Curvature and Osculating Circle

This lecture covers the concept of curvature in a plane curve, indicating how much the curve locally deviates from a straight line. It explains how to find the osculating circle of a curve at a given point, involving the center and radius of curvature. By taking two perpendiculars to the curve at two points infinitely close and calculating their intersection, the osculating circle can be determined. The lecture also discusses the evolute of a plane curve, which represents the locus of its centers of curvature. Parametric representations of curves and the cycloid trajectory are explored, along with the parametric equations of the evolute. Various examples and equations are provided to illustrate these geometric concepts.