Spherical harmonic series with two corrections to the coefficients. More...

#include <SphericalHarmonic2.hpp>

Public Types
enum	normalization { FULL, SCHMIDT }

Public Member Functions
CircularEngine	Circle (real tau1, real tau2, real p, real z, bool gradp) const

const SphericalEngine::coeff &	Coefficients () const

const SphericalEngine::coeff &	Coefficients1 () const

const SphericalEngine::coeff &	Coefficients2 () const

Math::real	operator() (real tau1, real tau2, real x, real y, real z) const

Math::real	operator() (real tau1, real tau2, real x, real y, real z, real &gradx, real &grady, real &gradz) const

	SphericalHarmonic2 (const std::vector< real > &C, const std::vector< real > &S, int N, const std::vector< real > &C1, const std::vector< real > &S1, int N1, const std::vector< real > &C2, const std::vector< real > &S2, int N2, real a, unsigned norm=FULL)

	SphericalHarmonic2 (const std::vector< real > &C, const std::vector< real > &S, int N, int nmx, int mmx, const std::vector< real > &C1, const std::vector< real > &S1, int N1, int nmx1, int mmx1, const std::vector< real > &C2, const std::vector< real > &S2, int N2, int nmx2, int mmx2, real a, unsigned norm=FULL)

	SphericalHarmonic2 ()

Private Types
typedef Math::real	real

Private Attributes
real	_a

SphericalEngine::coeff	_c [3]

unsigned	_norm

Detailed Description

Spherical harmonic series with two corrections to the coefficients.

This classes is similar to SphericalHarmonic, except that the coefficients C_nm are replaced by C_nm + tau' C'_nm + tau'' C''_nm (and similarly for S_nm).

Example of use:

// Example of using the GeographicLib::SphericalHarmonic2 class
#include <iostream>
#include <exception>
#include <vector>
#include <GeographicLib/SphericalHarmonic2.hpp>
using namespace std;
using namespace GeographicLib;
int main() {
  try {
    int N = 3, N1 = 2, N2 = 1;                     // The maxium degrees
    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 cb[] = {1, 2, 3, 4, 5, 6};
    vector<double> C1(cb, cb + (N1 + 1) * (N1 + 2) / 2);
    double sb[] = {3, 2, 1};
    vector<double> S1(sb, sb + N1 * (N1 + 1) / 2);
    double cc[] = {2, 1};
    vector<double> C2(cc, cc + (N2 + 1));
    vector<double> S2;
    double a = 1;
    SphericalHarmonic2 h(C, S, N, N, N, C1, S1, N1, N1, N1,
                         C2, S2, N2, N2, 0, a);
    double tau1 = 0.1, tau2 = 0.05, x = 2, y = 3, z = 1;
    double v, vx, vy, vz;
    v = h(tau1, tau2, 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 35 of file SphericalHarmonic2.hpp.

Member Typedef Documentation

typedef Math::real GeographicLib::SphericalHarmonic2::real

private

Definition at line 58 of file SphericalHarmonic2.hpp.

Member Enumeration Documentation

enum GeographicLib::SphericalHarmonic2::normalization

Supported normalizations for associate Legendre polynomials.

Enumerator
FULL	Fully normalized associated Legendre polynomials. See SphericalHarmonic::FULL for documentation.
SCHMIDT	Schmidt semi-normalized associated Legendre polynomials. See SphericalHarmonic::SCHMIDT for documentation.

Definition at line 40 of file SphericalHarmonic2.hpp.

Constructor & Destructor Documentation

GeographicLib::SphericalHarmonic2::SphericalHarmonic2	(	const std::vector< real > &	C,
		const std::vector< real > &	S,
		int	N,
		const std::vector< real > &	C1,
		const std::vector< real > &	S1,
		int	N1,
		const std::vector< real > &	C2,
		const std::vector< real > &	S2,
		int	N2,
		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]	C1	the coefficients C'_nm.
[in]	S1	the coefficients S'_nm.
[in]	N1	the maximum degree and order of the first correction coefficients C'_nm and S'_nm.
[in]	C2	the coefficients C''_nm.
[in]	S2	the coefficients S''_nm.
[in]	N2	the maximum degree and order of the second correction coefficients C'_nm and S'_nm.
[in]	a	the reference radius appearing in the definition of the sum.
[in]	norm	the normalization for the associated Legendre polynomials, either SphericalHarmonic2::FULL (the default) or SphericalHarmonic2::SCHMIDT.

Exceptions

GeographicErr	if N and N1 do not satisfy N ≥ N1 ≥ −1, and similarly for N2.
GeographicErr	if any of the vectors of coefficients is not large enough.

See SphericalHarmonic for the way the coefficients should be stored. N1 and N2 should satisfy N1 ≤ N and N2 ≤ N.

The class stores pointers to the first elements of C, S, C', S', C'', and S''. These arrays should not be altered or destroyed during the lifetime of a SphericalHarmonic object.

Definition at line 97 of file SphericalHarmonic2.hpp.

GeographicLib::SphericalHarmonic2::SphericalHarmonic2	(	const std::vector< real > &	C,
		const std::vector< real > &	S,
		int	N,
		int	nmx,
		int	mmx,
		const std::vector< real > &	C1,
		const std::vector< real > &	S1,
		int	N1,
		int	nmx1,
		int	mmx1,
		const std::vector< real > &	C2,
		const std::vector< real > &	S2,
		int	N2,
		int	nmx2,
		int	mmx2,
		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]	C1	the coefficients C'_nm.
[in]	S1	the coefficients S'_nm.
[in]	N1	the degree used to determine the layout of C' and S'.
[in]	nmx1	the maximum degree used for C' and S'.
[in]	mmx1	the maximum order used for C' and S'.
[in]	C2	the coefficients C''_nm.
[in]	S2	the coefficients S''_nm.
[in]	N2	the degree used to determine the layout of C'' and S''.
[in]	nmx2	the maximum degree used for C'' and S''.
[in]	mmx2	the maximum order used for C'' and S''.
[in]	a	the reference radius appearing in the definition of the sum.
[in]	norm	the normalization for the associated Legendre polynomials, either SphericalHarmonic2::FULL (the default) or SphericalHarmonic2::SCHMIDT.

Exceptions

GeographicErr	if the parameters do not satisfy N ≥ nmx ≥ mmx ≥ −1; N1 ≥ nmx1 ≥ mmx1 ≥ −1; N ≥ N1; nmx ≥ nmx1; mmx ≥ mmx1; and similarly for N2, nmx2, and mmx2.
GeographicErr	if any of the vectors of coefficients is not large enough.

The class stores pointers to the first elements of C, S, C', S', C'', and S''. These arrays should not be altered or destroyed during the lifetime of a SphericalHarmonic object.

Definition at line 154 of file SphericalHarmonic2.hpp.

GeographicLib::SphericalHarmonic2::SphericalHarmonic2 ( )

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 180 of file SphericalHarmonic2.hpp.

Member Function Documentation

CircularEngine GeographicLib::SphericalHarmonic2::Circle	(	real	tau1,
		real	tau2,
		real	p,
		real	z,
		bool	gradp
	)		const

inline

Create a CircularEngine to allow the efficient evaluation of several points on a circle of latitude at fixed values of tau1 and tau2.

Parameters

[in]	tau1	multiplier for correction coefficients C' and S'.
[in]	tau2	multiplier for correction coefficients C'' and S''.
[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.

SphericalHarmonic2::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}.. SphericalHarmonic2::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).

See SphericalHarmonic::Circle for an example of its use.

Definition at line 277 of file SphericalHarmonic2.hpp.

const SphericalEngine::coeff& GeographicLib::SphericalHarmonic2::Coefficients ( ) const

inline

Returns: the zeroth SphericalEngine::coeff object.

Definition at line 302 of file SphericalHarmonic2.hpp.

const SphericalEngine::coeff& GeographicLib::SphericalHarmonic2::Coefficients1 ( ) const

inline

Returns: the first SphericalEngine::coeff object.

Definition at line 307 of file SphericalHarmonic2.hpp.

const SphericalEngine::coeff& GeographicLib::SphericalHarmonic2::Coefficients2 ( ) const

inline

Returns: the second SphericalEngine::coeff object.

Definition at line 312 of file SphericalHarmonic2.hpp.

Math::real GeographicLib::SphericalHarmonic2::operator()	(	real	tau1,
		real	tau2,
		real	x,
		real	y,
		real	z
	)		const

inline

Compute a spherical harmonic sum with two correction terms.

Parameters

[in]	tau1	multiplier for correction coefficients C' and S'.
[in]	tau2	multiplier for correction coefficients C'' and S''.
[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 196 of file SphericalHarmonic2.hpp.

Math::real GeographicLib::SphericalHarmonic2::operator()	(	real	tau1,
		real	tau2,
		real	x,
		real	y,
		real	z,
		real &	gradx,
		real &	grady,
		real &	gradz
	)		const

inline

Compute a spherical harmonic sum with two correction terms and its gradient.

Parameters

[in]	tau1	multiplier for correction coefficients C' and S'.
[in]	tau2	multiplier for correction coefficients C'' and S''.
[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 234 of file SphericalHarmonic2.hpp.

Member Data Documentation

real GeographicLib::SphericalHarmonic2::_a

private

Definition at line 60 of file SphericalHarmonic2.hpp.

SphericalEngine::coeff GeographicLib::SphericalHarmonic2::_c[3]

private

Definition at line 59 of file SphericalHarmonic2.hpp.

unsigned GeographicLib::SphericalHarmonic2::_norm

private

Definition at line 61 of file SphericalHarmonic2.hpp.

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

SphericalHarmonic2.hpp

Public Types

Public Member Functions

Private Types

Private Attributes

Detailed Description

Member Typedef Documentation

Member Enumeration Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation