Planar point Location Search

Abstract

One of the more fundamental problems in computational geometry is that of locating a specific point in a given two-dimensional subdivision. It is thus necessary to define a subdivision appropriately. An embedding ∈ of a graph g maps each node v of g to a point ∈(v) in E ² and each arc a = {v,w} to a simple connected curve ∈(a) with endpoints ∈(v) and ∈(w). The embedding ∈ is plane if ∈(v)≠∈(w), for any two nodes v≠w of g, and if ∈(a)∩∈(b)=Ø, for any two arcs a≠b of g so we assume that a curve ∈(a) does not contain its endpoints. Graph g is planar if it admits a plane embedding of itself. The embedding of a node of g is called a vertex, the embedding of an arc is called an edge, and a connected component of E² reduced by all vertices and edges is called a region. For the sake of generality, we admit one node of g to be embedded at infinity; thus, all incident arcs correspond to unbounded edges of the subdivision, and all unbounded edges of the subdivision correspond to arcs incident upon this node. The embedding of a node at infinity is called an improper vertex. In formal terms, the point location search problem can now be defined as follows:

let S be a subdivision of E ²; for a given query point q, determine which of the regions, edges, and vertices of S contains point q.