A Review of Properties and Variations of Voronoi Diagrams

A REVIEW OF PROPERTIES AND VARIATIONS OF VORONOI DIAGRAMS29We will let T be the set of vertices of V l−1 that will be in V (p l ). Assumethat T is non-empty. Also, assume that V l−1 (T ) (the componentsof V l−1 in T ) is a tree. Following the aforementioned assumptions andgiven P = (p 1 , p 2 , . . . , p l−1 , p l ) and V l−1 , we will implement the followingprocedure to compute a new Voronoi diagram V l :(1) Find the generator p i (1 ≤ i ≤ l − 1) such that d(p i , p l ) is minimized.(2) Among the vertices q ijk on ∂V (p i ), find the one that gives thesmallest value of H(p i , p j , p k , p l ). Let T be the set consisting ofonly this vertex.(3) Repeat 3.a until T cannot be further augmented.(a) For each vertex q ijk connected by an edge to an existingelement of T , add q ijkto T if H(p i , p j , p k , p l ) < 0 andif the resultant T is a tree (satisfying one of our initialassumptions).(4) For every generator p i associated with a vertex q ijk that satisfiesassumption 4.6.7., draw the perpendicular bisector betweenp i and p l from e(p i , p j ) to e(p i , p k ). Let this line segment bee(p i , p l ), the interesection of e(p i , p l ) and e(p i , p j ) be the vertexq ijl , and the intersection of e(p i , p l ) and e(p i , p k ) be the vertexq ikl . In the case that q ijk lies on the outer circuit of V l−1 , draw