This article considers only curves in Euclidean space. Most of the notions presented here have analogues for curves in Riemannian and pseudo-Riemannian manifolds. For a discussion of curves in an arbitrary topological space, see the main article on curves.Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and in the Euclidean space by methods of differential and integral calculus. Starting in antiquity, many specific curves have been thoroughly investigated using the synthetic approach. Differential geometry takes another path: curves ar...