public static CubicResult CubicEquation(float a, float b, float c, float d)
{
//if a is zero???
CubicResult res = new CubicResult();
res.SetAll(-1);
//tried http://easycalculation.com/algebra/learn-cubic-equation.php, ugly due to complex numbers
//trying http://www.1728.org/cubic2.htm
float f = ((3 * c / a) - ((b * b) / (a * a))) / 3;
float g1 = 2 * Mathf.Pow(b, 3) / Mathf.Pow(a, 3);
float g2 = 9 * b * c / (Mathf.Pow(a, 2));
float g3 = 27 * d / a;
float g = (g1 - g2 + g3) / 27;
float h = (Mathf.Pow(g, 2) / 4) + (Mathf.Pow(f, 3) / 27);
if (Mathf.Approximately(h, 0) && Mathf.Approximately(g, 0) && Mathf.Approximately(f, 0))
{
float da = d / a;
res.SetAll(-1f * Mathf.Sign(da) * Mathf.Pow(Mathf.Abs(da), 1f / 3f));
}
else if (h > 0)
{
//1 root
float hsqrt = Mathf.Sqrt(h);
float r = (g / -2) + hsqrt;
float s = Mathf.Pow(r, 1f / 3f);
float t = (g / -2) - hsqrt;
//we only want the REAL roots, thx tho
float u = Mathf.Sign(t) * Mathf.Pow(Mathf.Abs(t), 1f / 3f);
res.r1.r = (s + u) - (b / (3 * a));
float realPart = -(s + u) / 2 - (b / (3 * a));
float iPart = (s - u) * SQRT3f / 2;
res.r2.r = realPart;
res.r2.i = iPart;
res.r3.r = realPart;
res.r3.i = -iPart;
}
else
{
//3 roots
float i = Mathf.Sqrt((Mathf.Pow(g, 2) / 4) - h);
float j = Mathf.Pow(i, 1.0f / 3.0f);
float k = Mathf.Acos(-(g / (2 * i)));
float k3 = k / 3;
float l = -j;
float m = Mathf.Cos(k3);
float n = SQRT3f * Mathf.Sin(k3);
float p = (b / (-3 * a));
res.r1.r = 2 * j * m + p;
res.r2.r = l * (m + n) + p;
res.r3.r = l * (m - n) + p;
}
return(res);
}
//http://www.1728.org/quartic2.htm
static public QuarticResult QuarticEquation(float a, float b, float c, float d, float e)
{
QuarticResult res = new QuarticResult();
res.SetAll(-1);
//divide through by a
b /= a;
c /= a;
d /= a;
e /= a;
a = 1;
float f = c - (3 * b * b / 8);
float g = d + (b * b * b / 8) - (b * c / 2);
float h = e - (3 * b * b * b * b / 256) + (b * b * c / 16) - (b * d / 4);
//determine coeffs for cubic
float acubic = 1;
float bcubic = f / 2;
float ccubic = ((f * f - 4 * h) / 16);
float dcubic = -1 * g * g / 64;
CubicResult cubicRes = CubicEquation(acubic, bcubic, ccubic, dcubic);
int numReals = cubicRes.GetNumReals();
if (numReals == 3)
{
//this method means we get 2 non zero roots, unless all 3 are 0 intersects
//try to grab first if not then second
float r1 = Mathf.Approximately(cubicRes.r1.r, 0) ? cubicRes.r2.r : cubicRes.r1.r;
//try to grab 3rd if not then second
float r2 = Mathf.Approximately(cubicRes.r3.r, 0) ? cubicRes.r2.r : cubicRes.r3.r;
float p = Mathf.Sqrt(r1);
float q = Mathf.Sqrt(r2);
float r = -g / (8 * p * q);
float s = b / (4 * a);
res.r1.r = p + q + r - s;
res.r2.r = p - q - r - s;
res.r3.r = -p + q - r - s;
res.r4.r = -p - q + r - s;
}
else
{
//this method means we get 2 img results
//try to grab first if not then second
ComplexNumber cr1 = cubicRes.r1.IsReal ? cubicRes.r2 : cubicRes.r1;
//try to grab 3rd if not then second
ComplexNumber cr2 = cubicRes.r3.IsReal ? cubicRes.r2 : cubicRes.r3;
ComplexNumber p = cr1.Sqrt();
ComplexNumber q = cr2.Sqrt();
// complex mul by real number zeros the complex part
ComplexNumber pq = p * q;
float rReal = -g / (8 * pq.r);
float sReal = b / (4 * a);
ComplexNumber r = (ComplexNumber)rReal;
ComplexNumber s = (ComplexNumber)sReal;
res.r1 = p + q + r - s;
res.r2 = p - q - r - s;
res.r3 = -p + q - r - s;
res.r4 = -p - q + r - s;
}
return(res);
}