ABSTRACT
Chapter 1 introduces the topic of differential geometry by studying the local theory of plane curves. A geometric theory is called local when it studies properties of geometric objects at specific points, that is to say properties who definition depends only an what happens near a given point. The chapter reviews the calculus of parametrized plane curves including tangent lines to curves at points. After presenting the curvature function of a curve at a point, we study the order of contact of a curve and the notion of the osculating circle to a curve at a point. The chapter ends with the Fundamental Theorem of Plane Curves, an application of the theorem on the uniqueness and existence of solutions to a differential equation, which has fundamental implications for the curvature function as a geometric descriptor of a curve.
