/// <summary>
/// Curvature for Uniform Rational B-Spline curve
/// An algorithm to compute the point on a B-spline curve and all derivatives up to and
/// including the dth at a fixed u value
/// Output is the array CK[], where CK [k] is the kth derivative, 0 ≦ k ≦ d.
/// K = (x'y" - y'x") / (x'^2 + y'^2)^(3/2)
/// Note: In order to make turning right has plus value, we use
/// K = (y'x" - x'y") / (x'^2 + y'^2)^(3/2)
/// Note: Calculating curvature for NURBS is complicated.
/// </summary>
/// <param name="u">interval parameter 0 ≦ u ≦ 1</param>
/// <returns>Curvature at u</returns>
public double Curvature(double u)
{
if (Degree < 3)
{
return(0);
}
int du = 2;
double[] CK_x = new double[du + 1];
double[] CK_y = new double[du + 1];
int span = NURBS.FindSpan(Degree, u, KnotVector);
double[][] nders = NURBS.DerBasisFuns(span, u, Degree, du, KnotVector);
for (int k = 1; k <= du; k++)
{
for (int j = 0; j <= Degree; j++)
{
CK_x[k] += nders[k][j] * ControlPoints[span - Degree + j].X;
CK_y[k] += nders[k][j] * ControlPoints[span - Degree + j].Y;
}
}
double curvature = (CK_y[1] * CK_x[2] - CK_x[1] * CK_y[2]) / Math.Pow(CK_x[1] * CK_x[1] + CK_y[1] * CK_y[1], 1.5);
return(curvature);
}
private double RoundTangent(double tangent)
{
if (NURBS.IsZero(tangent))
{
return(0.0);
}
else if (Math.Abs(tangent) > NURBS.MAX_VALUE)
{
return(NURBS.MAX_VALUE);
}
else
{
return(tangent);
}
}
/// <summary>
/// Non Rational B-Spline curve point
/// </summary>
/// <param name="u">interval parameter 0 ≦ u ≦ 1</param>
/// <returns></returns>
public GPoint NonRationalBSpline(double u)
{
int span = NURBS.FindSpan(Degree, u, KnotVector);
double[] basisFuns = NURBS.BasisFuns(span, Degree, KnotVector, u);
double x = 0.0;
double y = 0.0;
for (int i = 0; i <= Degree; i++)
{
x += basisFuns[i] * ControlPoints[span - Degree + i].X;
y += basisFuns[i] * ControlPoints[span - Degree + i].Y;
}
return(new GPoint(x, y));
}
/// <summary>
/// Rational B-Splines (NURBS) curve
/// </summary>
/// <param name="u">interval parameter 0 ≦ u ≦ 1</param>
/// <returns>curve point at u</returns>
public GPoint RationalBSpline(double u)
{
int span = NURBS.FindSpan(Degree, u, KnotVector);
double[] basisFuns = NURBS.BasisFuns(span, Degree, KnotVector, u);// CalcBasisFuns(span, u);
double x = 0;
double y = 0;
double rationalWeight = 0.0;
for (int i = 0; i <= Degree; i++)
{
GPoint cp = ControlPoints[span - Degree + i];
x += basisFuns[i] * cp.X * cp.W;
y += basisFuns[i] * cp.Y * cp.W;
rationalWeight += basisFuns[i] * cp.W;
}
return(new GPoint(x / rationalWeight, y / rationalWeight));
}
/// <summary>
/// Tangent value at u
/// </summary>
/// <param name="u">interval parameter 0 ≦ u ≦ 1</param>
/// <returns>Tangent at u</returns>
public double Tangent(double u)
{
if (Degree < 2)
{
return(0);
}
int span = NURBS.FindSpan(Degree, u, KnotVector);
int du = 1;
double[][] nders = NURBS.DerBasisFuns(span, u, Degree, du, KnotVector);
double c_x = 0.0;
double c_y = 0.0;
for (int j = 0; j <= Degree; j++)
{
c_x += nders[1][j] * ControlPoints[span - Degree + j].X;
c_y += nders[1][j] * ControlPoints[span - Degree + j].Y;
}
double tangent = c_y / c_x;
return(RoundTangent(tangent));
}
private double[] KnotVector; // Knot vector -> U
/// <summary>
/// Create BURBSCurve
/// </summary>
/// <param name="points">control points</param>
/// <param name="degree">degree of NURBS, order is p + 1</param>
public NURBSCurve(List <GPoint> points, int degree)
{
ControlPoints = points;
Degree = Math.Min(degree, points.Count - 1);
KnotVector = NURBS.CalcualteKnotVector(degree, points.Count);
}