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cou

I was at Grace Hopper and missed this lecture; could someone tell me the answer to this question?

keenan

@cou The answer is "no," and it takes a bit to work through; we didn't do it in class. Since you have 4 control points for a cubic Bezier curve, you have 4x4 = 16 control points for a cubic Bezier patch (see pictures on the previous slide). Suppose you want these patches to meet up with all four neighboring patches. Then you're basically pinning down the locations of the 12 "outer" control points, so that they match the locations of the neighboring patches boundary. This leaves you with four degrees of freedom, corresponding to the 2x2 control points on the interior of the patch.

Can someone come up with a clear argument for why these remaining control points give you enough flexibility to get tangent continuity if you have a regular patch layout (i.e., four patches around every vertex), and might not be enough if you have more/fewer patches around a vertex?