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So lets say I have a mesh and for each face I have the position of its $3$ vertices and the area of the face.

So let's say I have a point $p$ on this face and a vector $v$ that goes from $p$ to the camera/viewer. Let $x$ be its normalized projection onto the tangent plane to the surface $S$. What is the curvature at point $p$ in direction $x$?

First, how would I go about finding the tangent plane to this surface and projecting $v$ onto it? Then where do I go from there?

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Is a mesh a collection of triangles? Is $S$ the mesh? If not, what "surface" are you talking about? What do "tangent plane" and "curvature" mean for a collection of triangles? Are you talking about properties of some forgotten surface that the mesh approximates? –  bubba Mar 8 '13 at 12:42

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