/****************************************************************************
* VCGLib o o *
* Visual and Computer Graphics Library o o *
* _ O _ *
* Copyright(C) 2006 \/)\/ *
* Visual Computing Lab /\/| *
* ISTI - Italian National Research Council | *
* \ *
* All rights reserved. *
* *
* This program is free software; you can redistribute it and/or modify *
* it under the terms of the GNU General Public License as published by *
* the Free Software Foundation; either version 2 of the License, or *
* (at your option) any later version. *
* *
* This program is distributed in the hope that it will be useful, *
* but WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *
* GNU General Public License (http://www.gnu.org/licenses/gpl.txt) *
* for more details. *
* *
****************************************************************************/
#ifndef __VCGLIB_LEGENDRE_H
#define __VCGLIB_LEGENDRE_H
#include <vcg/math/base.h>
namespace vcg {
namespace math {
/*
* Contrary to their definition, the Associated Legendre Polynomials presented here are
* only computed for positive m values.
*
*/
template <typename ScalarType>
class Legendre {
protected :
/**
* Legendre Polynomial three term Recurrence Relation
*/
static inline ScalarType legendre_next(unsigned l, ScalarType P_lm1, ScalarType P_lm2, ScalarType x)
{
return ((2 * l + 1) * x * P_lm1 - l * P_lm2) / (l + 1);
}
/**
* Associated Legendre Polynomial three term Recurrence Relation.
* Raises the band index.
*/
static inline double legendre_next(unsigned l, unsigned m, ScalarType P_lm1, ScalarType P_lm2, ScalarType x)
{
return ((2 * l + 1) * x * P_lm1 - (l + m) * P_lm2) / (l + 1 - m);
}
/**
* Recurrence relation to compute P_m_(m+1) given P_m_m at the same x
*/
static inline double legendre_P_m_mplusone(unsigned m, ScalarType p_m_m, ScalarType x)
{
return x * (2.0 * m + 1.0) * p_m_m;
}
/**
* Starting relation to compute P_m_m according to the formula:
*
* pow(-1, m) * double_factorial(2 * m - 1) * pow(1 - x*x, abs(m)/2)
*
* which becomes, if x = cos(theta) :
*
* pow(-1, m) * double_factorial(2 * m - 1) * pow(sin(theta), abs(m)/2)
*/
static inline double legendre_P_m_m(unsigned m, ScalarType sin_theta)
{
ScalarType p_m_m = 1.0;
if (m > 0)
{
ScalarType fact = 1.0; //Double factorial here
for (unsigned i = 1; i <= m; ++i)
{
p_m_m *= fact * sin_theta; //raising sin_theta to the power of m/2
fact += 2.0;
}
if (m&1) //odd m
{
// Condon-Shortley Phase term
p_m_m *= -1;
}
}
return p_m_m;
}
static inline double legendre_P_l(unsigned l, ScalarType x)
{
ScalarType p0 = 1;
ScalarType p1 = x;
if (l == 0) return p0;
for (unsigned n = 1; n < l; ++n)
{
Swap(p0, p1);
p1 = legendre_next(n, p0, p1, x);
}
return p1;
}
/**
* Computes the Associated Legendre Polynomial for any given
* positive m and l, with m <= l and -1 <= x <= 1.
*/
static inline double legendre_P_l_m(unsigned l, unsigned m, ScalarType cos_theta, ScalarType sin_theta)
{
if(m > l) return 0;
if(m == 0) return legendre_P_l(l, cos_theta); //OK
else
{
ScalarType p_m_m = legendre_P_m_m(m, sin_theta); //OK
if (l == m) return p_m_m;
ScalarType p_m_mplusone = legendre_P_m_mplusone(m, p_m_m, cos_theta); //OK
if (l == m + 1) return p_m_mplusone;
unsigned n = m + 1;
while(n < l)
{
Swap(p_m_m, p_m_mplusone);
p_m_mplusone = legendre_next(n, m, p_m_m, p_m_mplusone, cos_theta);
++n;
}
return p_m_mplusone;
}
}
public :
static double Polynomial(unsigned l, ScalarType x)
{
assert (x <= 1 && x >= -1);
return legendre_P_l(l, x);
}
static double AssociatedPolynomial(unsigned l, unsigned m, ScalarType x)
{
assert (m <= l && x <= 1 && x >= -1);
return legendre_P_l_m(l, m, x, Sqrt(1.0 - x * x) );
}
static double AssociatedPolynomial(unsigned l, unsigned m, ScalarType cos_theta, ScalarType sin_theta)
{
assert (m <= l && cos_theta <= 1 && cos_theta >= -1 && sin_theta <= 1 && sin_theta >= -1);
return legendre_P_l_m(l, m, cos_theta, Abs(sin_theta));
}
};
template <typename ScalarType, int MAX_L>
class DynamicLegendre : public Legendre<ScalarType>
{
private:
ScalarType matrix[MAX_L][MAX_L]; //dynamic table
ScalarType _x; //table is conserved only across consistent x invocations
ScalarType _sin_theta;
void generate(ScalarType cos_theta, ScalarType sin_theta)
{
//generate all 'l's with m = 0
matrix[0][0] = 1;
matrix[0][1] = cos_theta;
for (unsigned l = 2; l < MAX_L; ++l)
{
matrix[0][l] = legendre_next(l-1, matrix[0][l-1], matrix[0][l-2], cos_theta);
}
for(unsigned l = 1; l < MAX_L; ++l)
{
for (unsigned m = 1; m <= l; ++m)
{
if (l == m) matrix[m][m] = legendre_P_m_m(m, sin_theta);
else if (l == m + 1) matrix[m][l] = legendre_P_m_mplusone(m, matrix[m][m], cos_theta);
else{
matrix[m][l] = legendre_next(l-1, m, matrix[m][l-1], matrix[m][l-2], cos_theta);
}
}
}
_x = cos_theta;
}
public :
DynamicLegendre() : _x(2), _sin_theta(2) {}
double AssociatedPolynomial(unsigned l, unsigned m, ScalarType x)
{
assert (m <= l && x <= 1 && x >= -1);
if (x != _x){
_sin_theta = Sqrt(1.0 - x * x);
generate(x, _sin_theta);
}
return matrix[m][l];
}
double AssociatedPolynomial(unsigned l, unsigned m, ScalarType cos_theta, ScalarType sin_theta)
{
assert (m <= l && cos_theta <= 1 && cos_theta >= -1 && sin_theta <= 1 && sin_theta >= -1);
if (cos_theta != _x){
_sin_theta = sin_theta;
generate(cos_theta, _sin_theta);
}
return matrix[m][l];
}
double Polynomial(unsigned l, ScalarType x)
{
assert (x <= 1 && x >= -1);
if (x != _x){
_sin_theta = Sqrt(1.0 - x * x);
generate(x, _sin_theta);
}
return matrix[0][l];
}
};
}} //vcg::math namespace
#endif