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

[Charles B. <cb...@cb...>]

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.

At 01:00 PM 7/26/2000 -0700, you wrote:
>>As for G1 vs. C1, G1 means that the tangents of the two connecting
>>patches are identical.  C1 means that the parametric velocities of
>>the two connecting patches are identical.  eg. if you had a particle
>>flying along a bezier curve at a constant speed in parametric
>>coordinates, then it would keep going the same direction at a G1
>>joint, but it would suddenly change speed (which it wouldn't do with
>>C1).  G1 or G2 is really what we care about for graphics, C1 and C2
>>are unecessarily strong constraints.
>
>So, if I have this right:
>
>Define g(x) = f(x)/|f(x)| for all |f(x)| != 0
>
>Then g C0 continuous at y => f is G0 at y?
>
>What would you call that property, i.e. if C0 == "continuous", what is G0?
>
>Tony Cox - DirectX Luminary
>Windows Gaming Developer Relations Group
>http://msdn.microsoft.com/directx 
>
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--------------------------------------
Charles Bloom           www.cbloom.com