/// <summary>
/// Fits a cubic spline to the list of control points.
/// </summary>
/// <param name="transferPoints">Control points to fit a cubic spline to.</param>
public void Calculate(List <T> controlPoints)
{
Debug.Assert(controlPoints.Count >= 3, "A cubic spline cannot be calculated with less " +
"than three points. Otherwise it is just a straight line (with two points).");
// We need to solve the equation taken from: http://mathworld.wolfram.com/CubicSpline.html,
//
// [2 1 0] [D[0]] [3(y[1] - y[0]) ]
// |1 4 1 | |D[1]| |3(y[2] - y[0]) |
// | 1 4 1 | | . | = | . |
// | ..... | | . | | . |
// | 1 4 1| | . | |3(y[n] - y[n-2])|
// [0 1 2] [D[n]] [3(y[n] - y[n-1])]
//
// where n is the number of cubics and D[i] are the derivatives at the control points.
// This linear system of equations happens to be tridiagonal and therefore can be solved
// using the Thomas algorithm (aka TDMA), which is a simplified form of Gaussian elimination
// that can obtain the solution in O(n) instead of O(n^3) required by Gaussian elimination.
// The Thomas algorithm was taken from: http://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm.
int numCubics = controlPoints.Count - 1;
T t = new T();
// Construct the coefficients a,b,c and the right-hand-side vector d.
T[,] matrixCoef = new T[numCubics + 1, 4];
matrixCoef[0, 0] = t.Zero;
matrixCoef[0, 1] = t.Two;
matrixCoef[0, 2] = t.One;
matrixCoef[0, 3] = t.Mult(t.Sub(controlPoints[1], controlPoints[0]), 3f);
for (int i = 1; i < numCubics; ++i)
{
matrixCoef[i, 0] = t.One;
matrixCoef[i, 1] = t.Four;
matrixCoef[i, 2] = t.One;
matrixCoef[i, 3] = t.Mult(t.Sub(controlPoints[i + 1], controlPoints[i - 1]), 3f);
}
matrixCoef[numCubics, 0] = t.One;
matrixCoef[numCubics, 1] = t.Two;
matrixCoef[numCubics, 2] = t.Zero;
matrixCoef[numCubics, 3] = t.Mult(t.Sub(controlPoints[numCubics], controlPoints[numCubics - 1]), 3f);
// Modify the coefficients (Thomas algorithm).
matrixCoef[0, 2] = t.Div(matrixCoef[0, 2], matrixCoef[0, 1]);
matrixCoef[0, 3] = t.Div(matrixCoef[0, 3], matrixCoef[0, 1]);
for (int i = 1; i < numCubics; ++i)
{
T val = t.Sub(matrixCoef[i, 1], t.Mult(matrixCoef[i - 1, 2], matrixCoef[i, 0]));
matrixCoef[i, 2] = t.Div(matrixCoef[i, 2], val);
matrixCoef[i, 3] = t.Div(t.Sub(matrixCoef[i, 3], t.Mult(matrixCoef[i - 1, 3], matrixCoef[i, 0])), val);
}
// Back substitute to solve for the derivaties (Thomas algorithm).
T[] derivatives = new T[numCubics + 1];
derivatives[numCubics - 1] = matrixCoef[numCubics - 1, 3];
for (int i = numCubics - 2; i >= 0; --i)
{
derivatives[i] = t.Sub(matrixCoef[i, 3], t.Mult(matrixCoef[i, 2], derivatives[i + 1]));
}
// Use the derivatives to solve for the coefficients of the cubics.
cubics = new Cubic <T> [numCubics];
for (int i = 0; i < numCubics; ++i)
{
T a = controlPoints[i];
T b = derivatives[i];
T c = t.Sub(t.Sub(t.Mult(t.Sub(controlPoints[i + 1], controlPoints[i]), 3f), t.Mult(derivatives[i], 2f)), derivatives[i + 1]);
T d = t.Add(t.Add(t.Mult(t.Sub(controlPoints[i], controlPoints[i + 1]), 2f), derivatives[i]), derivatives[i + 1]);
cubics[i] = new Cubic <T>(a, b, c, d);
}
}
