is not in [−90°, * 90°]. **********************************************************************/ void SetScale(real lat, real k = real(1)); /** * Forward projection, from geographic to Lambert conformal conic. * * @param[in] lon0 central meridian longitude (degrees). * @param[in] lat latitude of point (degrees). * @param[in] lon longitude of point (degrees). * @param[out] x easting of point (meters). * @param[out] y northing of point (meters). * @param[out] gamma meridian convergence at point (degrees). * @param[out] k scale of projection at point. * * The latitude origin is given by LambertConformalConic::LatitudeOrigin(). * No false easting or northing is added and \e lat should be in the range * [−90°, 90°]. The error in the projection is less than * about 10 nm (10 nanometers), true distance, and the errors in the * meridian convergence and scale are consistent with this. The values of * \e x and \e y returned for points which project to infinity (i.e., one * or both of the poles) will be large but finite. **********************************************************************/ void Forward(real lon0, real lat, real lon, real& x, real& y, real& gamma, real& k) const; /** * Reverse projection, from Lambert conformal conic to geographic. * * @param[in] lon0 central meridian longitude (degrees). * @param[in] x easting of point (meters). * @param[in] y northing of point (meters). * @param[out] lat latitude of point (degrees). * @param[out] lon longitude of point (degrees). * @param[out] gamma meridian convergence at point (degrees). * @param[out] k scale of projection at point. * * The latitude origin is given by LambertConformalConic::LatitudeOrigin(). * No false easting or northing is added. The value of \e lon returned is * in the range [−180°, 180°]. The error in the projection * is less than about 10 nm (10 nanometers), true distance, and the errors * in the meridian convergence and scale are consistent with this. **********************************************************************/ void Reverse(real lon0, real x, real y, real& lat, real& lon, real& gamma, real& k) const; /** * LambertConformalConic::Forward without returning the convergence and * scale. **********************************************************************/ void Forward(real lon0, real lat, real lon, real& x, real& y) const { real gamma, k; Forward(lon0, lat, lon, x, y, gamma, k); } /** * LambertConformalConic::Reverse without returning the convergence and * scale. **********************************************************************/ void Reverse(real lon0, real x, real y, real& lat, real& lon) const { real gamma, k; Reverse(lon0, x, y, lat, lon, gamma, k); } /** \name Inspector functions **********************************************************************/ ///@{ /** * @return \e a the equatorial radius of the ellipsoid (meters). This is * the value used in the constructor. **********************************************************************/ Math::real EquatorialRadius() const { return _a; } /** * @return \e f the flattening of the ellipsoid. This is the * value used in the constructor. **********************************************************************/ Math::real Flattening() const { return _f; } /** * @return latitude of the origin for the projection (degrees). * * This is the latitude of minimum scale and equals the \e stdlat in the * 1-parallel constructor and lies between \e stdlat1 and \e stdlat2 in the * 2-parallel constructors. **********************************************************************/ Math::real OriginLatitude() const { return _lat0; } /** * @return central scale for the projection. This is the scale on the * latitude of origin. **********************************************************************/ Math::real CentralScale() const { return _k0; } /** * \deprecated An old name for EquatorialRadius(). **********************************************************************/ GEOGRAPHICLIB_DEPRECATED("Use EquatorialRadius()") Math::real MajorRadius() const { return EquatorialRadius(); } ///@} /** * A global instantiation of LambertConformalConic with the WGS84 * ellipsoid, \e stdlat = 0, and \e k0 = 1. This degenerates to the * Mercator projection. **********************************************************************/ static const LambertConformalConic& Mercator(); }; } // namespace GeographicLib #endif // GEOGRAPHICLIB_LAMBERTCONFORMALCONIC_HPP