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I define a sheet (Monge's) surface as [tex]z=z(x,y)[/tex] where z(x,y) is a quadric defined by :

[tex]z=-\, \frac{1}{2} \mathbf{x}^T \mathbb{Q} \mathbf{x} [/tex]

where [tex]\mathbf{x}^T=[x, y] [/tex] and [tex]\mathbb{Q}[/tex] a (symetric but not diagonal) curvature matrix.

I've attached to this post a figure to describe the geometry.

My question is : how can I manage to evaluate s(x,y) (the length of the red curve), knowing x and y (the red curve is starting from the origin) ? I think s(x,y) can be evaluate by a curvilinear integral, but I don't know how to write it...

Thanks in advance for your answers.

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# Evaluate a length on a quadric surface

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