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Hi everyone

I have a fairly simple question about bezier curves: can you represent n bezier curves that have been continuously joined together by a single bezier curve of degree 3n?

My instinct is to just take the 3n+1 control points and use them for a degree 3n curve, but I'm not sure...

Thanks a lot

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    $\begingroup$ I think there might be an issue with regularity. A piecewise curve of $n$ pieces will only be piecewise differentiable (of some degree probably less than $n$, for example a piecewise cubic Bezier can be $C^1$ but not $C^2$), while a $3n$ degree Bezier curve is $3n$-times continuously differentiable. It's likely that choosing a high-enough degree would give a reasonably good approximation (think using a finite-element approximation (which is amusing in a way, since it is approximating what is often used as a finite-element)). $\endgroup$
    – Victor Dods
    Commented Apr 29, 2012 at 6:48

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No, you can't. The argument given in the comment is correct. Here's a simple example that might be more convincing. Suppose one of the original Bezier curves is linear, say with control points $(0,0)$, $(1,0)$, $(2,0)$, $(3,0)$, and the second one is curved, say with control points $(3,0)$, $(4,0)$, $(5,1)$, $(6,3)$. A parametric polynomial curve can not consist of a region that's linear and a region that's curved, so it can not represent these two original curves.

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