// Compute a length of a spline segment by using 5-point Legendre-Gauss quadrature
// https://medium.com/@all2one/how-to-compute-the-length-of-a-spline-e44f5f04c40
// https://en.wikipedia.org/wiki/Gaussian_quadrature
internal static float GetSplineSegmentLength(SplinePoint start, SplinePoint end)
{
Vector3 GetDerivative(float t)
{
// Cubic Hermite spline derivative coeffcients
Vector3 c0 = start.GetHandle(Direction.Forward);
Vector3 c1 = 6.0f * (end.position - start.position) - 4.0f * start.GetHandle(Direction.Forward) - 2.0f * end.GetHandle(Direction.Forward);
Vector3 c2 = 6.0f * (start.position - end.position) + 3.0f * (start.GetHandle(Direction.Forward) + end.GetHandle(Direction.Forward));
return(c0 + t * (c1 + t * c2));
}
GaussLengendreCoefficient[] coefficients =
{
new GaussLengendreCoefficient(0.0f, 0.5688889f),
new GaussLengendreCoefficient(-0.5384693f, 0.47862867f),
new GaussLengendreCoefficient(0.5384693f, 0.47862867f),
new GaussLengendreCoefficient(-0.90617985f, 0.23692688f),
new GaussLengendreCoefficient(0.90617985f, 0.23692688f),
};
float length = 0.0f;
foreach (GaussLengendreCoefficient coefficient in coefficients)
{
// This and the final (0.5 *) below are needed for a change of interval to [0, 1] from [-1, 1]
float t = 0.5f * (1.0f + coefficient.abscissa);
length += GetDerivative(t).magnitude *coefficient.weight;
}
return(0.5f * length);
}
internal static void DrawHandle(SplinePoint point, Direction direction, DebugColors.DebugColor color)
{
// Draw a Handle gizmo
var endPoint = point.GetHandle(direction);
Gizmos.color = color.wire;
Gizmos.DrawLine(point.transform.position, endPoint);
Gizmos.DrawSphere(endPoint, 0.05f);
}
internal static Vector3 EvaluateSplineSegment(SplinePoint pointA, SplinePoint pointB, float t)
{
var omt = 1.0f - t;
var omt2 = omt * omt;
var t2 = t * t;
// Get position in world space of value t between points A and B
return(pointA.transform.position * (omt2 * omt) +
pointA.GetHandle(Direction.Forward) * (3.0f * omt2 * t) +
pointB.GetHandle(Direction.Backward) * (3.0f * omt * t2) +
pointB.transform.position * (t2 * t));
}
internal static Vector3 EvaluateSplineSegmentNormal(SplinePoint pointA, SplinePoint pointB, float t)
{
// Evaluate the normal for value t between points A and B
return(((6.0f * (t * t)) - (6.0f * t)) * pointA.position + ((3.0f * (t * t)) - ((4.0f * t) + 1)) * pointA.GetHandle(Direction.Forward)
+ ((-6.0f * (t * t)) + (6.0f * t)) * pointB.position + ((3.0f * (t * t)) - (2.0f * t)) * pointB.GetHandle(Direction.Backward));
}