/// <summary>
/// Jacobi の楕円関数 cn u を求める。
/// </summary>
/// <param name="u">引数 u</param>
/// <param name="m">率 k の2乗</param>
/// <returns>cn(u, k)</returns>
public static double Cn(double u, double m)
{
if (m < 0.0 || m > 1.0)
{
return(double.NaN);
}
return(Math.Cos(Elliptic.Phi(u, m)));
}
/// <summary>
/// Jacobi の楕円関数 dn u を求める。
/// </summary>
/// <param name="u">引数 u</param>
/// <param name="m">率 k の2乗</param>
/// <returns>dn(u, k)</returns>
public static double Dn(double u, double m)
{
if (m < 0.0 || m > 1.0)
{
return(double.NaN);
}
double sn = Elliptic.Sn(u, m);
return(Math.Sqrt(1 - m * sn * sn));
}
/// <summary>
/// Jacobi の楕円関数(引数 u と率 k から振幅φおよび sn, cn, dn)を求める。
/// </summary>
/// <param name="u">引数 u</param>
/// <param name="m">率 k の2乗</param>
/// <param name="phi">振幅φ</param>
/// <param name="sn">sn(u, k)</param>
/// <param name="cn">cn(u, k)</param>
/// <param name="dn">dn(u, k)</param>
public static void Jacobi(double u, double m, out double phi, out double sn, out double cn, out double dn)
{
if (m < 0.0 || m > 1.0)
{
phi = double.NaN;
sn = double.NaN;
cn = double.NaN;
dn = double.NaN;
return;
}
if (m < EPSILON)
{
double t = Math.Sin(u);
double b = Math.Cos(u);
double ai = 0.25 * m * (u - t * b);
phi = u - ai;
sn = t - ai * b;
cn = b + ai * t;
dn = 1.0 - 0.5 * m * t * t;
return;
}
if (m >= 1 - EPSILON)
{
double ai = 0.25 * (1.0 - m);
double b = Math.Cosh(u);
double t = Math.Tanh(u);
double binv = 1.0 / b;
double twon = b * Math.Sinh(u);
phi = 2.0 * Math.Atan(Math.Exp(u)) - Constant.PI2 + ai * (twon - u) / b;
sn = t + ai * (twon - u) / (b * b);
ai *= t * phi;
cn = binv - ai * (twon - u);
dn = binv + ai * (twon + u);
return;
}
phi = Elliptic.Phi(u, m);
Elliptic.Jacobi(phi, m, out sn, out cn, out dn);
}
/// <summary>
/// 第1種不完全楕円積分。
/// </summary>
/// <param name="phi">振幅φ</param>
/// <param name="m">率 k の2乗</param>
/// <returns>積分値</returns>
public static double F(double phi, double m)
{
if (m == 0.0)
{
return(phi);
}
if (m == 1.0)
{
if (Math.Abs(phi) >= Math.PI / 2)
{
return(double.NaN);
}
return(Math.Log(Math.Tan((Math.PI / 2 + phi) / 2.0)));
}
int npio2 = (int)Math.Floor(phi / (Math.PI / 2));
if ((npio2 & 1) != 0)
{
++npio2;
}
double K = npio2 == 0 ? Elliptic.K(1.0 - m) : 0.0;
phi -= npio2 * Math.PI / 2;
int sign = phi < 0.0 ? -1 : 1;
phi = Math.Abs(phi);
double t = Math.Tan(phi);
if (Math.Abs(t) > 10.0)
{
/* Transform the amplitude */
double e = 1.0 / (Math.Sqrt(1.0 - m) * t);
/* ... but avoid multiple recursions. */
if (Math.Abs(e) < 10.0)
{
if (npio2 == 0)
{
K = Elliptic.K(1.0 - m);
}
double ret = K - Elliptic.F(Math.Atan(e), m);
return(sign * ret + npio2 * K);
}
}
double a = 1.0;
double b = Math.Sqrt(1.0 - m);
double c = Math.Sqrt(m);
int d = 1;
int mod = 0;
while (Math.Abs(c / a) > EPSILON)
{
double tmp = b / a;
phi = phi + (Math.Atan(t * tmp) + mod * Math.PI);
mod = (int)((phi + Math.PI / 2) / Math.PI);
t = t * (1.0 + tmp) / (1.0 - tmp * t * t);
c = (a - b) / 2.0;
tmp = Math.Sqrt(a * b);
a = (a + b) / 2.0;
b = tmp;
d += d;
}
return(sign * (Math.Atan(t) + mod * Math.PI) / (d * a) + npio2 * K);
}