Geometric Constructions in the Digital Plane

Abstract

We adapt several important properties from affine geometry so that they become applicable in the digital plane. Each affine property is first reformulated as a property about line transversals. Known results about transversals are then used to derive Helly type theorems for the digital plane. The main characteristic of a Helly type theorem is that it expresses a relation holding for a collection of geometric objects in terms of simpler relations holding for some of the subcollections. For example, we show that in the digital plane a collection of digital lines is parallel if and only if each of its 2-membered subcollections consists of two parallel digital lines. The derived Helly type theorems lead to many applications in digital image processing. For example, they provide an appropriate setting for verifying whether lines detected in a digital image satisfy the constraints imposed by a perspective projection. The results can be extended to higher dimensions or to other geometric systems, such as projective geometry.