Given a control polygon with five pairwise different points $d_0,...,d_4$ what is the minimum order of B-Spline curve for this polygon such that it is $C^2$ continuous ?
2026-03-26 01:28:44.1774488524
minimum order of bspline curve for C2 continuity
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The B-spline needs to be at least order 4 (i.e., degree 3) to have $C^2$ continuity for the entire curve.
A B-spline curve's continuity is decided upon is degree and its knot sequence. In general, its continuity is at best (degree-1). If the knot sequence has multiple interior knots such as [0.,0.,0.,0., 0.5, 0.5, 1., 1., 1., 1.], then the continuity will become (degree - knot multiplicity).
There are some special configuration of control poles that would actually make the continuity higher than the degree. For example, if we convert a straight line to a degree 1 B-spline curve or a circular arc into a rational quadratic B-spline curves, the continuity is not reduced as both a straight line and a circular arc is $C^{\infty}$. But these are special cases.