SphericalHamonics

class SphericalHarmonics : public SphericalBase

Public Functions

SphericalHarmonics(unsigned L_degree)

Sets up class for spherical harmonics basis based on legendre polynoms and associated legendre polynoms up to degree L. The basis then consists of N = L² +2L basis functions.

Parameters:

L_degree – - maximum degree of spherical harmonics basis, 0 <= L <= 1000 (upper bound due to numerical stability)

SphericalHarmonics(unsigned L_degree, unsigned short spatialDim)

Sets up class for monomial basis on sphere up to degree L. The basis then consists of N = L.

Parameters:

• L_degree – maximum degree of spherical harmonics basis, 0 <= L <= 1000 (upper bound due to numerical stability)

• spatialDim – spatial dimensioniality of the simulation

virtual Vector ComputeSphericalBasis(double my, double phi, double r = 1.0) override

Computes all N = L² +2L basis functions at point (my, phi)

Parameters:

• my – = cos(theta) - spherical coordinate, -1 <= x <= 1

• phi – - spherical coordinate, 0 <= phi <= 2*pi

Returns:

vector of basis functions at point (my, phi) with size N = L² +2L

virtual Vector ComputeSphericalBasisKarthesian(double x, double y, double z) override

Computes all N = L² +2L basis functions at point (x, y, z) on the unit sphere.

Parameters:

• x – coordinates on unit sphere

• y – coordinates on unit sphere

• z – coordinates on unit sphere

Returns:

vector of basis functions at point (x,y,z) with size N = L² +2L

std::vector<double> GetAssLegendrePoly(const double my)

Computes an entire set of (komplex congjugate) P_l^k and stores it in the vector _assLegendreP.

Parameters:

my – = cos(theta) - spherical coordinate, -1 <= my <= 1

Returns:

Associated Legendre Polynomial at my for all l and k

virtual unsigned GetBasisSize() final override

Returns length of the basis, i.e. number of elements of the basis.

virtual unsigned GetCurrDegreeSize(unsigned currDegree) final override

Return number of basis functions with degree equals to currDegree.

Parameters:

currDegree – degree of polynomials that are counted

virtual unsigned GetGlobalIndexBasis(int l_degree, int k_order) final override

helper function to get the global index for given k and l of the basis function Y_k^l.

Parameters:

• l_degree – - current degree of basis function, 0 <= l <= L

• k_order – - current order of basis function, -l <= k <= l

Protected Functions

inline unsigned GlobalIdxAssLegendreP(unsigned l_degree, unsigned k_order)

helper function to get the global index for given k and l of the associated legendre polynomial P_k^l.

Parameters:

• l_degree – - current degree of basis function, 0 <= l <= L

• k_order – - current order of basis function, 0 <= k <= l

void ComputeCoefficients()

computes values of a_param and b_param

void ComputeAssLegendrePoly(const double my)

Computes an entire set of (komplex congjugate) P_l^k and stores it in the vector _assLegendreP.

Parameters:

my – equals cos(theta) - spherical coordinate, -1 <= my <= 1

void ComputeYBasis(const double phi)

Computes the spherical harmonics basis function up to degree _LmaxDegree at polar coordinates (theta, psi) and stores the result in _YBasis;.

Parameters:

phi – spherical coordinate

unsigned GetBasisSizeUnprojected()

Returns length of the unprojected basis, i.e. number of elements of the basis in 3D.

Protected Attributes

unsigned _LMaxDegree

maximal degree of the spherical harmonics basis (this is “L” in the comments)

std::vector<double> _aParam

coefficients for the computations of the basis length of _aParam, _bParam : L + (L*(L+1))/2

std::vector<double> _assLegendreP

komplex conjugate of the associated legendre polynomials of degree 0 <= l <= L and order 0 <= k <= l length of _assLegendreP : L + (L*(L+1))/2

Vector _YBasis

spherical harmonic basis functions of degree 0 <= l <= L and order -l <= k <= l length : N = L² +2L