A curve in the plane is determined, up to orientation-preserving Euclidean motions, by its curvature function, $\kappa(s)$. Here is one of my favorite examples, from Alfred Gray's book, Modern
Lecture 17—Discrete Curvature II (Variational Viewpoint) – CS 15
Geodesics, geodesic curvature, geodesic parallels, geodesic
Distance distributions and inverse problems for metric measure
What are applications of convex sets and the notion of convexity
Differential Geometry: Curves — Surfaces — Manifolds, Third Edition
Differential Geometry: calculating Gaussian and Mean Curvature two
Read Differential Geometry of Curves and Surfaces by Kristopher Tapp available from Rakuten Kobo. This is a textbook on differential geometry
Differential Geometry of Curves and Surfaces
Principal curvature - Wikipedia
Differential Geometry: Lecture 15 part 3: Gaussian and Mean
Curvature of a surface, only using calculus