public Vector3d GeodesicTorsionAt(double t)
{
var c = Curve.DerivativeAt(t, 4);
var theta = Integrate.AdaptiveSimpson(u => TorsionAt(u), Curve.Domain.T0, t, 0.001);
var theta_1 = TorsionAt(t);
var a = Vector3d.CrossProduct(c[1], c[2]);
var a_1 = Vector3d.CrossProduct(c[1], c[3]) + Vector3d.CrossProduct(c[2], c[2]);
var d = a * c[3];
var d_1 = a * c[4];
var delta = c[1].Length;
var delta_1 = c[1] * c[2] / c[1].Length;
var alpha = a.Length;
var alpha_1 = a * a_1 / a.Length;
var T = c[1] / delta;
var T_1 = (delta * c[2] - c[1] * delta_1) / Math.Pow(delta, 2);
var B = a / alpha;
var B_1 = (alpha * a_1 - a * alpha_1) / Math.Pow(alpha, 2);
var N = Vector3d.CrossProduct(B, T);
var N_1 = Vector3d.CrossProduct(B, T_1) + Vector3d.CrossProduct(B_1, T);
var e2 = N * Math.Cos(theta) + B * Math.Sin(theta);
var e3 = B * Math.Cos(theta) - N * Math.Sin(theta);
var e3_1 = Math.Cos(theta) * (B_1 - N * theta_1) - Math.Sin(theta) * (N_1 + B * theta_1);
return(e3_1 * e2 * c[1] / c[1].SquareLength);
}
public Plane FrameAt(double t)
{
var t0 = Curve.Domain.T0;
var theta = Integrate.AdaptiveSimpson(TorsionAt, t0, t, 0.001);
var c = Curve.DerivativeAt(t, 3);
var a = Vector3d.CrossProduct(c[1], c[2]);
var d = a * c[3];
var delta = c[1].Length;
var alpha = a.Length;
var P = c[0];
var T = c[1] / delta;
var B = a / alpha;
var N = Vector3d.CrossProduct(B, T);
var e2 = N * Math.Cos(theta) + B * Math.Sin(theta);
var e3 = B * Math.Cos(theta) - N * Math.Sin(theta);
return(new Plane(new Point3d(P), e2, e3));
}
