/usr/include/GeographicLib/SphericalHarmonic.hpp is in libgeographic-dev 1.49-2.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 | /**
* \file SphericalHarmonic.hpp
* \brief Header for GeographicLib::SphericalHarmonic class
*
* Copyright (c) Charles Karney (2011) <charles@karney.com> and licensed under
* the MIT/X11 License. For more information, see
* https://geographiclib.sourceforge.io/
**********************************************************************/
#if !defined(GEOGRAPHICLIB_SPHERICALHARMONIC_HPP)
#define GEOGRAPHICLIB_SPHERICALHARMONIC_HPP 1
#include <vector>
#include <GeographicLib/Constants.hpp>
#include <GeographicLib/SphericalEngine.hpp>
#include <GeographicLib/CircularEngine.hpp>
namespace GeographicLib {
/**
* \brief Spherical harmonic series
*
* This class evaluates the spherical harmonic sum \verbatim
V(x, y, z) = sum(n = 0..N)[ q^(n+1) * sum(m = 0..n)[
(C[n,m] * cos(m*lambda) + S[n,m] * sin(m*lambda)) *
P[n,m](cos(theta)) ] ]
\endverbatim
* where
* - <i>p</i><sup>2</sup> = <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup>,
* - <i>r</i><sup>2</sup> = <i>p</i><sup>2</sup> + <i>z</i><sup>2</sup>,
* - \e q = <i>a</i>/<i>r</i>,
* - θ = atan2(\e p, \e z) = the spherical \e colatitude,
* - λ = atan2(\e y, \e x) = the longitude.
* - P<sub><i>nm</i></sub>(\e t) is the associated Legendre polynomial of
* degree \e n and order \e m.
*
* Two normalizations are supported for P<sub><i>nm</i></sub>
* - fully normalized denoted by SphericalHarmonic::FULL.
* - Schmidt semi-normalized denoted by SphericalHarmonic::SCHMIDT.
*
* Clenshaw summation is used for the sums over both \e n and \e m. This
* allows the computation to be carried out without the need for any
* temporary arrays. See SphericalEngine.cpp for more information on the
* implementation.
*
* References:
* - C. W. Clenshaw,
* <a href="https://doi.org/10.1090/S0025-5718-1955-0071856-0">
* A note on the summation of Chebyshev series</a>,
* %Math. Tables Aids Comput. 9(51), 118--120 (1955).
* - R. E. Deakin, Derivatives of the earth's potentials, Geomatics
* Research Australasia 68, 31--60, (June 1998).
* - W. A. Heiskanen and H. Moritz, Physical Geodesy, (Freeman, San
* Francisco, 1967). (See Sec. 1-14, for a definition of Pbar.)
* - S. A. Holmes and W. E. Featherstone,
* <a href="https://doi.org/10.1007/s00190-002-0216-2">
* A unified approach to the Clenshaw summation and the recursive
* computation of very high degree and order normalised associated Legendre
* functions</a>, J. Geodesy 76(5), 279--299 (2002).
* - C. C. Tscherning and K. Poder,
* <a href="http://cct.gfy.ku.dk/publ_cct/cct80.pdf">
* Some geodetic applications of Clenshaw summation</a>,
* Boll. Geod. Sci. Aff. 41(4), 349--375 (1982).
*
* Example of use:
* \include example-SphericalHarmonic.cpp
**********************************************************************/
class GEOGRAPHICLIB_EXPORT SphericalHarmonic {
public:
/**
* Supported normalizations for the associated Legendre polynomials.
**********************************************************************/
enum normalization {
/**
* Fully normalized associated Legendre polynomials.
*
* These are defined by
* <i>P</i><sub><i>nm</i></sub><sup>full</sup>(\e z)
* = (−1)<sup><i>m</i></sup>
* sqrt(\e k (2\e n + 1) (\e n − \e m)! / (\e n + \e m)!)
* <b>P</b><sub><i>n</i></sub><sup><i>m</i></sup>(\e z), where
* <b>P</b><sub><i>n</i></sub><sup><i>m</i></sup>(\e z) is Ferrers
* function (also known as the Legendre function on the cut or the
* associated Legendre polynomial) http://dlmf.nist.gov/14.7.E10 and \e k
* = 1 for \e m = 0 and \e k = 2 otherwise.
*
* The mean squared value of
* <i>P</i><sub><i>nm</i></sub><sup>full</sup>(cosθ)
* cos(<i>m</i>λ) and
* <i>P</i><sub><i>nm</i></sub><sup>full</sup>(cosθ)
* sin(<i>m</i>λ) over the sphere is 1.
*
* @hideinitializer
**********************************************************************/
FULL = SphericalEngine::FULL,
/**
* Schmidt semi-normalized associated Legendre polynomials.
*
* These are defined by
* <i>P</i><sub><i>nm</i></sub><sup>schmidt</sup>(\e z)
* = (−1)<sup><i>m</i></sup>
* sqrt(\e k (\e n − \e m)! / (\e n + \e m)!)
* <b>P</b><sub><i>n</i></sub><sup><i>m</i></sup>(\e z), where
* <b>P</b><sub><i>n</i></sub><sup><i>m</i></sup>(\e z) is Ferrers
* function (also known as the Legendre function on the cut or the
* associated Legendre polynomial) http://dlmf.nist.gov/14.7.E10 and \e k
* = 1 for \e m = 0 and \e k = 2 otherwise.
*
* The mean squared value of
* <i>P</i><sub><i>nm</i></sub><sup>schmidt</sup>(cosθ)
* cos(<i>m</i>λ) and
* <i>P</i><sub><i>nm</i></sub><sup>schmidt</sup>(cosθ)
* sin(<i>m</i>λ) over the sphere is 1/(2\e n + 1).
*
* @hideinitializer
**********************************************************************/
SCHMIDT = SphericalEngine::SCHMIDT,
};
private:
typedef Math::real real;
SphericalEngine::coeff _c[1];
real _a;
unsigned _norm;
public:
/**
* Constructor with a full set of coefficients specified.
*
* @param[in] C the coefficients <i>C</i><sub><i>nm</i></sub>.
* @param[in] S the coefficients <i>S</i><sub><i>nm</i></sub>.
* @param[in] N the maximum degree and order of the sum
* @param[in] a the reference radius appearing in the definition of the
* sum.
* @param[in] norm the normalization for the associated Legendre
* polynomials, either SphericalHarmonic::FULL (the default) or
* SphericalHarmonic::SCHMIDT.
* @exception GeographicErr if \e N does not satisfy \e N ≥ −1.
* @exception GeographicErr if \e C or \e S is not big enough to hold the
* coefficients.
*
* The coefficients <i>C</i><sub><i>nm</i></sub> and
* <i>S</i><sub><i>nm</i></sub> are stored in the one-dimensional vectors
* \e C and \e S which must contain (\e N + 1)(\e N + 2)/2 and \e N (\e N +
* 1)/2 elements, respectively, stored in "column-major" order. Thus for
* \e N = 3, the order would be:
* <i>C</i><sub>00</sub>,
* <i>C</i><sub>10</sub>,
* <i>C</i><sub>20</sub>,
* <i>C</i><sub>30</sub>,
* <i>C</i><sub>11</sub>,
* <i>C</i><sub>21</sub>,
* <i>C</i><sub>31</sub>,
* <i>C</i><sub>22</sub>,
* <i>C</i><sub>32</sub>,
* <i>C</i><sub>33</sub>.
* In general the (\e n,\e m) element is at index \e m \e N − \e m
* (\e m − 1)/2 + \e n. The layout of \e S is the same except that
* the first column is omitted (since the \e m = 0 terms never contribute
* to the sum) and the 0th element is <i>S</i><sub>11</sub>
*
* The class stores <i>pointers</i> to the first elements of \e C and \e S.
* These arrays should not be altered or destroyed during the lifetime of a
* SphericalHarmonic object.
**********************************************************************/
SphericalHarmonic(const std::vector<real>& C,
const std::vector<real>& S,
int N, real a, unsigned norm = FULL)
: _a(a)
, _norm(norm)
{ _c[0] = SphericalEngine::coeff(C, S, N); }
/**
* Constructor with a subset of coefficients specified.
*
* @param[in] C the coefficients <i>C</i><sub><i>nm</i></sub>.
* @param[in] S the coefficients <i>S</i><sub><i>nm</i></sub>.
* @param[in] N the degree used to determine the layout of \e C and \e S.
* @param[in] nmx the maximum degree used in the sum. The sum over \e n is
* from 0 thru \e nmx.
* @param[in] mmx the maximum order used in the sum. The sum over \e m is
* from 0 thru min(\e n, \e mmx).
* @param[in] a the reference radius appearing in the definition of the
* sum.
* @param[in] norm the normalization for the associated Legendre
* polynomials, either SphericalHarmonic::FULL (the default) or
* SphericalHarmonic::SCHMIDT.
* @exception GeographicErr if \e N, \e nmx, and \e mmx do not satisfy
* \e N ≥ \e nmx ≥ \e mmx ≥ −1.
* @exception GeographicErr if \e C or \e S is not big enough to hold the
* coefficients.
*
* The class stores <i>pointers</i> to the first elements of \e C and \e S.
* These arrays should not be altered or destroyed during the lifetime of a
* SphericalHarmonic object.
**********************************************************************/
SphericalHarmonic(const std::vector<real>& C,
const std::vector<real>& S,
int N, int nmx, int mmx,
real a, unsigned norm = FULL)
: _a(a)
, _norm(norm)
{ _c[0] = SphericalEngine::coeff(C, S, N, nmx, mmx); }
/**
* A default constructor so that the object can be created when the
* constructor for another object is initialized. This default object can
* then be reset with the default copy assignment operator.
**********************************************************************/
SphericalHarmonic() {}
/**
* Compute the spherical harmonic sum.
*
* @param[in] x cartesian coordinate.
* @param[in] y cartesian coordinate.
* @param[in] z cartesian coordinate.
* @return \e V the spherical harmonic sum.
*
* This routine requires constant memory and thus never throws an
* exception.
**********************************************************************/
Math::real operator()(real x, real y, real z) const {
real f[] = {1};
real v = 0;
real dummy;
switch (_norm) {
case FULL:
v = SphericalEngine::Value<false, SphericalEngine::FULL, 1>
(_c, f, x, y, z, _a, dummy, dummy, dummy);
break;
case SCHMIDT:
v = SphericalEngine::Value<false, SphericalEngine::SCHMIDT, 1>
(_c, f, x, y, z, _a, dummy, dummy, dummy);
break;
}
return v;
}
/**
* Compute a spherical harmonic sum and its gradient.
*
* @param[in] x cartesian coordinate.
* @param[in] y cartesian coordinate.
* @param[in] z cartesian coordinate.
* @param[out] gradx \e x component of the gradient
* @param[out] grady \e y component of the gradient
* @param[out] gradz \e z component of the gradient
* @return \e V the spherical harmonic sum.
*
* This is the same as the previous function, except that the components of
* the gradients of the sum in the \e x, \e y, and \e z directions are
* computed. This routine requires constant memory and thus never throws
* an exception.
**********************************************************************/
Math::real operator()(real x, real y, real z,
real& gradx, real& grady, real& gradz) const {
real f[] = {1};
real v = 0;
switch (_norm) {
case FULL:
v = SphericalEngine::Value<true, SphericalEngine::FULL, 1>
(_c, f, x, y, z, _a, gradx, grady, gradz);
break;
case SCHMIDT:
v = SphericalEngine::Value<true, SphericalEngine::SCHMIDT, 1>
(_c, f, x, y, z, _a, gradx, grady, gradz);
break;
}
return v;
}
/**
* Create a CircularEngine to allow the efficient evaluation of several
* points on a circle of latitude.
*
* @param[in] p the radius of the circle.
* @param[in] z the height of the circle above the equatorial plane.
* @param[in] gradp if true the returned object will be able to compute the
* gradient of the sum.
* @exception std::bad_alloc if the memory for the CircularEngine can't be
* allocated.
* @return the CircularEngine object.
*
* SphericalHarmonic::operator()() exchanges the order of the sums in the
* definition, i.e., ∑<sub><i>n</i> = 0..<i>N</i></sub>
* ∑<sub><i>m</i> = 0..<i>n</i></sub> becomes ∑<sub><i>m</i> =
* 0..<i>N</i></sub> ∑<sub><i>n</i> = <i>m</i>..<i>N</i></sub>.
* SphericalHarmonic::Circle performs the inner sum over degree \e n (which
* entails about <i>N</i><sup>2</sup> operations). Calling
* CircularEngine::operator()() on the returned object performs the outer
* sum over the order \e m (about \e N operations).
*
* Here's an example of computing the spherical sum at a sequence of
* longitudes without using a CircularEngine object \code
SphericalHarmonic h(...); // Create the SphericalHarmonic object
double r = 2, lat = 33, lon0 = 44, dlon = 0.01;
double
phi = lat * Math::degree<double>(),
z = r * sin(phi), p = r * cos(phi);
for (int i = 0; i <= 100; ++i) {
real
lon = lon0 + i * dlon,
lam = lon * Math::degree<double>();
std::cout << lon << " " << h(p * cos(lam), p * sin(lam), z) << "\n";
}
\endcode
* Here is the same calculation done using a CircularEngine object. This
* will be about <i>N</i>/2 times faster. \code
SphericalHarmonic h(...); // Create the SphericalHarmonic object
double r = 2, lat = 33, lon0 = 44, dlon = 0.01;
double
phi = lat * Math::degree<double>(),
z = r * sin(phi), p = r * cos(phi);
CircularEngine c(h(p, z, false)); // Create the CircularEngine object
for (int i = 0; i <= 100; ++i) {
real
lon = lon0 + i * dlon;
std::cout << lon << " " << c(lon) << "\n";
}
\endcode
**********************************************************************/
CircularEngine Circle(real p, real z, bool gradp) const {
real f[] = {1};
switch (_norm) {
case FULL:
return gradp ?
SphericalEngine::Circle<true, SphericalEngine::FULL, 1>
(_c, f, p, z, _a) :
SphericalEngine::Circle<false, SphericalEngine::FULL, 1>
(_c, f, p, z, _a);
break;
case SCHMIDT:
default: // To avoid compiler warnings
return gradp ?
SphericalEngine::Circle<true, SphericalEngine::SCHMIDT, 1>
(_c, f, p, z, _a) :
SphericalEngine::Circle<false, SphericalEngine::SCHMIDT, 1>
(_c, f, p, z, _a);
break;
}
}
/**
* @return the zeroth SphericalEngine::coeff object.
**********************************************************************/
const SphericalEngine::coeff& Coefficients() const
{ return _c[0]; }
};
} // namespace GeographicLib
#endif // GEOGRAPHICLIB_SPHERICALHARMONIC_HPP
|