A Survey on 2D Euclidean Curve Classes in Discrete Geometry with New Results

Classes of curves like par-regularity, \(\mu \) -reach, locally turn boundedness, quasi-regularity (and their generalizations) have been defined so as to guarantee geometrical or topological properties under discretization. An overview of their inter-relations is given. A focus is made on the Locally Turn Bounded (LTB) curves, a class having good discretization properties. It has already been shown that being LTB implies quasi-regularity. In this paper, it is shown that the LTB curves have a positive \(\mu \) -reach. Moreover, we show that a LTB curve having a Lipschitz turn is par-regular.