/// <summary>
/// Determines the derivatives of a curve at a given parameter.<br/>
/// <em>Corresponds to algorithm 3.2 from The NURBS Book by Piegl and Tiller.</em>
/// </summary>
/// <param name="curve">The curve object.</param>
/// <param name="parameter">Parameter on the curve at which the point is to be evaluated.</param>
/// <param name="numberDerivs">Integer number of basis functions - 1 = knots.length - degree - 2.</param>
/// <returns>The derivatives.</returns>
internal static List <Point4> CurveDerivatives(NurbsBase curve, double parameter, int numberDerivs)
{
List <Point4> curveHomogenizedPoints = curve.ControlPoints;
int n = curve.Knots.Count - curve.Degree - 2;
int derivateOrder = numberDerivs < curve.Degree ? numberDerivs : curve.Degree;
Point4[] ck = new Point4[numberDerivs + 1];
int knotSpan = curve.Knots.Span(n, curve.Degree, parameter);
List <Vector> derived2d = DerivativeBasisFunctionsGivenNI(knotSpan, parameter, curve.Degree, derivateOrder, curve.Knots);
for (int k = 0; k < derivateOrder + 1; k++)
{
for (int j = 0; j < curve.Degree + 1; j++)
{
double valToMultiply = derived2d[k][j];
Point4 pt = curveHomogenizedPoints[knotSpan - curve.Degree + j];
for (int i = 0; i < pt.Size; i++)
{
ck[k][i] = ck[k][i] + (valToMultiply * pt[i]);
}
}
}
return(ck.ToList());
}
/// <summary>
/// Inserts a collection of knots on a curve.<br/>
/// <em>Implementation of Algorithm A5.4 of The NURBS Book by Piegl and Tiller.</em>
/// </summary>
/// <param name="curve">The curve object.</param>
/// <param name="knotsToInsert">The set of knots.</param>
/// <returns>A curve with refined knots.</returns>
internal static NurbsBase KnotRefine(NurbsBase curve, IList <double> knotsToInsert)
{
if (knotsToInsert.Count == 0)
{
return(curve);
}
int degree = curve.Degree;
List <Point4> controlPoints = curve.ControlPoints;
KnotVector knots = curve.Knots;
// Initialize common variables.
int n = controlPoints.Count - 1;
int m = n + degree + 1;
int r = knotsToInsert.Count - 1;
int a = knots.Span(degree, knotsToInsert[0]);
int b = knots.Span(degree, knotsToInsert[r]);
Point4[] controlPointsPost = new Point4[n + r + 2];
double[] knotsPost = new double[m + r + 2];
// New control points.
for (int i = 0; i < a - degree + 1; i++)
{
controlPointsPost[i] = controlPoints[i];
}
for (int i = b - 1; i < n + 1; i++)
{
controlPointsPost[i + r + 1] = controlPoints[i];
}
// New knot vector.
for (int i = 0; i < a + 1; i++)
{
knotsPost[i] = knots[i];
}
for (int i = b + degree; i < m + 1; i++)
{
knotsPost[i + r + 1] = knots[i];
}
// Initialize variables for knot refinement.
int g = b + degree - 1;
int k = b + degree + r;
int j = r;
// Apply knot refinement.
while (j >= 0)
{
while (knotsToInsert[j] <= knots[g] && g > a)
{
controlPointsPost[k - degree - 1] = controlPoints[g - degree - 1];
knotsPost[k] = knots[g];
--k;
--g;
}
controlPointsPost[k - degree - 1] = controlPointsPost[k - degree];
for (int l = 1; l < degree + 1; l++)
{
int ind = k - degree + l;
double alfa = knotsPost[k + l] - knotsToInsert[j];
if (Math.Abs(alfa) < GSharkMath.Epsilon)
{
controlPointsPost[ind - 1] = controlPointsPost[ind];
}
else
{
alfa /= (knotsPost[k + l] - knots[g - degree + l]);
controlPointsPost[ind - 1] = (controlPointsPost[ind - 1] * alfa) +
(controlPointsPost[ind] * (1.0 - alfa));
}
}
knotsPost[k] = knotsToInsert[j];
--k;
--j;
}
return(new NurbsCurve(degree, knotsPost.ToKnot(), controlPointsPost.ToList()));
}