TobiiEyetracking/Source/TobiiInteractions/Private/TobiiRootFinders.h

/*
 *  Utility functions to find cubic and quartic roots,
 *  coefficients are passed like this:
 *
 *      c[0] + c[1]*x + c[2]*x^2 + c[3]*x^3 + c[4]*x^4 = 0
 *
 *  The functions return the number of non-complex roots and
 *  put the values into the s array.
 *
 *  Author:         Jochen Schwarze (schwarze@isa.de)
 *
 *  Jan 26, 1990    Version for Graphics Gems
 *  Oct 11, 1990    Fixed sign problem for negative q's in SolveQuartic
 *  	    	    (reported by Mark Podlipec),
 *  	    	    Old-style function definitions,
 *  	    	    IsZero() as a macro
 *  Nov 23, 1990    Some systems do not declare acos() and cbrt() in
 *                  <math.h>, though the functions exist in the library.
 *                  If large coefficients are used, EQN_EPS should be
 *                  reduced considerably (e.g. to 1E-30), results will be
 *                  correct but multiple roots might be reported more
 *                  than once.
 *
 * Graphics gems resources:
 * http://www.realtimerendering.com/resources/GraphicsGems/
 * 
 * License:
 * All graphics gems code can be used without restrictions. [See above link]
 */

#pragma once

#include    <math.h>
#ifndef M_PI
#define M_PI          3.14159265358979323846
#endif

/* epsilon surrounding for near zero values */

#define     EQN_EPS     1E-30
#define	    IsZero(x)	((x) > -EQN_EPS && (x) < EQN_EPS)
#define	    IsOne(x)	((x) > (1.0-EQN_EPS) && (x) < (1.0+EQN_EPS))

#ifdef NOCBRT
#define     cbrt(x)     ((x) > 0.0 ? pow((double)(x), 1.0/3.0) : \
                          ((x) < 0.0 ? -pow((double)-(x), 1.0/3.0) : 0.0))
#endif

int SolveQuadric(double c[3], double s[2])
{
    double p, q, D;

	if (IsZero(c[2]))
	{
		if (IsZero(c[1]))
		{
			return 0;
		}

		s[0] = -c[0] / c[1];
		return 1;
	}

    /* normal form: x^2 + px + q = 0 */

    p = c[ 1 ] / (2 * c[ 2 ]);
    q = c[ 0 ] / c[ 2 ];

    D = p * p - q;

    if (IsZero(D))
    {
		s[0] = -p;
		return 1;
    }
    else if (D < 0)
    {
		return 0;
    }
    else /* if (D > 0) */
    {
		double sqrt_D = sqrt(D);

		s[0] = sqrt_D - p;
		s[1] = -sqrt_D - p;
		return 2;
    }
}


int SolveCubic(double c[4], double s[3])
{
    int     i, num;
    double  sub;
    double  A, B, C;
    double  sq_A, p, q;
    double  cb_p, D;

	if (IsZero(c[3]))
	{
		SolveQuadric(c, s);
	}

	/* normal form: x^3 + Ax^2 + Bx + C = 0 */

	A = c[2];
	B = c[1];
	C = c[0];

	if (!IsOne(c[3]))
	{
		A /= c[3];
		B /= c[3];
		C /= c[3];
	}

    /*  substitute x = y - A/3 to eliminate quadric term:
	x^3 +px + q = 0 */

    sq_A = A * A;
    p = 1.0/3 * (- 1.0/3 * sq_A + B);
    q = 1.0/2 * (2.0/27 * A * sq_A - 1.0/3 * A * B + C);

    /* use Cardano's formula */

    cb_p = p * p * p;
    D = q * q + cb_p;

    if (IsZero(D))
    {
		if (IsZero(q)) /* one triple solution */
		{
			s[ 0 ] = 0;
			num = 1;
		}
		else /* one single and one double solution */
		{
			double u = cbrt(-q);
			s[ 0 ] = 2 * u;
			s[ 1 ] = - u;
			num = 2;
		}
    }
    else if (D < 0) /* Casus irreducibilis: three real solutions */
    {
		double phi = 1.0 / 3 * acos(-q / sqrt(-cb_p));
		double t = 2 * sqrt(-p);

		s[0] = t * cos(phi);
		s[1] = -t * cos(phi + M_PI / 3);
		s[2] = -t * cos(phi - M_PI / 3);
		num = 3;
    }
    else /* one real solution */
    {
		double sqrt_D = sqrt(D);
		double u = cbrt(sqrt_D - q);
		double v = - cbrt(sqrt_D + q);

		s[ 0 ] = u + v;
		num = 1;
    }

    /* resubstitute */

    sub = 1.0/3 * A;

    for (i = 0; i < num; ++i)
	s[ i ] -= sub;

    return num;
}


int SolveQuartic(double c[5], double s[4])
{
    double  coeffs[ 4 ];
    double  z, u, v, sub;
    double  A, B, C, D;
    double  sq_A, p, q, r;
    int     i, num;

	if (IsZero(c[4]))
	{
		SolveCubic(c, s);
	}

    /* normal form: x^4 + Ax^3 + Bx^2 + Cx + D = 0 */

	A = c[3];
	B = c[2];
	C = c[1];
	D = c[0];

	if (!IsOne(c[4]))
	{
		A /= c[4];
		B /= c[4];
		C /= c[4];
		D /= c[4];
	}

    /*  substitute x = y - A/4 to eliminate cubic term:
	x^4 + px^2 + qx + r = 0 */

    sq_A = A * A;
    p = - 3.0/8 * sq_A + B;
    q = 1.0/8 * sq_A * A - 1.0/2 * A * B + C;
    r = - 3.0/256*sq_A*sq_A + 1.0/16*sq_A*B - 1.0/4*A*C + D;

    if (IsZero(r))
    {
		/* no absolute term: y(y^3 + py + q) = 0 */

		coeffs[0] = q;
		coeffs[1] = p;
		coeffs[2] = 0;
		coeffs[3] = 1;

		num = SolveCubic(coeffs, s);

		s[num++] = 0;
    }
    else
    {
		/* solve the resolvent cubic ... */

		coeffs[ 0 ] = 1.0/2 * r * p - 1.0/8 * q * q;
		coeffs[ 1 ] = - r;
		coeffs[ 2 ] = - 1.0/2 * p;
		coeffs[ 3 ] = 1;

		(void) SolveCubic(coeffs, s);

		/* ... and take the one real solution ... */

		z = s[ 0 ];

		/* ... to build two quadric equations */

		u = z * z - r;
		v = 2 * z - p;

		if (IsZero(u))
			u = 0;
		else if (u > 0)
			u = sqrt(u);
		else
			return 0;

		if (IsZero(v))
			v = 0;
		else if (v > 0)
			v = sqrt(v);
		else
			return 0;

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

		num = SolveQuadric(coeffs, s);

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

		num += SolveQuadric(coeffs, s + num);
    }

    /* resubstitute */

    sub = 1.0/4 * A;

    for (i = 0; i < num; ++i)
		s[ i ] -= sub;

    return num;
}