ABSTRACT
This chapter outlines the theory and applications of various concepts arising in two rapidly growing, interrelated areas of geometry: discrete geometry (which deals with topics such as space filling, arrangements of geometric objects, and related combinatorial problems) and computational geometry (which deals with the many aspects of the design and analysis of geometric algorithms). The chapter introduces several categories of concepts: vector and point configurations, hyperplanes and half-spaces, convexity, and some combinatorial parameters associated with vector or point configurations, relevant to applications: signed bases, semi-spaces, k-sets, and centerpoints. Line arrangements and affine point configurations in the plane are related via polar-duality, a transformation which is better understood in terms of 3-dimensional vectors and central planes or using the projective and spherical models. As a consequence, theorems and algorithms on line arrangements follow directly from their counterparts on point configurations, and vice versa.
