  This paper shows that the well-known curve optimization problems which lead
to the straight line, the catenary curve, the brachistochrone, and the circle,
can all be handled using a unified formalism. Furthermore, from the general
differential equation fulfilled by these geodesics, we can guess additional
functions and the required metric. The parabola, for example, is a geodesic
under a metric guessed in this way. Numerical solutions are found for the
curves corresponding to geodesics in the various metrics using a ray-tracing
approach based on Fermat's principle.
