CPolySolver.h

Go to the documentation of this file.

//==============================================================================
/*
    Software License Agreement (BSD License)
    Copyright (c) 2003-2016, CHAI3D.
    (www.chai3d.org)

    All rights reserved.

    Redistribution and use in source and binary forms, with or without
    modification, are permitted provided that the following conditions
    are met:

    * Redistributions of source code must retain the above copyright
    notice, this list of conditions and the following disclaimer.

    * Redistributions in binary form must reproduce the above
    copyright notice, this list of conditions and the following
    disclaimer in the documentation and/or other materials provided
    with the distribution.

    * Neither the name of CHAI3D nor the names of its contributors may
    be used to endorse or promote products derived from this software
    without specific prior written permission.

    THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
    "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
    LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
    FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
    COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
    INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
    BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
    LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
    CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
    LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
    ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
    POSSIBILITY OF SUCH DAMAGE. 

    \author    <http://www.chai3d.org>
    \author    Francois Conti
    \version   3.2.0 $Rev: 2015 $
*/
//==============================================================================

//------------------------------------------------------------------------------
#ifndef CPolySolverH
#define CPolySolverH
//------------------------------------------------------------------------------
#include "math/CMaths.h"
//------------------------------------------------------------------------------

//------------------------------------------------------------------------------
namespace chai3d {
//------------------------------------------------------------------------------

//==============================================================================
//==============================================================================

//------------------------------------------------------------------------------
//------------------------------------------------------------------------------


//==============================================================================
//==============================================================================
inline int cSolveLinear(double a_coefficient[2], double a_solution[1])
{
    if (cZero(a_coefficient[1]))
    {
        return (0);
    }

    a_solution[0] = - a_coefficient[0] / a_coefficient[1];
    
    return (1);
}


//==============================================================================
//==============================================================================
inline int cSolveQuadric(double a_coefficient[3], double a_solution[2])
{
    double p, q, D;

    // make sure we have a d2 equation
    if (cZero(a_coefficient[2]))
    {
        return (cSolveLinear(a_coefficient, a_solution));
    }

    // normal for: x^2 + px + q
    p = a_coefficient[1] / (2.0 * a_coefficient[2]);
    q = a_coefficient[0] / a_coefficient[2];
    D = p * p - q;

    if (cZero(D))
    {
        // one double root
        a_solution[0] = a_solution[1] = -p;
        return (1);
    }

    if (D < 0.0)
    {
        // no real root
        return (0);
    }
    else
    {
        // two real roots
        double sqrt_D = sqrt(D);
        a_solution[0] = sqrt_D - p;
        a_solution[1] = -sqrt_D - p;
        return (2);
    }
}


//==============================================================================
//==============================================================================
inline int cSolveCubic(double a_coefficient[4], double a_solution[3])
{
    int i, num;
    double sub,
           A, B, C,
           sq_A, p, q,
           cb_p, D;

    // normalize the equation:x ^ 3 + Ax ^ 2 + Bx  + C = 0
    A = a_coefficient[2] / a_coefficient[3];
    B = a_coefficient[1] / a_coefficient[3];
    C = a_coefficient[0] / a_coefficient[3];

    // substitute x = y - A / 3 to eliminate the quadric term: x^3 + px + q = 0
    sq_A = A * A;
    p = 1.0/3.0 * (-1.0/3.0 * sq_A + B);
    q = 1.0/2.0 * (2.0/27.0 * A *sq_A - 1.0/3.0 * A * B + C);

    // use Cardano's formula
    cb_p = p * p * p;
    D = q * q + cb_p;

    if (cZero(D))
    {
        if (cZero(q))
        {
            // one triple solution
            a_solution[0] = 0.0;
            num = 1;
        }
        else
        {
            // one single and one double solution
            double u = cCbrt(-q);
            a_solution[0] = 2.0 * u;
            a_solution[1] = - u;
            num = 2;
        }
    }
    else
    {
        if (D < 0.0)
        {
            // casus irreductibilis: three real solutions
            double phi = 1.0/3.0 * acos(-q / sqrt(-cb_p));
            double t = 2.0 * sqrt(-p);
            a_solution[0] = t * cos(phi);
            a_solution[1] = -t * cos(phi + M_PI / 3.0);
            a_solution[2] = -t * cos(phi - M_PI / 3.0);
            num = 3;
        }
        else
        {
            // one real solution
            double sqrt_D = sqrt(D);
            double u = cCbrt(sqrt_D + fabs(q));
            if (q > 0.0)
            {
                a_solution[0] = - u + p / u ;
            }
            else
            {
                a_solution[0] = u - p / u;
            }
            num = 1;
        }
    }

    // resubstitute
    sub = 1.0 / 3.0 * A;
    for (i = 0; i < num; i++)
    {
        a_solution[i] -= sub;
    }

    return (num);
}


//==============================================================================
//==============================================================================
inline int cSolveQuartic(double a_coefficient[5], double a_solution[4])
{
    double  coeffs[4],
            z, u, v, sub,
            A, B, C, D,
            sq_A, p, q, r;
    int i, num;

    // normalize the equation:x ^ 4 + Ax ^ 3 + Bx ^ 2 + Cx + D = 0
    A = a_coefficient[3] / a_coefficient[4];
    B = a_coefficient[2] / a_coefficient[4];
    C = a_coefficient[1] / a_coefficient[4];
    D = a_coefficient[0] / a_coefficient[4];

    // subsitute x = y - A / 4 to eliminate the cubic term: x^4 + px^2 + qx + r = 0
    sq_A = A * A;
    p = -3.0 / 8.0 * sq_A + B;
    q = 1.0 / 8.0 * sq_A * A - 1.0 / 2.0 * A * B + C;
    r = -3.0 / 256.0 * sq_A * sq_A + 1.0 / 16.0 * sq_A * B - 1.0 / 4.0 * A * C + D;

    if (cZero(r))
    {
        // no absolute term:y(y ^ 3 + py + q) = 0
        coeffs[0] = q;
        coeffs[1] = p;
        coeffs[2] = 0.0;
        coeffs[3] = 1.0;

        num = cSolveCubic(coeffs, a_solution);
        a_solution[num++] = 0;
    }
    else
    {
        // solve the resolvent cubic...
        coeffs[0] = 1.0 / 2.0 * r * p - 1.0 / 8.0 * q * q;
        coeffs[1] = -r;
        coeffs[2] = -1.0 / 2.0 * p;
        coeffs[3] = 1.0;
        (void) cSolveCubic(coeffs, a_solution);

        // ...and take the one real solution...
        z = a_solution[0];

        // ...to build two quadratic equations
        u = z * z - r;
        v = 2.0 * z - p;

        if (cZero(u))
        {
            u = 0.0;
        }
        else if (u > 0.0)
        {
            u = sqrt(u);
        }
        else
        {
            return (0);
        }
        
        if (cZero(v))
        {
            v = 0;
        }
        else if (v > 0.0)
        {
            v = sqrt(v);
        }
        else
        {
            return (0);
        }

        coeffs[0] = z - u;
        coeffs[1] = q < 0 ? -v : v;
        coeffs[2] = 1.0;

        num = cSolveQuadric(coeffs, a_solution);

        coeffs[0] = z + u;
        coeffs[1] = q < 0 ? v : -v;
        coeffs[2] = 1.0;

        num += cSolveQuadric(coeffs, a_solution + num);
    }

    // resubstitute
    sub = 1.0 / 4 * A;
    for (i = 0; i < num; i++)
    {
        a_solution[i] -= sub;
    }

    return (num);
}


//------------------------------------------------------------------------------
}       // namespace chai3d
//------------------------------------------------------------------------------

//------------------------------------------------------------------------------
#endif  // CPolySolverH
//------------------------------------------------------------------------------