Edalat, A

Khanban, A

Spiral - Imperial College Digital Repository

CCCG 2003, Halifax, Nova Scotia, August 11–13, 2003Computing Delaunay Triangulation with Imprecise Input DataA. A. Khanban A. Edalatkhanban@doc.ic.ac.uk ae@doc.ic.ac.ukDepartment of Computing, Imperial College London, U.K.AbstractThe key step in the construction of the Delaunay trian-gulation of a finite set of planar points is to establishcorrectly whether a given point of this set is inside oroutside the circle determined by any other three points.We address the problem of formulating the in-circle testwhen the coordinates of the planar points are given onlyup to a given precision, which is usually the case in prac-tice. By modelling imprecise points as rectangles, andusing the idea of partial disc, we construct a reliablein-circle test that provides the best possible Delaunaytriangulation with the imprecise input data given byrectangles.keywords: robust algorithm, Delaunay triangulation,imprecise data, in-circle test, partial disc.1 IntroductionA Delaunay triangulation of a set of N planar pointsis a triangulation with the property that the interior ofthe circumcircle of each triangle contains no points ofthe set [3]. If there are no four cocircular points in theset, then the Delaunay Triangulation is unique. A di-vide and conquer algorithm [8] and an algorithm basedon sweepline technique [7] compute the Delaunay Tri-angulation in O(N log N), while a random incrementalalgorithm [3] has an average complexity of O(N log N).We present the simplest algorithm [12], which is O(N4),but illustrates the crucial role of the so-called in-circletest in computing the Delaunay Triangulation:Algorithm:for each of the possible N choose 3 “triangles” of points do{for each of the other N - 3 points do{if all the other points are outside of the circlecircumscribing the triangle (i.e. THE IN-CIRCLE TEST)then this triangle is in the unique Delaunay triangulationelse this triangle is not in the unique Delaunay triangulation}}Any algorithm for Delaunay Triangulation is based oncorrectly establishing whether a given point is inside oroutside the circumcircle of three given points. When thecoordinates of the N points are infinitely precise, thisin-circle test is determined in terms of finding the signsof various determinants. The Delaunay triangulation isdegenerate, i.e. is not unique, if at least one of thesedeterminants is itself zero or is computed to zero, giventhe precision of computation. In [13], degenerate casesin Delaunay triangulation are removed by modifying theinput data, whereas in [1] a tolerance value for the inputpoints is obtained, within which degenerate cases do notoccur.In practice, the input data is usually imprecise as thecoordinates of the points are either the result of somemeasurements with certain error or are actually floatingpoint numbers which represent rectangles rather thanpoints in the plane. The in-circle test then becomesnon-trivial.Ely and Leclerc [6] have formulated the following twoessential properties for the in-circle test. By reliableit is meant that the test must never make an error ifit does decide that the point is inside or outside thecircle. By sharp it is meant that the test should onlyfail (to decide whether the point is inside or outside ofthe circle) because of the underlying geometry of theimprecise points.The use of interval arithmetic [11] can provide reli-ability by ensuring that the successful result of the in-circle test is always correct, but this is almost always atthe expense of sharpness as the test may frequently failunjustifiably.In [6], Ely and Leclerc model an imprecise point bya disc and show that any three discs representing threeimprecise points determine eight circles in the plane,each tangent to the three discs, which are then used toformulate an in-circle test that is sharper than the naiveinterval arithmetic.However, in practice, an imprecise point is almost al-ways given by two imprecise x and y coordinates, rep-resenting two horizontal and vertical line segments re-spectively, which therefore give rise to a rectangle ratherthan a disc.115th Canadian Conference on Computational Geometry, 2003In [10], the notion of partial Delaunay triangulationand the partial Voronoi diagram of N partial points,represented by N rectangles, in the plane were in-troduced and shown to be effectively computable inan order-theoretic framework for computational geom-etry [4]. This approach, i.e. representing the imprecisepoints by rectangles, has already been used in solvingthe problem of computing the convex hull for impreciseinput [5], which computes an interior and an exterior forthe partial convex hull. As the rectangles converge theexact points, the interior and the exterior of the partialconvex hull converge to the interior and the exterior ofthe classical convex hull for the exact points.In this paper, following the above approach, we rep-resent imprecise points by rectangles and, using the no-tion of partial disc established in [10], we construct areliable in-circle test which is completely sharp: if theoutput is neither “inside” nor “outside”, then we havepossible degeneracy with the given impreciseness of theinput, i.e. there exist four cocircular points, one in eachof four rectangles.In the classical case, the Delaunay triangulation ofN points is equivalent to the Voronoi diagram of theseN points (sites) [3, 2]. Given the Delaunay triangu-lation of the N points, one can find the Voronoi dia-gram by drawing the perpendicular bisectors of all theedges of the triangles. Conversely, given the Voronoi di-agram of the N points, the Delaunay triangulation canbe constructed by connecting the sites in each pair ofneighbouring regions. Therefore, solving either of theseproblems will also solve the other. But we note that theproblem of Delaunay triangulation with imprecise in-put, addressed here, is equivalent to the partial Voronoidiagram considered in [10], which is different from theproblem of the Voronoi diagram of N rectangles in theplane. In the problem of Voronoi diagram of N rect-angles, one has to determine N regions of the plane,each containing exactly one of the rectangles, such thatthe rectangle in that region is the closest rectangle toany point in that region. In fact, these two problems,i.e. partial Voronoi diagram and Voronoi diagram forrectangles, are based on different distance functions asexplained in [10]. See also [9] for the similar case of theVoronoi diagram of finite number of line segments in theplane.2 Partial DiscWe first recall the notion of a partial disc [10], whichis the generalization of a disc. Any three noncollinearpoints x, y, z in the plane determine a disc Dxyz whichhas x, y, z on its boundary. Now assume that all weknow is that the three points x, y, z lie respectivelyin the three rectangles R1, R2, R3, where each rect-angle R = [a, b] × [c, d] is the product of two ratio-Figure 1: The six centresnal intervals [a, b] and [c, d]. We say that the threerectangles are noncollinear if there is no straight linewhich intersects all three rectangles. Assuming the non-collinearity condition, R1, R2, R3 will determine a par-tial disc that is given by a pair of disjoint open subsets(I(R1, R2, R3), E(R1, R2, R3)) with the interiorI(R1, R2, R3) = ({Dxyz | x ∈ R1, y ∈ R2, z ∈ R3})◦,and the exteriorE(R1, R2, R3) = ({Dxyz | x ∈ R1, y ∈ R2, z ∈ R3})c,where A◦ and Ac are respectively the interior and com-plement of the set A. In words, I := I(R1, R2, R3) is thelargest open set contained in the interior of any circlewhich intersects all three rectangles R1, R2, R3, whereasE := E(R1, R2, R3) is the largest open set contained inthe exterior of any such circle.The sets I and E are computed as follows. Let δij bethe locus of all points in R2 with the furthest distancefrom points in Ri equal to the closest distance to Rj ,i.e.δij = {x ∈ R2 | maxp∈Rid(x, p) = minp∈Rj d(x, p)}.Each δij is a continuous curve made up of a number oflinear and parabolic segments as in Figure 1.Proposition 2.1 The following intersectionsδ12∩δ13, δ21∩δ23, δ31∩δ32, δ21∩δ31, δ12∩δ32, δ13∩δ23are either singleton or empty. They are all singletonsiff the three rectangles are noncollinear, otherwise someof these intersections will be empty.2CCCG 2003, Halifax, Nova Scotia, August 11–13, 2003Assuming the noncollinearity condition, we define sixcentre points as follows:{oF CC} = δ12 ∩ δ13, {oCF F } = δ21 ∩ δ31,{oCF C} = δ21 ∩ δ23, {oF CF } = δ12 ∩ δ32,{oCCF} = δ31 ∩ δ32, {oF F C} = δ13 ∩ δ23.(1)These points have been depicted in Figure 1. Note thatoCCFis the centre of a circle with radius rCCF, whichpasses through these three points:(i) the point of R1 closest to oCCF ,(ii) the point of R2 closest to oCCF ,(iii) the point of R3 furthest from oCCF ,hence the subscript in oCCF, and similarly for the othervertices. If any one or more of the rectangles R1, R2and R3 are in fact singletons then some of the abovecentre points will coincide.Let D(o, r) denote the closed disc with centre o andradius r. The interior I and the exterior E of the partialdisc of R1, R2, R3 are given by:I = (D1 ∩ D2 ∩ D3)◦, E = (D′1 ∪ D′2 ∪ D′3)c,whereD1 = D(oF CC , rF CC ), D′1 = D(oCF F , rCF F ),D2 = D(oCF C , rCF C ), D′2 = D(oF CF , rF CF ),D3 = D(oCCF , rCCF ), D′3 = D(oF F C , rF F C ).In Figure 2, the boundaries of the discs D1, D2 and D3are depicted with dotted lines, those of D′1, D′2 and D′3with solid lines and the boundaries of the sets I and Ewith dashed lines. The closed region bounded betweenI and E, i.e. bounded between the two closed dashedcurves, is the boundary of the partial disc.Note that there are two other discs, one which passesthrough the closest point of each of the three rectan-gles and one which passes through the furthest pointsof these rectangles. The interior of the partial disc iscontained in the interiors of these two discs, and alsothe exterior of the partial disc is contained in the exte-riors of these two discs. Thus, they lie completely insidethe boundary of the partial disc and do not contributeany extra information for the construction of the partialdisc.3 The In-Circle TestWe start by first computing the δij boundaries as in [10].Then, we find the intersection points as in (1), whichgive us the centres of the circles. When all the three im-precise points are non-degenerate rectangles, there willbe three centres for the interior circles and three centresfor the exterior circles. The following formula gives theradius r of any of the six circles, which make up theFigure 2: The interior and the exterior of a partial discinterior and the exterior of the partial disc, in terms ofits centre o:r = min {{max d(o, p) | p is a vertex of Ri} | i = 1, 2, 3} .Using the following containment predicate, we canverify that a given imprecise point or rectangle R is in-side, outside, or on the boundary of a partial disc givenby three rectangles R1, R2, R3.Con(R, (R1, R2, R3)) =inside if R ⊂ I(R1, R2, R3)outside if R ⊂ E(R1, R2, R3)boundary otherwise.Since the coordinates of the centres of the six interiorand exterior circles are intersections of straight lines orparabolas with rational coefficients, the output of thecontainment predicate can be correctly computed usingrational arithmetic.Theorem 3.1 If the three rectangles are collinear, thenthe in-circle algorithm will verify it using Proposi-tion 2.1. Otherwise, with the noncollinearity condition,the algorithm outputs the correct value of the contain-ment predicate.We finally note that computing the partial disc pass-ing through three imprecise points and applying the in-circle test for the fourth imprecise point is of order O(1)and does not affect the complexity of any algorithmwhich uses this test.315th Canadian Conference on Computational Geometry, 2003References[1] Manuel Abellanas, Ferran Hurtado, and Pedro A.Ramos, Structural tolerance and Delaunay triangu-lation, Information Processing Letters 71 (1999),no. 5–6, 221–227.[2] F. Aurenhammer and R. Klein, Voronoi diagrams,Handbook of computational geometry (J.-R. Sackand J. Urrutia, eds.), Elsevire Science B.V., 1999,pp. 201–290.[3] M. de Berg, M. van Kreveld, M. Overmars, andO. Schwarzkopf, Computational geometry, algo-rithms and applications, 2nd ed., Springer, 2000.[4] A. Edalat and A. Lieutier, Foundation of a com-putable solid modelling, Theoretical Computer Sci-ence 284(2) (2002), 319–345.[5] A. Edalat, A. Lieutier, and E. Kashefi, The con-vex hull in a new model of computation, Proc. 13thCanad. Conf. Comput. Geom., 2001, pp. 93–96.[6] J. S. Ely and A. P. Leclerc, Correct Delaunay tri-angulation in the presence of inexact inputs andarithmetic, Reliable Computing 6 (2000), 23–38.[7] S. Fortune, A sweepline algorithm for Voronoi di-agram, Algorithmica 2 (1987), 153–174.[8] Leonidas Guibas and Jorge Stolfi, Primitives forthe manipulation of general subdivisions and thecomputation of Voronoi, ACM Transactions onGraphics (TOG) 4 (1985), no. 2, 74–123.[9] Martin Held, VRONI: An engineering approach tothe reliable and efficient computation of voronoi di-agrams of points and line segments, ComputationalGeometry 18 (2001), no. 2, 95–123.[10] A. A. Khanban, A. Edalat, and A. Lieutier, Com-putability of partial Delaunay triangulation andVoronoi diagram, Electronic Notes in Theoreti-cal Computer Science (Vasco Brattka, MatthiasSchröder, and Klaus Weihrauch, eds.), vol. 66, El-sevier Science Publishers, 2002.[11] R. E. Moore, Methods and applications of inter-val analysis, SIAM Studies in Applied Mathemat-ics (W. F. Ames, ed.), SIAM, 1979.[12] J. O’Rourke, Computational geometry in C, Cam-bridge University Press, New York, 1994.[13] K. Sugihara, A simple method for avoiding nu-merical errors and degeneracy in voroni diagramconstruction, IEICE Trans. Fundamentals E75-A(1992), no. 4, 468–477.4

Computing Delaunay triangulation with imprecise input data

Abstract

Similar works

Full text

Available Versions

Spiral - Imperial College Digital Repository