Hi everyone,(adsbygoogle = window.adsbygoogle || []).push({});

First of all I am not sure if I have chosen the right category for this posting but this looked the most reasonable out.

I have a problem that I would like to solve but I am not sure where to look for answers. It seems like something other people might have worked with before but I am not sure what these types of problems are called.

(Please do excuse the abuse of any notation since I am only learning now about diff. geometry and such).

Basically it is an optimization problem. I have a 2d surface y= F(σ,κ) in ℝ3 and on that surface I have a curve γ. I do not have a parameterisation for the curve, but it is continuous and I can sample it at any point P (see illustration).

http://img694.imageshack.us/img694/9179/surfacekq.png [Broken]

My task is:

Given a random initial point P0 on γ, to find the minimum of F along γ.

I can only move along this (blue) curve γ, backwards or forwards. So the only thing I can adjust is my step length λ.

Now of course I can use various heuristics to solve this, but there must be a more elegant way to solve this problem and someone must have seen these class of problems before.

I do have a closed form analytic expression for the whole F(σ,κ) as well as the first and second partial derivatives at each point. Also note that I have expressions for geodesics, arc elements, Riemmanian metric etc.

Can someone point me to the right direction?

Many thanks

V.

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# Optimisation along a curve on a surface

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