GeographicLib
1.21
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Spherical Harmonic series. More...
#include <GeographicLib/SphericalHarmonic.hpp>
Public Types | |
enum | normalization { FULL, SCHMIDT } |
Public Member Functions | |
SphericalHarmonic (const std::vector< real > &C, const std::vector< real > &S, int N, real a, unsigned norm=FULL) | |
SphericalHarmonic (const std::vector< real > &C, const std::vector< real > &S, int N, int nmx, int mmx, real a, unsigned norm=FULL) | |
SphericalHarmonic () | |
Math::real | operator() (real x, real y, real z) const throw () |
Math::real | operator() (real x, real y, real z, real &gradx, real &grady, real &gradz) const throw () |
CircularEngine | Circle (real p, real z, bool gradp) const |
const SphericalEngine::coeff & | Coefficients () const throw () |
Spherical Harmonic series.
This class evaluates the spherical harmonic sum
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)) ] ]
where
Two normalizations are supported for Pnm
Clenshaw summation is used for the sums over both n and 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:
Example of use:
// Example of using the GeographicLib::SphericalHarmonic class // $Id: b7e4a45d66787db49d40aa7bc991ab686cc32d44 $ #include <iostream> #include <exception> #include <vector> #include <GeographicLib/SphericalHarmonic.hpp> using namespace std; using namespace GeographicLib; int main() { try { int N = 3; // The maxium degree double ca[] = {10, 9, 8, 7, 6, 5, 4, 3, 2, 1}; // cosine coefficients vector<double> C(ca, ca + (N + 1) * (N + 2) / 2); double sa[] = {6, 5, 4, 3, 2, 1}; // sine coefficients vector<double> S(sa, sa + N * (N + 1) / 2); double a = 1; SphericalHarmonic h(C, S, N, a); double x = 2, y = 3, z = 1; double v, vx, vy, vz; v = h(x, y, z, vx, vy, vz); cout << v << " " << vx << " " << vy << " " << vz << "\n"; } catch (const exception& e) { cerr << "Caught exception: " << e.what() << "\n"; return 1; } return 0; }
Supported normalizations for the associated Legendre polynomials.
FULL |
Fully normalized associated Legendre polynomials. These are defined by Pnmfull(z) = (-1)m sqrt(k (2n + 1) (n - m)! / (n + m)!) Pnm(z), where Pnm(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 k = 1 for m = 0 and k = 2 otherwise. The mean squared value of Pnmfull(cos theta) cos(m lambda) and Pnmfull(cos theta) sin(m lambda) over the sphere is 1. |
SCHMIDT |
Schmidt semi-normalized associated Legendre polynomials. These are defined by Pnmschmidt(z) = (-1)m sqrt(k (n - m)! / (n + m)!) Pnm(z), where Pnm(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 k = 1 for m = 0 and k = 2 otherwise. The mean squared value of Pnmschmidt(cos theta) cos(m lambda) and Pnmschmidt(cos theta) sin(m lambda) over the sphere is 1/(2n + 1). |
Definition at line 71 of file SphericalHarmonic.hpp.
GeographicLib::SphericalHarmonic::SphericalHarmonic | ( | const std::vector< real > & | C, |
const std::vector< real > & | S, | ||
int | N, | ||
real | a, | ||
unsigned | norm = FULL |
||
) | [inline] |
Constructor with a full set of coefficients specified.
[in] | C | the coefficients Cnm. |
[in] | S | the coefficients Snm. |
[in] | N | the maximum degree and order of the sum |
[in] | a | the reference radius appearing in the definition of the sum. |
[in] | norm | the normalization for the associated Legendre polynomials, either SphericalHarmonic::full (the default) or SphericalHarmonic::schmidt. |
The coefficients Cnm and Snm are stored in the one-dimensional vectors C and S which must contain (N + 1)(N + 2)/2 and N (N + 1)/2 elements, respectively, stored in "column-major" order. Thus for N = 3, the order would be: C00, C10, C20, C30, C11, C21, C31, C22, C32, C33. In general the (n,m) element is at index m*N - m*(m - 1)/2 + n. The layout of S is the same except that the first column is omitted (since the m = 0 terms never contribute to the sum) and the 0th element is S11
The class stores pointers to the first elements of C and S. These arrays should not be altered or destroyed during the lifetime of a SphericalHarmonic object.
Definition at line 159 of file SphericalHarmonic.hpp.
GeographicLib::SphericalHarmonic::SphericalHarmonic | ( | const std::vector< real > & | C, |
const std::vector< real > & | S, | ||
int | N, | ||
int | nmx, | ||
int | mmx, | ||
real | a, | ||
unsigned | norm = FULL |
||
) | [inline] |
Constructor with a subset of coefficients specified.
[in] | C | the coefficients Cnm. |
[in] | S | the coefficients Snm. |
[in] | N | the degree used to determine the layout of C and S. |
[in] | nmx | the maximum degree used in the sum. The sum over n is from 0 thru nmx. |
[in] | mmx | the maximum order used in the sum. The sum over m is from 0 thru min(n, mmx). |
[in] | a | the reference radius appearing in the definition of the sum. |
[in] | norm | the normalization for the associated Legendre polynomials, either SphericalHarmonic::FULL (the default) or SphericalHarmonic::SCHMIDT. |
The class stores pointers to the first elements of C and S. These arrays should not be altered or destroyed during the lifetime of a SphericalHarmonic object.
Definition at line 186 of file SphericalHarmonic.hpp.
GeographicLib::SphericalHarmonic::SphericalHarmonic | ( | ) | [inline] |
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.
Definition at line 199 of file SphericalHarmonic.hpp.
Math::real GeographicLib::SphericalHarmonic::operator() | ( | real | x, |
real | y, | ||
real | z | ||
) | const throw () [inline] |
Compute the spherical harmonic sum.
[in] | x | cartesian coordinate. |
[in] | y | cartesian coordinate. |
[in] | z | cartesian coordinate. |
This routine requires constant memory and thus never throws an exception.
Definition at line 212 of file SphericalHarmonic.hpp.
Math::real GeographicLib::SphericalHarmonic::operator() | ( | real | x, |
real | y, | ||
real | z, | ||
real & | gradx, | ||
real & | grady, | ||
real & | gradz | ||
) | const throw () [inline] |
Compute a spherical harmonic sum and its gradient.
[in] | x | cartesian coordinate. |
[in] | y | cartesian coordinate. |
[in] | z | cartesian coordinate. |
[out] | gradx | x component of the gradient |
[out] | grady | y component of the gradient |
[out] | gradz | z component of the gradient |
This is the same as the previous function, except that the components of the gradients of the sum in the x, y, and z directions are computed. This routine requires constant memory and thus never throws an exception.
Definition at line 245 of file SphericalHarmonic.hpp.
CircularEngine GeographicLib::SphericalHarmonic::Circle | ( | real | p, |
real | z, | ||
bool | gradp | ||
) | const [inline] |
Create a CircularEngine to allow the efficient evaluation of several points on a circle of latitude.
[in] | p | the radius of the circle. |
[in] | z | the height of the circle above the equatorial plane. |
[in] | gradp | if true the returned object will be able to compute the gradient of the sum. |
SphericalHarmonic::operator()() exchanges the order of the sums in the definition, i.e., sum(n = 0..N)[sum(m = 0..n)[...]] becomes sum(m = 0..N)[sum(n = m..N)[...]]. SphericalHarmonic::Circle performs the inner sum over degree n (which entails about N2 operations). Calling CircularEngine::operator()() on the returned object performs the outer sum over the order m (about N operations). This routine may throw a bad_alloc exception in the CircularEngine constructor.
Here's an example of computing the spherical sum at a sequence of longitudes without using a CircularEngine object
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"; }
Here is the same calculation done using a CircularEngine object. This will be about N/2 times faster.
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"; }
Definition at line 312 of file SphericalHarmonic.hpp.
Referenced by GeographicLib::GravityModel::Circle().
const SphericalEngine::coeff& GeographicLib::SphericalHarmonic::Coefficients | ( | ) | const throw () [inline] |
Definition at line 336 of file SphericalHarmonic.hpp.
Referenced by GeographicLib::GravityModel::GravityModel().