GeographicLib 1.52
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JacobiConformal.hpp
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1/**
2 * \file JacobiConformal.hpp
3 * \brief Header for GeographicLib::JacobiConformal class
4 *
5 * <b>NOTE:</b> This is just sample code. It is not part of GeographicLib
6 * itself.
7 *
8 * Copyright (c) Charles Karney (2014-2020) <charles@karney.com> and licensed
9 * under the MIT/X11 License. For more information, see
10 * https://geographiclib.sourceforge.io/
11 **********************************************************************/
12
14
15namespace GeographicLib {
16 /**
17 * \brief Jacobi's conformal projection of a triaxial ellipsoid
18 *
19 * <b>NOTE:</b> This is just sample code. It is not part of GeographicLib
20 * itself.
21 *
22 * This is a conformal projection of the ellipsoid to a plane in which
23 * the grid lines are straight; see Jacobi,
24 * <a href="https://books.google.com/books?id=ryEOAAAAQAAJ&pg=PA212">
25 * Vorlesungen &uuml;ber Dynamik, &sect;28</a>. The constructor takes the
26 * semi-axes of the ellipsoid (which must be in order). Member functions map
27 * the ellipsoidal coordinates &omega; and &beta; separately to \e x and \e
28 * y. Jacobi's coordinates have been multiplied by
29 * (<i>a</i><sup>2</sup>&minus;<i>c</i><sup>2</sup>)<sup>1/2</sup> /
30 * (2<i>b</i>) so that the customary results are returned in the cases of
31 * a sphere or an ellipsoid of revolution.
32 *
33 * The ellipsoid is oriented so that the large principal ellipse, \f$Z=0\f$,
34 * is the equator, \f$\beta=0\f$, while the small principal ellipse,
35 * \f$Y=0\f$, is the prime meridian, \f$\omega=0\f$. The four umbilic
36 * points, \f$\left|\omega\right| = \left|\beta\right| = \frac12\pi\f$, lie
37 * on middle principal ellipse in the plane \f$X=0\f$.
38 *
39 * For more information on this projection, see \ref jacobi.
40 **********************************************************************/
42 typedef Math::real real;
43 real _a, _b, _c, _ab2, _bc2, _ac2;
44 EllipticFunction _ex, _ey;
45 static void norm(real& x, real& y) {
46 using std::hypot;
47 real z = hypot(x, y); x /= z; y /= z;
48 }
49 public:
50 /**
51 * Constructor for a trixial ellipsoid with semi-axes.
52 *
53 * @param[in] a the largest semi-axis.
54 * @param[in] b the middle semi-axis.
55 * @param[in] c the smallest semi-axis.
56 *
57 * The semi-axes must satisfy \e a &ge; \e b &ge; \e c > 0 and \e a >
58 * \e c. This form of the constructor cannot be used to specify a
59 * sphere (use the next constructor).
60 **********************************************************************/
61 JacobiConformal(real a, real b, real c)
62 : _a(a), _b(b), _c(c)
63 , _ab2((_a - _b) * (_a + _b))
64 , _bc2((_b - _c) * (_b + _c))
65 , _ac2((_a - _c) * (_a + _c))
66 , _ex(_ab2 / _ac2 * Math::sq(_c / _b), -_ab2 / Math::sq(_b),
67 _bc2 / _ac2 * Math::sq(_a / _b), Math::sq(_a / _b))
68 , _ey(_bc2 / _ac2 * Math::sq(_a / _b), +_bc2 / Math::sq(_b),
69 _ab2 / _ac2 * Math::sq(_c / _b), Math::sq(_c / _b))
70 {
71 using std::isfinite;
72 if (!(isfinite(_a) && _a >= _b && _b >= _c && _c > 0))
73 throw GeographicErr("JacobiConformal: axes are not in order");
74 if (!(_a > _c))
75 throw GeographicErr
76 ("JacobiConformal: use alternate constructor for sphere");
77 }
78 /**
79 * Alternate constructor for a triaxial ellipsoid.
80 *
81 * @param[in] a the largest semi-axis.
82 * @param[in] b the middle semi-axis.
83 * @param[in] c the smallest semi-axis.
84 * @param[in] ab the relative magnitude of \e a &minus; \e b.
85 * @param[in] bc the relative magnitude of \e b &minus; \e c.
86 *
87 * This form can be used to specify a sphere. The semi-axes must
88 * satisfy \e a &ge; \e b &ge; c > 0. The ratio \e ab : \e bc must equal
89 * (<i>a</i>&minus;<i>b</i>) : (<i>b</i>&minus;<i>c</i>) with \e ab
90 * &ge; 0, \e bc &ge; 0, and \e ab + \e bc > 0.
91 **********************************************************************/
92 JacobiConformal(real a, real b, real c, real ab, real bc)
93 : _a(a), _b(b), _c(c)
94 , _ab2(ab * (_a + _b))
95 , _bc2(bc * (_b + _c))
96 , _ac2(_ab2 + _bc2)
97 , _ex(_ab2 / _ac2 * Math::sq(_c / _b),
98 -(_a - _b) * (_a + _b) / Math::sq(_b),
99 _bc2 / _ac2 * Math::sq(_a / _b), Math::sq(_a / _b))
100 , _ey(_bc2 / _ac2 * Math::sq(_a / _b),
101 +(_b - _c) * (_b + _c) / Math::sq(_b),
102 _ab2 / _ac2 * Math::sq(_c / _b), Math::sq(_c / _b))
103 {
104 using std::isfinite;
105 if (!(isfinite(_a) && _a >= _b && _b >= _c && _c > 0 &&
106 ab >= 0 && bc >= 0))
107 throw GeographicErr("JacobiConformal: axes are not in order");
108 if (!(ab + bc > 0 && isfinite(_ac2)))
109 throw GeographicErr("JacobiConformal: ab + bc must be positive");
110 }
111 /**
112 * @return the quadrant length in the \e x direction.
113 **********************************************************************/
114 Math::real x() const { return Math::sq(_a / _b) * _ex.Pi(); }
115 /**
116 * The \e x projection.
117 *
118 * @param[in] somg sin(&omega;).
119 * @param[in] comg cos(&omega;).
120 * @return \e x.
121 **********************************************************************/
122 Math::real x(real somg, real comg) const {
123 real somg1 = _b * somg, comg1 = _a * comg; norm(somg1, comg1);
124 return Math::sq(_a / _b)
125 * _ex.Pi(somg1, comg1, _ex.Delta(somg1, comg1));
126 }
127 /**
128 * The \e x projection.
129 *
130 * @param[in] omg &omega; (in degrees).
131 * @return \e x (in degrees).
132 *
133 * &omega; must be in (&minus;180&deg;, 180&deg;].
134 **********************************************************************/
135 Math::real x(real omg) const {
136 real somg, comg;
137 Math::sincosd(omg, somg, comg);
138 return x(somg, comg) / Math::degree();
139 }
140 /**
141 * @return the quadrant length in the \e y direction.
142 **********************************************************************/
143 Math::real y() const { return Math::sq(_c / _b) * _ey.Pi(); }
144 /**
145 * The \e y projection.
146 *
147 * @param[in] sbet sin(&beta;).
148 * @param[in] cbet cos(&beta;).
149 * @return \e y.
150 **********************************************************************/
151 Math::real y(real sbet, real cbet) const {
152 real sbet1 = _b * sbet, cbet1 = _c * cbet; norm(sbet1, cbet1);
153 return Math::sq(_c / _b)
154 * _ey.Pi(sbet1, cbet1, _ey.Delta(sbet1, cbet1));
155 }
156 /**
157 * The \e y projection.
158 *
159 * @param[in] bet &beta; (in degrees).
160 * @return \e y (in degrees).
161 *
162 * &beta; must be in (&minus;180&deg;, 180&deg;].
163 **********************************************************************/
164 Math::real y(real bet) const {
165 real sbet, cbet;
166 Math::sincosd(bet, sbet, cbet);
167 return y(sbet, cbet) / Math::degree();
168 }
169 };
170
171} // namespace GeographicLib
Header for GeographicLib::EllipticFunction class.
Elliptic integrals and functions.
Math::real Delta(real sn, real cn) const
Exception handling for GeographicLib.
Definition: Constants.hpp:315
Jacobi's conformal projection of a triaxial ellipsoid.
Math::real y(real bet) const
Math::real x(real omg) const
JacobiConformal(real a, real b, real c)
Math::real y(real sbet, real cbet) const
Math::real x(real somg, real comg) const
JacobiConformal(real a, real b, real c, real ab, real bc)
Mathematical functions needed by GeographicLib.
Definition: Math.hpp:76
static T degree()
Definition: Math.hpp:159
static void sincosd(T x, T &sinx, T &cosx)
Definition: Math.cpp:126
static T sq(T x)
Definition: Math.hpp:171
Namespace for GeographicLib.
Definition: Accumulator.cpp:12