Evaluate a length on a quadric surface

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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.

 

[pat_connell]

The answer to your question is that you need to use the formula for arc length which is:s(x,y) = \int_{0}^{y} \sqrt{1 + (\frac{\partial z}{\partial x})^2 + (\frac{\partial z}{\partial y})^2}dxWhere z=z(x,y) is your Monge's surface. You can then use the chain rule to evaluate the partial derivatives of z with respect to x and y.Hope this helps!

 

[tacman]

Hello,

Thank you for providing the definition and figure for the quadric surface you are working with. Evaluating a length on a quadric surface can be done using a curvilinear integral, as you mentioned. The general formula for a curvilinear integral is:

\int_C ds = \int_a^b \sqrt{dx^2 + dy^2}

where C is the curve, a and b are the starting and ending points, and ds is the infinitesimal length element of the curve.

In your case, the curve you are interested in is the red curve starting from the origin. To evaluate the length of this curve, you will need to first parameterize the curve. This can be done by defining x and y in terms of a parameter t, such as x(t) and y(t). Then, you can plug these parameterized values into the general formula for the curvilinear integral to find the length of the curve.

For example, if your parameterization is x(t) = t and y(t) = t^2, then the length of the curve from t=0 to t=1 would be:

\int_0^1 \sqrt{(dx/dt)^2 + (dy/dt)^2} dt = \int_0^1 \sqrt{1 + 4t^2} dt

This integral can then be evaluated using integration techniques to find the length of the curve.

I hope this helps you to evaluate the length of the red curve on your quadric surface. Best of luck in your calculations!