Differential Geometry of Plane Curves — Selected Topics

Description. This is a topics course covering classical aspects of the differential geometry of plane curves, as well as some contemporary variations and developments (as time permits). The emphasis is less on definitions and more on interesting examples and nice results: the 4-vertex theorem, the Tait–Kneser nesting theorem, evolutes and involutes, bicycle trajectories, the classical curves (cycloid, catenary, tractrix, elasticae, …). The prerequisites are a standard undergraduate vector calculus course, curiosity, and a mind open to new ideas.

Sessions:
1: The osculating circle, the Tait-Knesser Theorem --> Notes
2: Envelopes of curves, The 4 vertex theorem --> Notes
3: The tractrix, bicycle mathematics --> Notes
4: The geometry of Kepler orbits --> Article