# Boundary Value Problems for Cone Geodesics

Boundary Value Problems for Cone Geodesics

Geodesics are an important concept in differential geometry. Roughly speaking, a geodesic is locally the shortest path joining two points on a surface. However, extending a geodesic does not necessarily give the shortest path between two endpoints. For example, on a sphere the geodesics are arcs of great circles, which are the unique shortest paths between their endpoints only when the arc is smaller than a semicircle.

Two points are chosen at random on the cone and the Demonstration draws the geodesic between them. The geodesic is solved from the Euler–Lagrange equations as a boundary value problem (BVP). Then a golf-like game is devised to find the geodesic connecting the two points solving the canonical geodesic equations as an initial value problem (IVP). The IVP and BVP solutions can be made to coincide; however, there may be another solution that connects the two points. Therefore, geodesics should not be mistaken as the shortest path between two points on a surface.