legendre.h

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/****************************************************************************
* 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 :

        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);
        }

        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);
        }

        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;
        }

        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)
                {
                        std::swap(p0, p1);
                        p1 = legendre_next(n, p0, p1, x);
                }

                return p1;
        }

        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)
                        {
                                std::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