RE: [Algorithms] C1 vs G1 (Was: "N-Patches")

[Tony C. <to...@mi...>]

>I don't thing there is any meaning of "G0".  In equations,
>C(n) and G(n) conditions for connecting parametric lines
>are:
>
>let V(u) be a vector-valued function of the scalar parameter u,
>with u in [0,1]
>similarly for W(u).
>
>Then C0/G0 is V(1) = W(0)
>
>C1 is V'(1) = W'(0)
>
>where a prime (') indicates differentiation with respect to the
>parameter, and V'(1) implicitly means
>
>  [ d/du V(u) ] @ u=1
>
>Then G1 is
>
>Normalize[ V'(1) ] = Normalize[ W'(0) ]
>
>Similarly for Cn, just do n differentiations.

I'm just trying to get a picture in my head of this. I've always used a
somewhat more general definition of what C(n) continutity means:

For a function f:A->B where A and B are metric spaces:

f is C(n) continuous at y <=> f is n-times differentiable at y and the nth
derivative is continuous.

I think what you're saying is the same thing as what I'm saying. I'd just
like the neatness of being able to say G(n) means C(n-1) continuous with the
added condition that the n-1th derivative satisfies the condition I gave for
G0 continuity (with the obvious norm induced on B via the metric).

I guess the difference is that my background is in pure maths, where we
would use a more general definition of C(n) continuity (anything that
involves talking about a specific range of a finite-dimensional parameter
space seems a little sloppy to me), but where I never came across the G(n)
beast. Whereas the G(n) concept is clearly useful in computer graphics
(which I'm guessing is your background) in relation to parametric surfaces
so that is the way you wind up defining it.

Tony Cox - DirectX Luminary
Windows Gaming Developer Relations Group
http://msdn.microsoft.com/directx