public static CubicBezier[] BuildCardinalSpline(Coordinates[] pts, Coordinates?m0, Coordinates?mn, double tension)
{
if (pts.Length < 2)
{
throw new ArgumentException("There must be at least 2 points to construct cardinal spline", "pts");
}
// compute reasonable starting and ending tangents if not supplied
if (m0 == null)
{
m0 = tension * (pts[1] - pts[0]);
}
if (mn == null)
{
mn = tension * (pts[pts.Length - 1] - pts[pts.Length - 2]);
}
if (pts.Length == 2)
{
return(new CubicBezier[] { CubicBezier.FromCubicHermite(pts[0], m0.Value, pts[1], mn.Value) });
}
// we have > 2 points, so we will have n-1 beziers
CubicBezier[] beziers = new CubicBezier[pts.Length - 1];
// fill in the first bezier
beziers[0] = CubicBezier.FromCubicHermite(pts[0], m0.Value, pts[1], tension * (pts[2] - pts[0]));
// iterate through and fill in all but the last
for (int i = 1; i < pts.Length - 2; i++)
{
beziers[i] = CubicBezier.FromCubicHermite(pts[i], tension * (pts[i + 1] - pts[i - 1]), pts[i + 1], tension * (pts[i + 2] - pts[i]));
}
// fill int the last bezier
beziers[pts.Length - 2] = CubicBezier.FromCubicHermite(pts[pts.Length - 2], tension * (pts[pts.Length - 1] - pts[pts.Length - 3]), pts[pts.Length - 1], mn.Value);
// return the beziers
return(beziers);
}
public static CubicBezier[] BuildC2Spline(Coordinates[] pts, Coordinates?m0, Coordinates?mn, double tension)
{
if (pts.Length < 2)
{
throw new ArgumentException("There must be at least 2 points to construct cardinal spline", "pts");
}
// compute reasonable starting and ending tangents if not supplied
if (m0 == null)
{
m0 = tension * (pts[1] - pts[0]);
}
if (mn == null)
{
mn = tension * (pts[pts.Length - 1] - pts[pts.Length - 2]);
}
if (pts.Length == 2)
{
return(new CubicBezier[] { CubicBezier.FromCubicHermite(pts[0], m0.Value, pts[1], mn.Value) });
}
int n = pts.Length - 1;
// we have > 2 points, so we will have n-1 beziers
CubicBezier[] beziers = new CubicBezier[n];
// build up constraint matrix
Matrix A = new Matrix(2 * n, 2 * n, 0);
Matrix b = new Matrix(2 * n, 2);
// matrix row index value
int idx = 0;
// add starting/ending tangent constraint
Coordinates p01 = m0.Value / 3.0 + pts[0];
A[idx, 0] = 1;
b[idx, 0] = p01.X; b[idx, 1] = p01.Y;
idx++;
Coordinates pn2 = -mn.Value / 3.0 + pts[n];
A[idx, 2 * n - 1] = 1;
b[idx, 0] = pn2.X; b[idx, 1] = pn2.Y;
idx++;
// add C1 constraints
for (int i = 0; i < n - 1; i++)
{
A[idx, i * 2 + 1] = 1;
A[idx, (i + 1) * 2] = 1;
b[idx, 0] = pts[i + 1].X * 2; b[idx, 1] = pts[i + 1].Y * 2;
idx++;
}
// add C2 constraints
for (int i = 0; i < n - 1; i++)
{
A[idx, i * 2] = 1;
A[idx, i * 2 + 1] = -2;
A[idx, (i + 1) * 2] = 2;
A[idx, (i + 1) * 2 + 1] = -1;
b[idx, 0] = 0; b[idx, 1] = 0;
idx++;
}
// build the LUDecomposition of A to solve system A*P = b;
LuDecomposition lu = new LuDecomposition(A);
Matrix P = lu.Solve(b);
// work back the Parameters
for (int i = 0; i < n; i++)
{
beziers[i] = new CubicBezier(pts[i], new Coordinates(P[2 * i, 0], P[2 * i, 1]), new Coordinates(P[2 * i + 1, 0], P[2 * i + 1, 1]), pts[i + 1]);
}
return(beziers);
}