GeographicLib::SphericalHarmonic2 Class Reference

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

`#include <GeographicLib/SphericalHarmonic2.hpp>`

## Public Types

enum | normalization { FULL , SCHMIDT }

## Public Member Functions

| 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 ()
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
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

## Detailed Description

Spherical harmonic series with two corrections to the coefficients.

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

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;
}
}
main
int main(int argc, const char *const argv[])
Definition: CartConvert.cpp:29
SphericalHarmonic2.hpp
Header for GeographicLib::SphericalHarmonic2 class.
GeographicLib::SphericalHarmonic2::SphericalHarmonic2
SphericalHarmonic2()
Definition: SphericalHarmonic2.hpp:180
GeographicLib
Namespace for GeographicLib.
Definition: Accumulator.cpp:12
std
Definition: NearestNeighbor.hpp:812

Definition at line 35 of file SphericalHarmonic2.hpp.

## Member Enumeration Documentation

## normalization

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

## SphericalHarmonic2() [1/3]

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 Cnm.
  [in] | S | the coefficients Snm.
  [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.

## SphericalHarmonic2() [2/3]

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 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] | 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.

## SphericalHarmonic2() [3/3]

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

## operator()() [1/2]

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.

References FULL, and SCHMIDT.

## operator()() [2/2]

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 235 of file SphericalHarmonic2.hpp.

References FULL, and SCHMIDT.

## Circle()

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 279 of file SphericalHarmonic2.hpp.

References FULL, and SCHMIDT.

## Coefficients()

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

inline

Returns
  the zeroth SphericalEngine::coeff object.

Definition at line 304 of file SphericalHarmonic2.hpp.

## Coefficients1()

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

inline

Returns
  the first SphericalEngine::coeff object.

Definition at line 309 of file SphericalHarmonic2.hpp.

## Coefficients2()

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

inline

Returns
  the second SphericalEngine::coeff object.

Definition at line 314 of file SphericalHarmonic2.hpp.

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

- SphericalHarmonic2.hpp