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Determining how sharply curved a mesh is at each point

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If we have a surface represented as a mesh with a normal at each vertex, and/or a normal map, either way so that a normal at any point can be obtained...

 

... how might we calculate how flat/curved the surface is at each point? Sorry for the loose wording, but I don't even know exactly what it is I want to measure! In 2D it's just the rate of change of the normal to a line, but in 3D...

 

It sounds like something there will be common methods used which basically give a reasonable approach. I want to apply this to the skin on a human body. For example on fingers, these are approximately cylindrical so in one direction they are totally flat but in the other they are very tightly curved. I think I'd want to take the most curved dimension in this scenario.

 

Any thoughts?

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