Comments by "EebstertheGreat" (@EebstertheGreat) on "The Continuity of Splines" video.

  1. There is also a nice way to define geometric continuity for general curves, not just splines. A parametric curve is in the geometric continuity class GCⁿ iff it is rectifiable and it becomes Cⁿ when reparameterized by arclength. Or equivalently, if there is any regular reparameterization that is Cⁿ (a reparameterization is "regular" if its first derivative is never the zero vector). Sometimes it is also required that the parameterizations be orientation-preserving. So if a given curve c ∈ R^m has parametric representations p : [a,b]→R^m and q : [c,d]→R^m, then these preserve orientation iff there is a function φ: [a,b]→[c,d] such that p = φ ∘ q and φ(t) > 0 for all t in [a,b]. If there is such a q and orientation-preserving φ for which q ∈ Cⁿ , then p ∈ GCⁿ. I don't know how to generalize this to parametric surfaces, though.
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