Curvature, Turning Numbers and Winding Numbers - Differential Geometry

Explore the advanced concepts of planar curvature in differential geometry, including turning numbers, winding numbers, and their connections to complex analysis. Delve into the Dog on a Leash theorem and its applications. Examine parametric curves, with a focus on uniform circular motion and the role of circular functions. Learn about reparametrization techniques for achieving unit speed curves. Gain insights into the degree of maps from circles to circles and their significance in mathematics.

Introduction
Turning number
Rational parameterization
Winding numbers
Dog on a leash theorem
Degrees
Constant motion
Circular motion