GeographicLib  2.1.2
LambertConformalConic.hpp
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1 /**
2  * \file LambertConformalConic.hpp
3  * \brief Header for GeographicLib::LambertConformalConic class
4  *
5  * Copyright (c) Charles Karney (2010-2022) <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_LAMBERTCONFORMALCONIC_HPP)
11 #define GEOGRAPHICLIB_LAMBERTCONFORMALCONIC_HPP 1
12 
14 
15 namespace GeographicLib {
16 
17  /**
18  * \brief Lambert conformal conic projection
19  *
20  * Implementation taken from the report,
21  * - J. P. Snyder,
22  * <a href="http://pubs.er.usgs.gov/usgspubs/pp/pp1395"> Map Projections: A
23  * Working Manual</a>, USGS Professional Paper 1395 (1987),
24  * pp. 107--109.
25  *
26  * This is a implementation of the equations in Snyder except that divided
27  * differences have been used to transform the expressions into ones which
28  * may be evaluated accurately and that Newton's method is used to invert the
29  * projection. In this implementation, the projection correctly becomes the
30  * Mercator projection or the polar stereographic projection when the
31  * standard latitude is the equator or a pole. The accuracy of the
32  * projections is about 10 nm (10 nanometers).
33  *
34  * The ellipsoid parameters, the standard parallels, and the scale on the
35  * standard parallels are set in the constructor. Internally, the case with
36  * two standard parallels is converted into a single standard parallel, the
37  * latitude of tangency (also the latitude of minimum scale), with a scale
38  * specified on this parallel. This latitude is also used as the latitude of
39  * origin which is returned by LambertConformalConic::OriginLatitude. The
40  * scale on the latitude of origin is given by
41  * LambertConformalConic::CentralScale. The case with two distinct standard
42  * parallels where one is a pole is singular and is disallowed. The central
43  * meridian (which is a trivial shift of the longitude) is specified as the
44  * \e lon0 argument of the LambertConformalConic::Forward and
45  * LambertConformalConic::Reverse functions.
46  *
47  * This class also returns the meridian convergence \e gamma and scale \e k.
48  * The meridian convergence is the bearing of grid north (the \e y axis)
49  * measured clockwise from true north.
50  *
51  * There is no provision in this
52  * class for specifying a false easting or false northing or a different
53  * latitude of origin. However these are can be simply included by the
54  * calling function. For example the Pennsylvania South state coordinate
55  * system (<a href="https://www.spatialreference.org/ref/epsg/3364/">
56  * EPSG:3364</a>) is obtained by:
57  * \include example-LambertConformalConic.cpp
58  *
59  * <a href="ConicProj.1.html">ConicProj</a> is a command-line utility
60  * providing access to the functionality of LambertConformalConic and
61  * AlbersEqualArea.
62  **********************************************************************/
64  private:
65  typedef Math::real real;
66  real eps_, epsx_, ahypover_;
67  real _a, _f, _fm, _e2, _es;
68  real _sign, _n, _nc, _t0nm1, _scale, _lat0, _k0;
69  real _scbet0, _tchi0, _scchi0, _psi0, _nrho0, _drhomax;
70  static const int numit_ = 5;
71  static real hyp(real x) {
72  using std::hypot;
73  return hypot(real(1), x);
74  }
75  // Divided differences
76  // Definition: Df(x,y) = (f(x)-f(y))/(x-y)
77  // See:
78  // W. M. Kahan and R. J. Fateman,
79  // Symbolic computation of divided differences,
80  // SIGSAM Bull. 33(2), 7-28 (1999)
81  // https://doi.org/10.1145/334714.334716
82  // http://www.cs.berkeley.edu/~fateman/papers/divdiff.pdf
83  //
84  // General rules
85  // h(x) = f(g(x)): Dh(x,y) = Df(g(x),g(y))*Dg(x,y)
86  // h(x) = f(x)*g(x):
87  // Dh(x,y) = Df(x,y)*g(x) + Dg(x,y)*f(y)
88  // = Df(x,y)*g(y) + Dg(x,y)*f(x)
89  // = Df(x,y)*(g(x)+g(y))/2 + Dg(x,y)*(f(x)+f(y))/2
90  //
91  // hyp(x) = sqrt(1+x^2): Dhyp(x,y) = (x+y)/(hyp(x)+hyp(y))
92  static real Dhyp(real x, real y, real hx, real hy)
93  // hx = hyp(x)
94  { return (x + y) / (hx + hy); }
95  // sn(x) = x/sqrt(1+x^2): Dsn(x,y) = (x+y)/((sn(x)+sn(y))*(1+x^2)*(1+y^2))
96  static real Dsn(real x, real y, real sx, real sy) {
97  // sx = x/hyp(x)
98  real t = x * y;
99  return t > 0 ? (x + y) * Math::sq( (sx * sy)/t ) / (sx + sy) :
100  (x - y != 0 ? (sx - sy) / (x - y) : 1);
101  }
102  // Dlog1p(x,y) = log1p((x-y)/(1+y))/(x-y)
103  static real Dlog1p(real x, real y) {
104  using std::log1p;
105  real t = x - y; if (t < 0) { t = -t; y = x; }
106  return t != 0 ? log1p(t / (1 + y)) / t : 1 / (1 + x);
107  }
108  // Dexp(x,y) = exp((x+y)/2) * 2*sinh((x-y)/2)/(x-y)
109  static real Dexp(real x, real y) {
110  using std::sinh; using std::exp;
111  real t = (x - y)/2;
112  return (t != 0 ? sinh(t)/t : 1) * exp((x + y)/2);
113  }
114  // Dsinh(x,y) = 2*sinh((x-y)/2)/(x-y) * cosh((x+y)/2)
115  // cosh((x+y)/2) = (c+sinh(x)*sinh(y)/c)/2
116  // c=sqrt((1+cosh(x))*(1+cosh(y)))
117  // cosh((x+y)/2) = sqrt( (sinh(x)*sinh(y) + cosh(x)*cosh(y) + 1)/2 )
118  static real Dsinh(real x, real y, real sx, real sy, real cx, real cy)
119  // sx = sinh(x), cx = cosh(x)
120  {
121  // real t = (x - y)/2, c = sqrt((1 + cx) * (1 + cy));
122  // return (t ? sinh(t)/t : real(1)) * (c + sx * sy / c) /2;
123  using std::sinh; using std::sqrt;
124  real t = (x - y)/2;
125  return (t != 0 ? sinh(t)/t : 1) * sqrt((sx * sy + cx * cy + 1) /2);
126  }
127  // Dasinh(x,y) = asinh((x-y)*(x+y)/(x*sqrt(1+y^2)+y*sqrt(1+x^2)))/(x-y)
128  // = asinh((x*sqrt(1+y^2)-y*sqrt(1+x^2)))/(x-y)
129  static real Dasinh(real x, real y, real hx, real hy) {
130  // hx = hyp(x)
131  using std::asinh;
132  real t = x - y;
133  return t != 0 ?
134  asinh(x*y > 0 ? t * (x + y) / (x*hy + y*hx) : x*hy - y*hx) / t :
135  1 / hx;
136  }
137  // Deatanhe(x,y) = eatanhe((x-y)/(1-e^2*x*y))/(x-y)
138  real Deatanhe(real x, real y) const {
139  real t = x - y, d = 1 - _e2 * x * y;
140  return t != 0 ? Math::eatanhe(t / d, _es) / t : _e2 / d;
141  }
142  void Init(real sphi1, real cphi1, real sphi2, real cphi2, real k1);
143  public:
144 
145  /**
146  * Constructor with a single standard parallel.
147  *
148  * @param[in] a equatorial radius of ellipsoid (meters).
149  * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
150  * Negative \e f gives a prolate ellipsoid.
151  * @param[in] stdlat standard parallel (degrees), the circle of tangency.
152  * @param[in] k0 scale on the standard parallel.
153  * @exception GeographicErr if \e a, (1 &minus; \e f) \e a, or \e k0 is
154  * not positive.
155  * @exception GeographicErr if \e stdlat is not in [&minus;90&deg;,
156  * 90&deg;].
157  **********************************************************************/
158  LambertConformalConic(real a, real f, real stdlat, real k0);
159 
160  /**
161  * Constructor with two standard parallels.
162  *
163  * @param[in] a equatorial radius of ellipsoid (meters).
164  * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
165  * Negative \e f gives a prolate ellipsoid.
166  * @param[in] stdlat1 first standard parallel (degrees).
167  * @param[in] stdlat2 second standard parallel (degrees).
168  * @param[in] k1 scale on the standard parallels.
169  * @exception GeographicErr if \e a, (1 &minus; \e f) \e a, or \e k1 is
170  * not positive.
171  * @exception GeographicErr if \e stdlat1 or \e stdlat2 is not in
172  * [&minus;90&deg;, 90&deg;], or if either \e stdlat1 or \e
173  * stdlat2 is a pole and \e stdlat1 is not equal \e stdlat2.
174  **********************************************************************/
175  LambertConformalConic(real a, real f, real stdlat1, real stdlat2, real k1);
176 
177  /**
178  * Constructor with two standard parallels specified by sines and cosines.
179  *
180  * @param[in] a equatorial radius of ellipsoid (meters).
181  * @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
182  * Negative \e f gives a prolate ellipsoid.
183  * @param[in] sinlat1 sine of first standard parallel.
184  * @param[in] coslat1 cosine of first standard parallel.
185  * @param[in] sinlat2 sine of second standard parallel.
186  * @param[in] coslat2 cosine of second standard parallel.
187  * @param[in] k1 scale on the standard parallels.
188  * @exception GeographicErr if \e a, (1 &minus; \e f) \e a, or \e k1 is
189  * not positive.
190  * @exception GeographicErr if \e stdlat1 or \e stdlat2 is not in
191  * [&minus;90&deg;, 90&deg;], or if either \e stdlat1 or \e
192  * stdlat2 is a pole and \e stdlat1 is not equal \e stdlat2.
193  *
194  * This allows parallels close to the poles to be specified accurately.
195  * This routine computes the latitude of origin and the scale at this
196  * latitude. In the case where \e lat1 and \e lat2 are different, the
197  * errors in this routines are as follows: if \e dlat = abs(\e lat2 &minus;
198  * \e lat1) &le; 160&deg; and max(abs(\e lat1), abs(\e lat2)) &le; 90
199  * &minus; min(0.0002, 2.2 &times; 10<sup>&minus;6</sup>(180 &minus; \e
200  * dlat), 6 &times 10<sup>&minus;8</sup> <i>dlat</i><sup>2</sup>) (in
201  * degrees), then the error in the latitude of origin is less than 4.5
202  * &times; 10<sup>&minus;14</sup>d and the relative error in the scale is
203  * less than 7 &times; 10<sup>&minus;15</sup>.
204  **********************************************************************/
205  LambertConformalConic(real a, real f,
206  real sinlat1, real coslat1,
207  real sinlat2, real coslat2,
208  real k1);
209 
210  /**
211  * Set the scale for the projection.
212  *
213  * @param[in] lat (degrees).
214  * @param[in] k scale at latitude \e lat (default 1).
215  * @exception GeographicErr \e k is not positive.
216  * @exception GeographicErr if \e lat is not in [&minus;90&deg;,
217  * 90&deg;].
218  **********************************************************************/
219  void SetScale(real lat, real k = real(1));
220 
221  /**
222  * Forward projection, from geographic to Lambert conformal conic.
223  *
224  * @param[in] lon0 central meridian longitude (degrees).
225  * @param[in] lat latitude of point (degrees).
226  * @param[in] lon longitude of point (degrees).
227  * @param[out] x easting of point (meters).
228  * @param[out] y northing of point (meters).
229  * @param[out] gamma meridian convergence at point (degrees).
230  * @param[out] k scale of projection at point.
231  *
232  * The latitude origin is given by LambertConformalConic::LatitudeOrigin().
233  * No false easting or northing is added and \e lat should be in the range
234  * [&minus;90&deg;, 90&deg;]. The error in the projection is less than
235  * about 10 nm (10 nanometers), true distance, and the errors in the
236  * meridian convergence and scale are consistent with this. The values of
237  * \e x and \e y returned for points which project to infinity (i.e., one
238  * or both of the poles) will be large but finite.
239  **********************************************************************/
240  void Forward(real lon0, real lat, real lon,
241  real& x, real& y, real& gamma, real& k) const;
242 
243  /**
244  * Reverse projection, from Lambert conformal conic to geographic.
245  *
246  * @param[in] lon0 central meridian longitude (degrees).
247  * @param[in] x easting of point (meters).
248  * @param[in] y northing of point (meters).
249  * @param[out] lat latitude of point (degrees).
250  * @param[out] lon longitude of point (degrees).
251  * @param[out] gamma meridian convergence at point (degrees).
252  * @param[out] k scale of projection at point.
253  *
254  * The latitude origin is given by LambertConformalConic::LatitudeOrigin().
255  * No false easting or northing is added. The value of \e lon returned is
256  * in the range [&minus;180&deg;, 180&deg;]. The error in the projection
257  * is less than about 10 nm (10 nanometers), true distance, and the errors
258  * in the meridian convergence and scale are consistent with this.
259  **********************************************************************/
260  void Reverse(real lon0, real x, real y,
261  real& lat, real& lon, real& gamma, real& k) const;
262 
263  /**
264  * LambertConformalConic::Forward without returning the convergence and
265  * scale.
266  **********************************************************************/
267  void Forward(real lon0, real lat, real lon,
268  real& x, real& y) const {
269  real gamma, k;
270  Forward(lon0, lat, lon, x, y, gamma, k);
271  }
272 
273  /**
274  * LambertConformalConic::Reverse without returning the convergence and
275  * scale.
276  **********************************************************************/
277  void Reverse(real lon0, real x, real y,
278  real& lat, real& lon) const {
279  real gamma, k;
280  Reverse(lon0, x, y, lat, lon, gamma, k);
281  }
282 
283  /** \name Inspector functions
284  **********************************************************************/
285  ///@{
286  /**
287  * @return \e a the equatorial radius of the ellipsoid (meters). This is
288  * the value used in the constructor.
289  **********************************************************************/
290  Math::real EquatorialRadius() const { return _a; }
291 
292  /**
293  * @return \e f the flattening of the ellipsoid. This is the
294  * value used in the constructor.
295  **********************************************************************/
296  Math::real Flattening() const { return _f; }
297 
298  /**
299  * @return latitude of the origin for the projection (degrees).
300  *
301  * This is the latitude of minimum scale and equals the \e stdlat in the
302  * 1-parallel constructor and lies between \e stdlat1 and \e stdlat2 in the
303  * 2-parallel constructors.
304  **********************************************************************/
305  Math::real OriginLatitude() const { return _lat0; }
306 
307  /**
308  * @return central scale for the projection. This is the scale on the
309  * latitude of origin.
310  **********************************************************************/
311  Math::real CentralScale() const { return _k0; }
312  ///@}
313 
314  /**
315  * A global instantiation of LambertConformalConic with the WGS84
316  * ellipsoid, \e stdlat = 0, and \e k0 = 1. This degenerates to the
317  * Mercator projection.
318  **********************************************************************/
319  static const LambertConformalConic& Mercator();
320  };
321 
322 } // namespace GeographicLib
323 
324 #endif // GEOGRAPHICLIB_LAMBERTCONFORMALCONIC_HPP
Header for GeographicLib::Constants class.
#define GEOGRAPHICLIB_EXPORT
Definition: Constants.hpp:67
GeographicLib::Math::real real
Definition: GeodSolve.cpp:31
Lambert conformal conic projection.
void Forward(real lon0, real lat, real lon, real &x, real &y) const
void Reverse(real lon0, real x, real y, real &lat, real &lon) const
static T sq(T x)
Definition: Math.hpp:212
static T eatanhe(T x, T es)
Definition: Math.cpp:205
Namespace for GeographicLib.
Definition: Accumulator.cpp:12