We describe surfaces and geodesics without assuming prior knowledge of differential geometry. This involves selecting and presenting basic definitions and theorems. Included in this discussion are definitions of surface, coordinate patch, curvature, geodesic, etc. This summary closes with a proof of the length-minimizing properties of geodesics. Examples of surfaces of constant gaussian curvature are given and plotted in Mathematica. We also describe geodesics on these surfaces and plot select examples.

Author Bio

I am currently a senior math major at Columbia University. I am also in adouble degree program with the Jewish Theological Seminary of America,where I am a modern Jewish studies major. This article began as a finalproject for a Differential Geometry class taught by Professor Charles Doran inthe fall of 2000. Following graduation, I plan to teach mathematics in the New York City public schools, then return to school to become a mathematician.