vcglib/vcg/math/legendre.h

/****************************************************************************
* 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 {

private :
		
	/**
	 * 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));
	}
};

}} //vcg::math namespace

#endif