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I am trying to detect when a closed parametric surface intersect itself. My surface is described as a triplet of parametric functions $x(u,v)$, $y(u,v)$ and $z(u,v)$ where $u,v\in[0,1]$. For that purpose I'm trying to use the Gauss-Bonnet theorem. This theorem relates the integral of the Gaussian curvature over the surface to the Euler characteristic (which is an invariant of a regular surface).

My questions is the following: will the Euler characteristic computed by the Gauss-Bonnet theorem change if the surface self-intersects, and if so, in what manner?


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The Gauss-Bonnet theorem relates intrinsic properties of the surface, properties that are independent of how it is immersed in the ambient space. You will not be able to detect self-intersections this way. I don't have a good suggestion for how to do this detection.

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I see. In my case I have a Bézier closed surface $M$ without self-intersection, and I'd like to detect the configuration of points that lead to a self-intersecting surface. If I compute the Euler characteristic using the expression $\chi(M)=\int_{M}K dA$ (ignoring the border term) I'll always obtain the correct number when the surface no not self-intersect. But, if the number is not correct, I can guarantee that something went wrong :) – user26400 Mar 8 '12 at 21:49
As I see it, neither side of the equation will change as you deform the surface, regardless of whether or not a self intersection occurs. Of course if $\chi(M)$ were computed from the image itself, rather than from the parameter domain, then it could change, but if you could do that, you would already know whether or not there was a self-intersection. – yasmar Mar 9 '12 at 9:59

To start with can we detect self intersection of regular self-intersecting curves e.g., Lissajou's figures, hypocycloids .. in the plane? If so, the 2-D overwraps could also perhaps be obtained in a similar manner working through the first fundamental form.

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