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I have two bicubic Bezier surfaces that will intersect.

According to this paper: http://nishitalab.org/user/nis/cdrom/cad/CAGD91geometric.pdf At the end of page 1. The general genus of intersection curve of these two patches is 433. It is stated this is bad since its greater than 2.

Could you explain in simple english what is the meaning of genus 433?

Does it have any geometric meaning?

Is it number of possible separate intersections of two surfaces?

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There are several more-or-less equivalent definitions of the term "genus". You can Google "genus" or "topological genus" to find out more. None of the definitions have a clear relationship with any intuitive idea of "complexity" (as far as I can see), so I don't find them very helpful.

In the reference you cited, the "genus = 433" statement is not very significant. The point is only that the genus is not zero, and therefore the curve can not be represented exactly by parametric equations using rational functions. This justifies the use of spline approximations, instead, which is the subject of the paper.

If you want to understand why the genus is 433, read the 1988 paper by Katz and Sederberg that's referenced in the one you cited.

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