experimental/Intersection/CubicUtilities.cpp

#include "CubicUtilities.h"
#include "DataTypes.h"
#include "QuadraticUtilities.h"

void coefficients(const double* cubic, double& A, double& B, double& C, double& D) {
    A = cubic[6]; // d
    B = cubic[4] * 3; // 3*c
    C = cubic[2] * 3; // 3*b
    D = cubic[0]; // a
    A -= D - C + B;     // A =   -a + 3*b - 3*c + d
    B += 3 * D - 2 * C; // B =  3*a - 6*b + 3*c
    C -= 3 * D;         // C = -3*a + 3*b
}

// cubic roots

const double PI = 4 * atan(1);

static bool is_unit_interval(double x) {
    return x > 0 && x < 1;
}

// from SkGeometry.cpp (and Numeric Solutions, 5.6)
int cubicRoots(double A, double B, double C, double D, double t[3]) {
    if (approximately_zero(A)) {  // we're just a quadratic
        return quadraticRoots(B, C, D, t);
    }
    double a, b, c;
    {
        double invA = 1 / A;
        a = B * invA;
        b = C * invA;
        c = D * invA;
    }
    double a2 = a * a;
    double Q = (a2 - b * 3) / 9;
    double R = (2 * a2 * a - 9 * a * b + 27 * c) / 54;
    double Q3 = Q * Q * Q;
    double R2MinusQ3 = R * R - Q3;
    double adiv3 = a / 3;
    double* roots = t;
    double r;

    if (R2MinusQ3 < 0)   // we have 3 real roots
    {
        double theta = acos(R / sqrt(Q3));
        double neg2RootQ = -2 * sqrt(Q);

        r = neg2RootQ * cos(theta / 3) - adiv3;
        if (is_unit_interval(r))
            *roots++ = r;

        r = neg2RootQ * cos((theta + 2 * PI) / 3) - adiv3;
        if (is_unit_interval(r))
            *roots++ = r;

        r = neg2RootQ * cos((theta - 2 * PI) / 3) - adiv3;
        if (is_unit_interval(r))
            *roots++ = r;
    }
    else                // we have 1 real root
    {
        double A = fabs(R) + sqrt(R2MinusQ3);
        A = cube_root(A);
        if (R > 0) {
            A = -A;
        }
        if (A != 0) {
            A += Q / A;
        }
        r = A - adiv3;
        if (is_unit_interval(r))
            *roots++ = r;
    }
    return (int)(roots - t);
}