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

Math::real	operator() (real x, real y, real z, real &gradx, real &grady, real &gradz) const

CircularEngine	Circle (real p, real z, bool gradp) const

const SphericalEngine::coeff &	Coefficients () const

Detailed Description

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

p² = x² + y²,
r² = p² + z²,
q = a/r,
θ = atan2(p, z) = the spherical colatitude,
λ = atan2(y, x) = the longitude.
P_nm(t) is the associated Legendre polynomial of degree n and order m.

Two normalizations are supported for P_nm

fully normalized denoted by SphericalHarmonic::FULL.
Schmidt semi-normalized denoted by SphericalHarmonic::SCHMIDT.

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:

C. W. Clenshaw, A note on the summation of Chebyshev series, 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 unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions, J. Geodesy 76(5), 279–299 (2002).
C. C. Tscherning and K. Poder, Some geodetic applications of Clenshaw summation, Boll. Geod. Sci. Aff. 41(4), 349–375 (1982).

Example of use:

// Example of using the GeographicLib::SphericalHarmonic class
 
#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;
  }
}

Definition at line 69 of file SphericalHarmonic.hpp.

Member Enumeration Documentation

◆ normalization

enum GeographicLib::SphericalHarmonic::normalization

Supported normalizations for the associated Legendre polynomials.

Enumerator

FULL

Fully normalized associated Legendre polynomials.

These are defined by P_nm^full(z) = (−1)^m sqrt(k (2n + 1) (n − m)! / (n + m)!) P_n^m(z), where P_n^m(z) is Ferrers function (also known as the Legendre function on the cut or the associated Legendre polynomial) https://dlmf.nist.gov/14.7.E10 and k = 1 for m = 0 and k = 2 otherwise.

The mean squared value of P_nm^full(cosθ) cos(mλ) and P_nm^full(cosθ) sin(mλ) over the sphere is 1.

SCHMIDT

Schmidt semi-normalized associated Legendre polynomials.

These are defined by P_nm^schmidt(z) = (−1)^m sqrt(k (n − m)! / (n + m)!) P_n^m(z), where P_n^m(z) is Ferrers function (also known as the Legendre function on the cut or the associated Legendre polynomial) https://dlmf.nist.gov/14.7.E10 and k = 1 for m = 0 and k = 2 otherwise.

The mean squared value of P_nm^schmidt(cosθ) cos(mλ) and P_nm^schmidt(cosθ) sin(mλ) over the sphere is 1/(2n + 1).

Definition at line 74 of file SphericalHarmonic.hpp.

Constructor & Destructor Documentation

◆ SphericalHarmonic() [1/3]

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.

Parameters

[in]	C	the coefficients C_nm.
[in]	S	the coefficients S_nm.
[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.

Exceptions

GeographicErr	if N does not satisfy N ≥ −1.
GeographicErr	if C or S is not big enough to hold the coefficients.

The coefficients C_nm and S_nm 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 167 of file SphericalHarmonic.hpp.

◆ SphericalHarmonic() [2/3]

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.

Parameters

[in]	C	the coefficients C_nm.
[in]	S	the coefficients S_nm.
[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.

Exceptions

GeographicErr	if N, nmx, and mmx do not satisfy N ≥ nmx ≥ mmx ≥ −1.
GeographicErr	if C or S is not big enough to hold the coefficients.

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 198 of file SphericalHarmonic.hpp.

◆ SphericalHarmonic() [3/3]

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 211 of file SphericalHarmonic.hpp.

Member Function Documentation

◆ operator()() [1/2]

Math::real GeographicLib::SphericalHarmonic::operator()	(	real	x,
		real	y,
		real	z
	)		const

inline

Compute the spherical harmonic sum.

Parameters

[in]	x	cartesian coordinate.
[in]	y	cartesian coordinate.
[in]	z	cartesian coordinate.

Returns: V the spherical harmonic sum.

This routine requires constant memory and thus never throws an exception.

Definition at line 224 of file SphericalHarmonic.hpp.

◆ operator()() [2/2]

Math::real GeographicLib::SphericalHarmonic::operator()	(	real	x,
		real	y,
		real	z,
		real &	gradx,
		real &	grady,
		real &	gradz
	)		const

inline

Compute a spherical harmonic sum and its gradient.

Parameters

[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

Returns: 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 x, y, and z directions are computed. This routine requires constant memory and thus never throws an exception.

Definition at line 258 of file SphericalHarmonic.hpp.

◆ Circle()

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.

Parameters

[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.

Exceptions

std::bad_alloc if the memory for the CircularEngine can't be allocated.

Returns: the CircularEngine object.

SphericalHarmonic::operator()() exchanges the order of the sums in the definition, i.e., ∑_{n = 0..N} ∑_{m = 0..n} becomes ∑_{m = 0..N} ∑_{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).

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 326 of file SphericalHarmonic.hpp.

Referenced by GeographicLib::GravityModel::Circle().

◆ Coefficients()

const SphericalEngine::coeff & GeographicLib::SphericalHarmonic::Coefficients ( ) const

inline

Returns: the zeroth SphericalEngine::coeff object.

Definition at line 350 of file SphericalHarmonic.hpp.

Referenced by GeographicLib::GravityModel::GravityModel().

The documentation for this class was generated from the following file:

SphericalHarmonic.hpp

Public Types

Public Member Functions

Detailed Description

Member Enumeration Documentation

◆ normalization

Constructor & Destructor Documentation

◆ SphericalHarmonic() [1/3]

◆ SphericalHarmonic() [2/3]

◆ SphericalHarmonic() [3/3]

Member Function Documentation

◆ operator()() [1/2]

◆ operator()() [2/2]

◆ Circle()

◆ Coefficients()