Geodesics on a cone Solve the problem of finding a shortest path over the surface of a cone of semi-angle α by the calculus of variations. Take the equation of the path in the form ρ=ρ(θ), where ρ is distance from the vertex O and θ is the cylindrical polar angle measured around the axis of the cone. Obtain the general expression for the path length and find the extremal that satisfies the end conditions ρ(−π/2)=ρ(π/2)=a.
Verify that this extremal is the same as the shortest path that would be obtained by developing the cone on to a plane.