  Consider a frictionless surface S in a gravitational field that need not be
uniform. Given two points A and B on S, what curve is traced out by a particle
that starts at A and reaches B in the shortest time? This paper considers this
problem on simple surfaces such as surfaces of revolution and solves the
problem two ways: First, we use conservation of mechanical energy and the
Euler-Lagange equation; second, we use geometrical optics and the eikonal
equation. We conclude with a discussion of the relativistic effects at
relativistic velocities.
