Geodesics on Some Surfaces
This Demonstration shows geodesic curves on some selected surfaces. Mathematica's built-in function NDSolve is used to solve the appropriate differential equation and functions are defined to calculate Christoffel coefficients and the second fundamental form (not actually needed, but it could be used for possible extensions for asymptotic curves). You can choose from 11 surfaces; the program is easily extendable to other surfaces.
 E. Abbena, S. Salamon, and A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd ed., Boca Raton, FL: Chapman & Hall/CRC, 2006.