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Hello,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.
[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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