# Evaluate a length on a quadric surface

1. Jul 19, 2006

### Hash

Hello,

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

$$z=-\, \frac{1}{2} \mathbf{x}^T \mathbb{Q} \mathbf{x}$$

where $$\mathbf{x}^T=[x, y]$$ and $$\mathbb{Q}$$ 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...

#### Attached Files:

• ###### geometrie_conforme.png
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Last edited: Jul 19, 2006