public void EllipticFSpecialCases()
{
foreach (double phi in TestUtilities.GenerateUniformRealValues(0.0, Math.PI / 2.0, 4))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.EllipticF(phi, 0.0), phi));
}
}
public void EllipticFBetaRelationship()
{
foreach (double phi in TestUtilities.GenerateUniformRealValues(0.0, Math.PI / 2.0, 8))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.EllipticF(phi, 1.0 / Math.Sqrt(2.0)),
AdvancedMath.Beta(1.0 / 2.0, 1.0 / 4.0, 1.0 - MoreMath.Pow(Math.Cos(phi), 4)) / Math.Sqrt(8.0)
));
}
}
public void EllipticFCompleteAgreement()
{
foreach (double k in TestUtilities.GenerateRealValues(0.01, 1.0, 8))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.EllipticF(Math.PI / 2.0, k),
AdvancedMath.EllipticK(k)
));
}
}
/// <summary>
/// See this paper: http://archive.bridgesmathart.org/2016/bridges2016-179.pdf
/// This logically belongs in HyperbolicModels.cs, but I didn't want the dependency on Meta.Numerics in R3.Core.
/// </summary>
public static Vector3D SquareToPoincare(Vector3D s)
{
double K_e = AdvancedMath.EllipticF(Math.PI / 2, 1.0 / Math.Sqrt(2));
Complex a = new Complex(1, -1) / Math.Sqrt(2);
Complex b = new Complex(1, 1) / 2;
Complex z = s;
Complex result = a * JacobiCn(K_e * (b * z - 1), 1 / Math.Sqrt(2));
return(Vector3D.FromComplex(result));
}
public void JacobiAsInverse()
{
foreach (double k in TestUtilities.GenerateRealValues(1.0E-2, 1.0, 3))
{
foreach (double u in TestUtilities.GenerateRealValues(1.0E-1, Math.PI, 3))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.EllipticF(Math.Asin(AdvancedMath.JacobiSn(u, k)), k), u
));
}
}
}
public void EllipticFIntegral()
{
foreach (double k in TestUtilities.GenerateRealValues(1.0E-2, 1.0, 8))
{
foreach (double phi in TestUtilities.GenerateUniformRealValues(0.0, Math.PI / 2.0, 4))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
FunctionMath.Integrate(t => AdvancedMath.EllipticF(t, k) / Math.Sqrt(1.0 - MoreMath.Pow(k * Math.Sin(t), 2)), Interval.FromEndpoints(0.0, phi)),
MoreMath.Pow(AdvancedMath.EllipticF(phi, k), 2) / 2.0
));
}
}
}
public void EllipticFDisplacement()
{
foreach (int m in TestUtilities.GenerateUniformIntegerValues(-100, 100, 4))
{
foreach (double phi in TestUtilities.GenerateUniformRealValues(-Math.PI / 2.0, +Math.PI / 2.0, 4))
{
foreach (double k in TestUtilities.GenerateRealValues(1.0E-4, 1.0, 4))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.EllipticF(m * Math.PI + phi, k),
2 * m * AdvancedMath.EllipticK(k) + AdvancedMath.EllipticF(phi, k)
));
}
}
}
}
public void EllipticFIntegration()
{
foreach (double k in TestUtilities.GenerateRealValues(1.0E-2, 1.0, 8))
{
Func <double, double> f = delegate(double t) {
double z = k * Math.Sin(t);
return(1.0 / Math.Sqrt(1.0 - z * z));
};
foreach (double phi in TestUtilities.GenerateUniformRealValues(0.0, Math.PI / 2.0, 8))
{
Interval i = Interval.FromEndpoints(0.0, phi);
Assert.IsTrue(TestUtilities.IsNearlyEqual(
FunctionMath.Integrate(f, i), AdvancedMath.EllipticF(phi, k)
));
}
}
}
