Citation Context

...bdivisions in such asway that their topological structure is completely determined by that of theirsgraphs. For example, if the manifold is restricted to be a sphere and the graph isstriply connected =-=[8]-=-, then the subdivision is determined up to equivalence.sHowever, any set of conditions strong enough to achieve this goal would probablysoutlaw “degeneracies” such as loops, multiple edges with the sa...

Citation Context

...ing the currently best availablesalgorithms, the Voronoi diagram of n points can be computed in 0( n log n) timesand stored in O(n) space; these bounds have been shown to be optimal in thesworst case =-=[18]-=-. Once we have the Voronoi diagram, we can construct in linearsfurther time a structure with which we can do point location in a planarssubdivision in O(log n) time [ 111.sShamos [18] first pointed ou...

Citation Context

... important properties of Voronoisdiagrams and their duals, with an emphasis on the results we will need later on.sFor a fuller treatment of these topics the reader should consult refs. [13], [18],sor =-=[19]-=-.sIf we are given only two sites, then the associated Voronoi regions are simplysthe two (open) half-planes delimited by the bisector of the two sites. Moresgenerally, when n sites are given, the regi...

Citation Context

...dges, and faces. Once the topological properties of the diagram aresknown, its geometrical properties (coordinates, lengths, angles, etc.) can bescomputed in time linear in the number of sites. Boots =-=[2,20]-=- was apparently thesfirst to observe that the computation of a Voronoi diagram can be greatlyssimplified by working with its dual, which is known as the Delaunay diagram ofsthe given sites. This allow...

Citation Context

...Guibas and J. StolfisFig. 5. The edge functions.sc-3 NedsA number of useful properties can be deduced from these, as for examplese Flip-l = e Flip,se Sym = e Rot2,se Rot-’ = e Rot3 = e Flip Rot Flip, =-=(1)-=-se Dual = e Flip Rot,se One&-’ = e Rot Onext Rot = e Flip Onext Flip,sand so forth. For added convenience in talking about subdivisions, we introducessome derived functions. By analogy with e Lnext an...

Citation Context

... the Voronoi diagram is that we need notscompute straight-line intersections unless the coordinates of Voronoi vertices aresneeded. Our algorithm follows closely the one proposed by Lee and Schachters=-=[14]-=- and is the dual of that described by Shamos and Hoey [19]. Like theirs, itsruns in time 0 (n log n) and uses linear storage. The reasons for including it heresare twofold. First of all we wanted to i...

Citation Context

...al high-level language and correctly handles degen-serate cases. For completeness, we apply the same methodology to a simpler (butsasymptotically slower) incremental algorithm due to Green and Sibson =-=[7]-=-.sOur algorithms are built using essentially two primitives: a geometric predicatesand a topological operator for manipulating the structure of the diagrams. Thesgeometrical primitive, which we call t...

Citation Context

...ent of L and R.sFrom the above discussion it is apparent that our divide-and-conquer algorithmsis computing the convex hull of the lifted images of the sites. It is in fact exactlysthe Preparata-Hong =-=[17]-=- agorithm for computing the convex hull of n points insthree dimensions. If the InCircle test is replaced by a “positive volume” test, assobtained by substituting in .%i (A, B, C, D) the third column ...

Citation Context

...ord pointers + 12 bits). The simplification for orientablesmanifolds reduces those 12 bits to 8. This compares favorably with the winged-sedge representation [l] and with the Muller-Preparata variant =-=[16]-=-. Indeed, allsthree representations use essentially the same pointers: each edge is connectedsto the four “immediately adjacent” ones (One&, Oprev, Dnext, Dprev), and thesfour Data fields of our struc...

Citation Context

...ve fun verifying that our expanding circles passingsthrough a chord become rotating supporting planes around the lifted image ofsthe chord. Thus we are computing the “sleeve” discussed in [17]. Brown =-=[4]-=- hassobserved that a similar correspondence can be obtained by lifting the sitessstereographically onto a sphere.s10. AN INCREMENTAL ALGORITHMsThe algorithm of the previous section assumes that all po...

Citation Context

...htforward programming exercise, given an auxiliary stack of size O( ] J&S’ ( )sand a Boolean mark bit on each directed edge [12]. With a few more bits persedge, we can do away with the stack entirely =-=[6]-=-. A slight modification of thosesalgorithms can be used to enumerate the vertices of the subdivision, in the sensesof visiting exactly one edge out of every vertex. If we take the dual subdivision,swe...

Citation Context

...ome of the most important properties of Voronoisdiagrams and their duals, with an emphasis on the results we will need later on.sFor a fuller treatment of these topics the reader should consult refs. =-=[13]-=-, [18],sor [19].sIf we are given only two sites, then the associated Voronoi regions are simplysthe two (open) half-planes delimited by the bisector of the two sites. Moresgenerally, when n sites are ...

Citation Context

... e. is the canonicalsrepresentative of the group to which e belongs.sThe group of edges containing e is represented in the data structure by onesedge record e, divided into four parts e [0] through e =-=[3]-=-. Part e [r] correspondssto the edge e. Rot’. See Figure 7a. A generic edge e = e. Rot’Flipf is representedsby the triplet (e r, f ), called an edge reference. We may think of this triplet as aspointe...

Citation Context

...dges, and faces. Once the topological properties of the diagram aresknown, its geometrical properties (coordinates, lengths, angles, etc.) can bescomputed in time linear in the number of sites. Boots =-=[2,20]-=- was apparently thesfirst to observe that the computation of a Voronoi diagram can be greatlyssimplified by working with its dual, which is known as the Delaunay diagram ofsthe given sites. This allow...

Citation Context

...z. qsTherefore, the topological structure of a subdivision is completely and uniquelyscharacterized by its edge algebra. An analogous theorem seems to have beensdiscovered independently by Damphousse =-=[5]-=-. Theorems 3.3 and 3.7 also implysthat all completions of a subdivision are equivalent and that two subdivisions aresequivalent if and only if their duals are equivalent. Therefore, the dual of a simp...

Citation Context

... as possible and leads to a clean theorysand data structure.sWe assume the reader is familiar with a few basic concepts of point-setstopology, such as topological space, continuity, and homeomorphism =-=[9]-=-. Twossubsets A and B of a topological space M are said to be separable if somesneighborhood of A is disjoint from some neighborhood of B; otherwise, they aressaid to be incident on each other. A line...

Citation Context

...systematic enumeration of all edges in a (connected) subdivision is asstraightforward programming exercise, given an auxiliary stack of size O( ] J&S’ ( )sand a Boolean mark bit on each directed edge =-=[12]-=-. With a few more bits persedge, we can do away with the stack entirely [6]. A slight modification of thosesalgorithms can be used to enumerate the vertices of the subdivision, in the sensesof visitin...