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Sylvain Petitjean
To cite this version:
Sylvain Petitjean. Collection of abstracts of the 24th European Workshop on Computational Geome- try. Sylvain Petitjean. INRIA-LORIA, pp.270, 2008. �inria-00595116�
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March 18-20, 2008
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LORIA
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Sylvain.Petitjean@loria.fr
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Compilation copyright c2008 by Sylvain Petitjean.
Copyrights of individual papers retained by the authors.
Preface
The24th European Workshop on Computational Geomety(EuroCG’08) was held at the Lab- oratoire Lorrain de Recherche en Informatique et ses Applications (LORIA) on March 18-20, 2008. It was preceded by a one-day workshop entitled “CGAL Innovations and Applications: Robust Geometric Software for Complex Shapes” held on March 17, 2008. More information about both events can be found at http://eurocg08.loria.fr(see alsohttp://www.eurocg.org for previous workshops).
The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop. It is also available electronically from the workshop’s web site at http:
//eurocg08.loria.fr/EuroCG08Abstracts.pdf. This year’s record of 72 submissions with authors from 22 different countries, covering a wide range of topics, shows that Computational Geometry is a lively and still growing research field in Europe.
Following the tradition of the workshop, many contributions present ongoing research, and it is expected that most of them will appear in a more complete version in scientific journals. Selected papers from the workshop will be invited to a special issue of Computational Geometry: Theory and Applications.
We thank the editors-in-chief, Kurt Mehlhorn and J¨org-R¨udiger Sack, for their cooperation.
We would also like to thank all the authors for submitting papers and presenting their results at the workshop. We are especially grateful to our keynote speakers Pierre Alliez, Jean Ponce, and Fabrice Rouillier for accepting our invitation. Special thanks go to our sub-referees and to Bettina Speckmann for providing us with the LATEX class used to format this collection.
Finally, we are grateful to LORIAfor providing the necessary infrastructure, and to our sponsors for their support: INRIA,Universit´e Henri Poincar´e,Universit´e Nancy 2, Institut National Polytechnique de Lorraine, GDR Informatique Math´ematique of CNRS, Communaut´e Urbaine du Grand Nancy, Conseil G´en´eral de Meurthe-et-Moselle, Conseil R´egional de Lorraine,Dassault Syst`emes andInstitut Fran¸cais du P´etrole.
Note that the next edition of EuroCG will be held in 2009 in Brussels, Belgium.
Sylvain Petitjean (Editor)
Program Committee
Laurent Dupont Hazel Everett Xavier Goaoc Sylvain Lazard (chair)
Sylvain Petitjean Marc Pouget
Organizing Committee
Anne-Lise Charbonnier Sylvain Lazard (chair) Julien Demouth Luis Pe˜naranda Laurent Dupont Sylvain Petitjean
Hazel Everett Marc Pouget Xavier Goaoc Mirsada Tihic
Table of Contents
Tuesday, March 18
9:30 - 10:30, Session 1
Delaunay Edge Flips in Dense Surface Triangulations S.-W. Cheng and T. Dey
1
Decomposing Non-Convex Fat Polyhedra M. de Berg and C. Gray
5
Schnyder Woods for Higher Genus Triangulated Surfaces L. Castelli Aleardi, E. Fusy and T. Lewiner
9
Seed Polytopes for Incremental Approximation
O. Aichholzer, F. Aurenhammer, T. Hackl, B. Kornberger, S. Plantinga, G. Rote, A. Sturm, and G. Vegter
13
10:50 - 11:50, Session 2
On the Reliability of Practical Point-in-Polygon Strategies S. Schirra
17
Minimizing the Symmetric Difference Distance in Conic Spline Approximation S. Ghosh and G. Vegter
21
Mixed Volume Techniques for Embeddings of Laman Graphs R. Steffens and T. Theobald
25
Geometric Analysis of Algebraic Surfaces Based on Planar Arrangements E. Berberich, M. Kerber and M. Sagraloff
29
12:00 - 13:00, Invited Talk 1
The Challenge of 3D Photo/Cinematography to Computational Geometry J. Ponce
33
14:30 - 15:50, Session 3A
Improved Upper Bounds on the Number of Vertices of Weight≤kin Particular Arrangements of Pseudocircles
R. Ortner
35
Helly-Type Theorems for Approximate Covering J. Demouth, O. Devillers, M. Glisse and X. Goaoc
39
Dynamic Free-Space Detection for Packing Algorithms
T. Baumann, M. Jans, E. Sch¨omer, C. Schweikert and N. Wolpert
43
On Computing the Vertex Centroid of a Polytope H. R. Tiwary
47
14:30 - 15:50, Session 3B
Space-Filling Curve Properties for Efficient Spatial Index Structures H. Haverkort and F. van Walderveen
51
Optimizing Active Ranges for Consistent Dynamic Map Labeling K. Been, M. N¨ollenburg, S.-H. Poon and A. Wolff
55
Order-kTriangulations of Convex Inclusion Chains in the Plane W. El-Oraiby and D. Schmitt
59
Constructing the Segment Delaunay Triangulation by Flip M. Br´evilliers, N. Chevallier and D. Schmitt
63
16:10 - 17:30, Session 4A
Intersection Graphs of Pseudosegments and Chordal Graphs: An Application of Ramsey Theory C. Dangelmayr, S. Felsner and W. T. Trotter
67
Augmenting the Connectivity of Planar and Geometric Graphs I. Rutter and A. Wolff
71
Colour Patterns for Polychromatic Four-Colourings of Rectangular Subdivisions H. Haverkort, M. L¨offler, E. Mumford, M. O’Meara, J. Snoeyink and B. Speckmann
75
Polychromatic 4-Coloring of Rectangular Partitions D. Dimitrov, E. Horev and R. Krakovski
79
16:10 - 17:30, Session 4B
Exact Implementation of Arrangements of Geodesic Arcs on the Sphere with Applications E. Fogel, O. Setter and D. Halperin
83
Voronoi Diagram of Ellipses in CGAL
I. Z. Emiris, E. Tsigaridas and G. M. Tzoumas
87
A CGAL-Based Univariate Algebraic Kernel and Application to Arrangements S. Lazard, L. Pe˜naranda and E. Tsigaridas
91
Generic Implementation of a Data Structure for 3D Regular Complexes A. Bru and M. Teillaud
95
Wednesday, March 19
9:30 - 10:30, Session 5
Online Uniformity of Integer Points on a Line T. Asano
99
Edge-Unfolding Medial Axis Polyhedra J. O’Rourke
103
Inducing Polygons of Line Arrangements E. Mumford, L. Scharf and M. Scherfenberg
107
Coloring Geometric Range Spaces
G. Aloupis, J. Cardinal, S. Collette, S. Langerman and S. Smorodinsky
111
10:50 - 11:50, Session 6
A Lower Bound for the Transformation of Compatible Perfect Matchings A. Razen
115
Edge-Removal and Non-Crossing Configurations in Geometric Graphs
O. Aichholzer, S. Cabello, R. Fabila-Monroy, D. Flores-Pe˜naloza, T. Hackl, C. Huemer, F. Hur- tado and D. R. Wood
119
Computing the Dilation of Edge-Augmented Graphs in Metric Spaces C. Wulff-Nilsen
123
Approximating the Minimum Spanning Tree of Set of Points in the Hausdorff Metric V. Alvarez and R. Seidel
127
12:00 - 13:00, Invited Talk 2
Optimization Techniques for Geometry Processing P. Alliez
131
14:30 - 15:50, Session 7A
Geometry with Imprecise Lines M. L¨offler and M. van Kreveld
133
The Linear Parametric Geometric Uncertainty Model: Points, Lines and their Relative Positioning Y. Myers and L. Joskowicz
137
Smoothing Imprecise 1-Dimensional Terrains C. Gray, M. L¨offler and R. Silveira
141
Noisy Bottleneck Colored Point Set Matching in 3D Y. Diez and J. A. Sellar`es
145
14:30 - 15:50, Session 7B
Pareto Envelopes in Simple Polygons
V. Chepoi, K. Nouioua, E. Thiel and Y. Vax`es
149
Shortest Inspection-Path Queries in Simple Polygons C. Knauer, G. Rote and L. Schlipf
153
A Search for Medial Axes in Straight Skeletons K. Vyatkina
157
On Computing Integral Minimum Link Paths in Simple Polygons W. Ding
161
16:10 - 17:30, Session 8A
Constant-Working-Space Image Scan with a Given Angle T. Asano
165
Consistent Digital Rays
J. Chun, M. Korman, M. N¨ollenburg and T. Tokuyama
169
Matching a Straight Line on a Two-Dimensional Integer Domain E. Charrier and L. Buzer
173
Exploring Simple Triangular and Hexagonal Grid Polygons Online D. Herrmann, T. Kamphans and E. Langetepe
177
16:10 - 17:30, Session 8B
Manifold Homotopy via the Flow Complex B. Sadri
181
Surface Deformation on a Discrete Model for a CAD System I.-G. Ciuciu, F. Danesi, Y. Gardan and E. Perrin
185
Optimal Insertion of a Segment Highway in a City Metric M. Korman and T. Tokuyama
189
Algorithms for Graphs of Bounded Treewidth via Orthogonal Range Searching S. Cabello and C. Knauer
193
Thursday, March 20
9:30 - 10:30, Session 9A
A Tight Bound for the Delaunay Triangulation of Points on a Polyhedron N. Amenta, D. Attali and O. Devillers
197
Discrete Voronoi Diagrams on Surface Triangulations and a Sampling Condition for Topological Guarantee
M. Moriguchi and K. Sugihara
201
On the Locality of Extracting a 2-Manifold inR3 D. Dumitriu, S. Funke, M. Kutz and N. Milosavljevic
205
9:30 - 10:30, Session 9B
Arrangements on Surfaces of Genus One: Tori and Dupin Cyclides E. Berberich and M. Kerber
209
On the Topology of Planar Algebraic Curves
J. Cheng, S. Lazard, L. Pe˜naranda, M. Pouget, S. Lazard, F. Rouillier, E. Tsigaridas
213
Topological Considerations for the Incremental Computation of Voronoi Diagrams of Circular Arcs
M. Held and S. Huber
217
10:50 - 11:50, Session 10A
The Entropic Centers of Multivariate Normal Distributions F. Nielsen and R. Nock
221
Quantum Voronoi Diagrams F. Nielsen and R. Nock
225
Triangulating the 3D Periodic Space M. Caroli, N. Kruithof and M. Teillaud
229
10:50 - 11:50, Session 10B
Realizability of Solids from Three Silhouettes T. Ohgami and K. Sugihara
233
Good Visibility Maps on Polyhedral Terrains N. Coll, N. Madern and J. A. Sellar`es
237
Directly Visible Pairs and Illumination by Reflections in Orthogonal Polygons M. Aanjaneya, A. Bishnu and S. P. Pal
241
12:00 - 13:00, Invited Talk 3
Computer Algebra and Computational Geometry F. Rouillier
245
14:30 - 15:30, Session 11
On Planar Visibility Polygon Simplification A. Zarei and M. Ghodsi
247
The Kinetic Facility Location Problem B. Degener, J. Gehweiler and C. Lammersen
251
Probabilistic Matching of Polygons H. Alt, L. Scharf and D. Schymura
255
Delaunay Edge Flips in Dense Surface Triangulations
∗Siu-Wing Cheng† Tamal K. Dey‡
Abstract
We study the conversion of a surface triangulation to a subcomplex of the Delaunay triangulation with edge flips. We show that the surface triangulations which closely approximate a smooth surface with uniform density can be transformed to a Delaunay triangu- lation with a simple edge flip algorithm. The con- dition on uniformity becomes less stringent with in- creasing density of the triangulation. If the condition is dropped, the output surface triangulation becomes
“almost Delaunay” instead of exactly Delaunay.
1 Introduction
The importance of computing Delaunay triangula- tions in applications of science and engineering can- not be overemphasized. Among the different Delau- nay triangulation algorithms, flip based algorithms are most popular and perhaps the most dominant ap- proach in practice. The sheer elegance and simplicity of this approach make it attractive to implement.
InR2, if the circumcircle of a trianglet contains a vertex of another trianglet′sharing an edgeewith it, flipping e means replacing ewith the other diagonal edge contained in the union oftandt′. A well-known elegant result is that this process terminates and pro- duces the Delaunay triangulation. In higher dimen- sions, the edge flips can be naturally extended to bi- stellar flips. Edelsbrunner and Shah [8] showed that bi-stellar flips can be used with incremental point in- sertion to construct weighted Delaunay triangulations in three and higher dimensions.
Given the increasing demand of computing surface triangulations that are sub-complexes of Delaunay tri- angulations [1, 6, 7], it is natural to ask if a surface triangulation can be converted to a Delaunay one by edge flips and, if so, under what conditions. Once the surface triangulation is made Delaunay, a number of tools that exploit Delaunay properties can be used for further processing.
We show that a dense triangulation can be flipped
∗Research supported by NSF grants CCF-0430735 and CCF-0635008 and Research Grant Council, Hong Kong, China (612107).
†Department of Computer Science and Engineering, HKUST, Clear Water Bay, Hong Kong,scheng@cse.ust.hk
‡Department of Computer Science and Engineering, The Ohio State University, Columbus, OH 43210, USA, tamaldey@cse.ohio-state.edu
to a Delaunay triangulation if the density is uniform in some sense. The practical implication of this result is that reasonably dense triangulations can be con- verted to Delaunay triangulations with a simple edge flip algorithm. Furthermore, the results in this paper have been used for a recent algorithm on maintaining deforming meshes with provable guarantees [5]. What happens if we do not have the uniformity condition?
We show that the flip algorithm still terminates but the output surface may not be Delaunay. Nonetheless, this surface is “almost Delaunay” in the sense that the diametric ball of each triangle shrunk by a small amount remains empty. These approximate Delaunay triangulations may find applications where exact De- launay triangulations are not required; for example, see the work by Bandyopadhyay and Snoeyink [3].
2 Preliminaries
Surface. Let Σ ⊂R3 be a smooth compact surface without boundary. Themedial axis is the set of cen- ters of all maximally empty balls. Thereachγof Σ is the infimum over Euclidean distances of all points in Σ to its medial axis. Letnx denote the outward unit normal of Σ at a pointx∈Σ.
Triangulation. We sayT is a triangulation of a sur- face Σ if vertices ofTlie in Σ and its underlying space
|T|is homeomorphic to Σ. For any trianglet∈T,nt
denotes the outward unit normal oft.
The triangulation T has a consistent orientation if for any triangle t ∈ T and for any vertex q of t,
∠nt,nq≤ π2.
If a trianglet∈T shares an edgepqwith a triangle pqs, we calls aneighbor vertex oft. Letρ(t) denote the circumradius oft. The ratio ofρ(t) to the shortest edge length oftis called theradius-edge ratio. We call the maximum radius-edge ratio of triangles inT the radius-edge ratio ofT.
We call T ε-dense for some ε < 1 if ρ(t)≤εγ for each trianglet∈TandThas a consistent orientation.
Also, if the distance between any two vertices inT is at leastδεγ for someδ <1, we callT (ε, δ)-dense.
Stab and flip. LetB(c, r) denote the ball with center c and radius r. Acircumscribing ball of a triangle t is any ball that has the vertices ofton its boundary.
The diametric ball is the smallest such ball and we denote it byDt.
A vertexv ofT stabsa ballB ifv lies insideB. A triangle t∈T is stabbedif Dt is stabbed by a vertex of T. The triangle t islocally stabbed if the stabbing vertex is one of the three neighbor vertices of t.
The common edgepqbetween two trianglespqrand pqsinT isflippableifpqris stabbed bys(i.e., locally stabbed). We will show later that this definition is symmetric. Flippingpq means replacingpqr andpqs by the trianglesprsandqrs. If the new triangulation is T′ we write T →pq T′.
Power distance. Given a pointxand a ballB(c, r), the power distance pow(x, B(c, r)) is kc−xk2−r2. Given two balls B1 and B2, their bisector C(B1, B2) consists of points at equal power distances from B1
and B2. It turns out that C(B1, B2) is a plane. If B1 and B2 intersect, C(B1, B2) contains the circle
∂B1∩∂B2.
Background results. The following previous results on normal approximations will be useful.
Lemma 1 ([2, 4]) For any two pointsxand y inΣ such thatkx−yk ≤εγfor someε≤13,∠nx,ny ≤1−εε
and ∠nx,(y−x)≥arccos(ε2).
Combining Lemmas 1 and a result in [7], we get:
Corollary 2 Let T be an ε-dense triangulation for some ε < 0.1. For any vertex q of a triangle t ∈ T,
∠nt,nq ≤7ε.
Define thedihedral anglebetween two adjacent tri- anglespqrandqrsas∠npqr,nqrs. Corollary 2 implies that:
Corollary 3 Let T be an ε-dense triangulation for some ε <0.1. For any two adjacent trianglespqr and qrs inT,∠npqr,nqrs ≤14ε.
3 Flip algorithm
The flip algorithm that we consider is very simple: as long as there is a flippable edge, flip it.
MeshFlip(T)
1. If there is a flippable edge e∈ T then flipeelse outputT;
2. T :=T′ whereT →e T′; go to step 1.
There are two issues. First, under what condition doesMeshFlipterminate? Second, what triangulation doesMeshFlipproduce? In this section, we show that MeshFlip terminates if T is an ε-dense triangulation.
We address the second issue later.
The following lemma establishes the symmetry in local stabbing.
Lemma 4 Letpqrandpqsbe two adjacent triangles such thatsstabspqr. If∠npqr,npqs<π2,rstabspqs.
Proof. It can be shown that the bisector Cpq sep- arates r and s if pqr and pqs make an angle larger than π2 or equivalently ∠npqr,npqs < π2. LetCpq+ be the half-space supported byCpqand containings. De- fine Cpq− similarly as the half-space not containings.
Clearly,Dpqs∩Cpq+ ⊂Dpqr∩Cpq+ assis on the bound- ary of Dpqs. So Dpqr∩Cpq− ⊂ Dpqs∩Cpq−. But Cpq− containsrwhich is on the boundary ofDpqr. Soris
insideDpqs.
Next, we show that an edge flip produces two tri- angles with a smaller maximum circumradius.
Lemma 5 LetTbe a triangulation with dihedral an- gles less than π2. Let pqr, pqs ∈ T be triangles such thatsstabs pqr. Thenρ(qrs)≤max{ρ(pqr), ρ(pqs)} andρ(prs)≤max{ρ(pqr), ρ(pqs)}.
Proof. We prove the lemma forρ(qrs). The analysis for ρ(prs) is similar. Consider the bisectors Cqr = C(Dpqr, Dqrs) and Cqs =C(Dpqs, Dqrs). LetCqr+ be the half-space supported byCqrcontainings. LetCqs+ be the half-space supported byCqs containingp.
By assumption the dihedral angle betweenpqrand pqs is at most π2. Then, Lemma 4 applies to claim thatrstabs pqs.
Clearly, the center of Dqrs lies in the union Cqr+ ∪ Cqs+. First, assume that Cqr+ contains the center of Dqrs. Clearly, Dqrs∩Cqr+ ⊂ Dpqr∩Cqr+ as s is con- tained in Dpqr by the assumption that s stabs pqr.
This implies that Dpqr∩Cqr+ contains the center of Dqrs. So Dqrs is smaller than Dpqr establishing the claim. If Cqs+ contains the center of Dqrs, the above argument can be repeated by replacingCqr+ withCqs+,
Dpqr withDpqs, andswithr.
Since the maximum circumradius decreases mono- tonically by the edge flips, the triangles can still be oriented consistently with Σ and a homeomorphism using closest point map [7] can be established between Σ and the new triangulation. Hence, the new trian- gulation satisfies the conditions for beingε-dense.
Corollary 6 If T →e T′ for a flippable edgeeand T isε-dense for someε <1, then T′ is alsoε-dense.
Lemma 7 If T is ε-dense for some ε < 0.1, then MeshFlip(T)terminates.
Proof. LetR1, R2, .., Rn be the decreasing sequence of the radii of the diametric balls of the triangles at any instant of the flip process. First of all, an edge flip preserves the number of triangles in the triangulation.
An edge flip may change the entries in this sequence
of radii, but not its length. We claim that after a flip the new radii sequenceR′1, R2′, ..., R′ndecreases lexico- graphically, that is, there is aj such thatRi =R′ifor all 1≤i≤j andRj+1> R′j+1. Letj+ 1 be the first index where Rj+16=R′j+1. Since each flip maintains ε-density (Corollary 6), the dihedral angles between adjacent triangles remain at most 14εby Corollary 3.
This angle is less than π2 for ε < 0.1. One can ap- ply Lemma 5 to each intermediate triangulation. By this lemma, the maximum of the two radii before a flip decreases after the flip. It means that the trian- gle corresponding to the radiusRj+1has been flipped and its place has been taken by a triangle whose cir- cumradius is smaller than Rj+1. So the new radii sequence is smaller lexicographically. It follows that the same triangulation cannot appear twice during the flip sequence. As there are finitely many possible triangulations,MeshFlipmust terminate.
4 Uniform dense triangulation
We prove thatMeshFlipcan turn an (ε, δ)-dense trian- gulation to a Gabriel triangulation where no diametric ball of any triangle is stabbed.
Lemma 8 Assume that a vertex v stabs a triangle pqr in an ε-dense triangulation for someε <0.1. Let
¯
v be the point inpqrclosest tov. The angle between vv¯and the support line ofnpqr is at least π2 −26ε.
Proof. LetT be anε-dense triangulation of a surface Σ with reach γ. Since v stabs Dpqr, we have kp− vk ≤ 2εγ which implies that kv −v¯k ≤ 2εγ. Walk from v towards ¯v and let abcbe the first triangle in T that we hit. Let y be the point in abc that we hit. (The triangle abc could possibly be pqr.) We have kv−yk ≤ kv −v¯k ≤ 2εγ. By the ε-density assumption, we have ka−yk ≤2εγ. It follows that ka−vk ≤ ka−yk+kv−yk ≤4εγ. Then,∠nv,na<
8ε by Lemma 1, and∠nabc,na ≤7εby Corollary 2.
Therefore,∠nv,nabc<8ε+ 7ε= 15ε.
Letℓbe an oriented line throughv and ¯vsuch that ℓenters the polyhedron bounded byT aty∈abcand then exits atv. Assume to the contrary thatℓmakes an angle less than π2−26εwithnpqr. Sincekp−vk ≤ 2εγ, Lemma 1 and Corollary 2 imply that∠nv,npqr≤ 4ε+7ε= 11ε. Thus,ℓmakes an angle less thanπ2−15ε withnv. Since∠nv,nabc<15ε,ℓmust make an angle less than π2 withnabc. Becauseℓenters atyand then exits at v,∠nv,nabc is greater than π−(π2 −15ε)−
π
2 = 15ε, contradicting the previous deduction that
∠nv,nabc<15ε.
Lemma 9 Assume that a vertex v stabs a triangle pqr in an ε-dense triangulation for some ε < 0.1.
There exists an edge, say pq, such that r and v are separated by the plane Hpq that contains pq and is perpendicular topqr.
Proof. By Lemma 8,v¯v makes a positive angle with the line of npqr. It follows that v does not project orthogonally onto a point inside pqr. Hence, there exists an edgepqsuch thatHpqseparatesrandv.
q s p
r v
!
vpq
> z
q npqr
p v v Cpq
r
q p
Hpq
Hpq
Figure 1: (left) : trianglepqris stabbed byv. Bothv ands lie on the same side ofHpq and Cpq. The case of v being in the thin wedge between Hpq and Cpq
is eliminated if pqr has bounded radius-edge ratio.
(middle) : the worst case for angle∠vpq. (right): the planes of Hpq and vpq make large angle ensuring v andsare on the same side ofCpq.
Lemma 10 Assume that a vertex v stabs a tri- angle pqr in an ε-dense triangulation with radius- edge ratio a < 2 sin 24ε1 . If ε < 72π, pqr is locally stabbed orvstabs a triangletsuch thatpow(v, Dt)<
pow(v, Dpqr).
Proof. By Lemma 9, there is a plane Hpq through the edgepq and perpendicular topqr such thatHpq
separatesrandv. Letpqsbe the other triangle inci- dent topq. Ifslies insideDpqr,pqris locally stabbed and we are done. So assume thatsdoes not lie inside Dpqr. By Corollary 3,∠npqr,npqs≤14ε, which is less than π2 for ε < 72π. Therefore, Hpq separates r and stoo. It means that v ands lie on the same side of Hpq; see Figure 1.
LetCpqdenote the bisectorC(Dpqr, Dpqs). LetCpq+ be the half-space bound by Cpq containings. It fol- lows that Dpqr∩Cpq+ ⊂ Dpqs∩Cpq+ as s lies outside Dpqr. Suppose thatCpq+ containsv. Then,vlies inside Dpqs as v lies insideDpqr. This immediately implies that v stabs pqs and pow(v, Dpqs) < pow(v, Dpqr).
Therefore, the lemma holds if we can show thatCpq+ contains v. This is exactly where we need bounded aspect ratios for triangles.
Let ¯sand ¯vbe the orthogonal projections ofsandv respectively onto the line ofpq. Consider the following facts.
(i) The acute angle between s¯s and npqr is equal to π2 −∠npqr,npqs, which is at least π2 −14ε by Corollary 3.
(ii) The angle betweenHpqandCpqcannot be larger than∠npqr,npqswhich is at most 14ε.
(iii) We prove that ∠npqr, v¯v > ∠npqr,npqs ≥
∠Hpq, Cpq.
These facts imply that v and s on the same side of Cpq asHpq. Therefore, it suffices to prove (iii).
First, observe that if ¯v is the closest point ofv in pq, we have by Lemma 8
∠npqr, v¯v≥ π
2 −26ε≥14ε≥∠npqr,npqs. So, assume the contrary. In that case, the closest point of v in pq is either por q. Assume it to bep.
Since ¯v lies outsidepq, the angle∠vpq is obtuse. We claim that this angle cannot be arbitrarily close to π. In fact, this angle cannot be more than the max- imum obtuse angle pq makes with the tangent plane of Dpqr at p. Simple calculation (Figure 1(middle)) shows that this angle is π2 + arccoskp−qk/2ρ(pqr) giving
∠vpq≤ π
2 + arccos 1 2a.
Since Dpqr containsv, kv−pk ≤2εγ. By Lemma 1,
∠np, vp≥arccosε. Applying Corollary 2, we get
∠npqr, vp≥∠np, vp−∠npqr,np≥arccosε−7ε.
Let zp k v¯v (Figure 1(right)). Then, ∠vv, vp¯ =
∠vpz =∠vpq−π2 ≤arccos2a1. One has∠npqr, v¯v≥
∠npqr, vp−∠v¯v, vp=∠npqr, vp−∠vpz≥arccosε− 7ε−arccos2a1 ≥ π2 −10ε −arccos2a1 for ε < 72π. We are now left to show that π2 −10ε−arccos2a1 >
∠npqr,npqs which requires π2 −24ε > arccos2a1 or a < 2 sin 24ε1 . This is precisely the condition required
by the lemma.
We are ready to prove the main results of this sec- tion.
Theorem 11 For any ε < 72π and δ = 2 sin 24ε, an (ε, δ)-dense triangulation has a stabbed triangle if and only if it has a locally stabbed triangle.
Proof. The ‘if’ part is obvious. Consider the ‘only if’ part. Let pqr be stabbed by v. As δ = 2 sin 24ε, the radius-edge ratio is at most 1/(2 sin 24ε). By Lemma 10,pqr is locally stabbed orvstabs a triangle twhere pow(v, Dt)<pow(v, Dpqr). In the latter case, repeat the argument witht. We must reach a locally stabbed triangle since the power distance of v from the diametric balls cannot decrease indefinitely.
Theorem 12 For any ε < 72π and δ = 2 sin 24ε, an (ε, δ)-dense triangulation can be flipped to a Gabriel triangulation.
Proof. The maximum circumradius decreases after each flip and the nearest neighbor distance cannot be decreased by flips. So MeshFlip maintains the (ε, δ)- dense conditions after each flip. By Theorem 11, all triangles are Gabriel upon termination.
5 Dense triangulations
We also study the effect of MeshFlip on an ε-dense triangulationT without the uniformity condition.
Take a trianglet∈T. Letcbe its circumcenter. A β-balloftis a circumscribing ball centered atc+βnt. The triangletisβ-stabbed if a vertex stabs theβ-ball and (−β)-ball of t. We call t locally β-stabbedif the stabbing vertex is one of the three neighbor vertices oft.
If we decrease the radius of Dt by β, we get a smaller concentric ball which we denote byDβt. We call T β-Gabriel if for each trianglet∈T,Dβt is not stabbed by any vertex ofT.
Theorem 13 For anyε <0.1, an ε-dense triangula- tion of a surface with reachγcontains aβ-stabbed tri- angle only if it contains a locally(β−88ε2γ)-stabbed triangle.
By choosingβ= 88ε2γ, Theorem 13 implies that no triangle is 88ε2γ-stabbed at the termination ofMesh- Flip. So the output triangulation is 88ε2γ-Gabriel.
Theorem 14 For any ε < 0.1, an ε-dense triangu- lation of a surface with reach γ can be flipped to a 88ε2γ-Gabriel triangulation.
The omitted details can be found in the full version available at the authors’ webpages.
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Decomposing Non-Convex Fat Polyhedra
Mark de Berg∗ Chris Gray∗
Abstract
We show that any locally-fat polyhedron with nver- tices and convex fat faces can be decomposed into O(n) tetrahedra. We also show that the additional restriction that the faces are fat is necessary: there are fat polyhedra without fat faces that require Ω(n2) pieces in any convex decomposition. Finally, we show that if we want the tetrahedra in the decomposition to be fat themselves, then the number of tetrahedra cannot be bounded as a function ofn.
1 Introduction
Polyhedra and their planar equivalent, polygons, play an important role in many geometric problems. From an algorithmic point of view, however, general poly- hedra are unwieldy to handle directly: several algo- rithms can only handleconvex polyhedra, preferably of constant complexity. Hence, there has been exten- sive research into decomposing polyhedra into tetra- hedra or other constant-complexity convex pieces.
The two main issues in developing decomposition al- gorithms are (i) to keep the number of pieces in the decomposition small, and (ii) to compute the decom- position quickly.
In the planar setting the number of pieces is, in fact, not an issue if the pieces should be triangles:
any polygon admits a triangulation, and any triangu- lation of a polygon withnvertices hasn−2 triangles.
Hence, research focused on developing fast triangula- tion algorithms, culminating in Chazelle’s linear-time triangulation algorithm [7]. An extensive survey of algorithms for decomposing polygons and their appli- cations is given by Keil [10].
For 3-dimensional polyhedra, however, the situa- tion is much less rosy. First of all, not every non- convex polyhedron admits a tetrahedralization: there are polyhedra that cannot be decomposed into tetra- hedra without using Steiner points. Moreover, decid- ing whether a polyhedron admits a tetrahedralization without Steiner points is NP-complete [12]. Thus we have to settle for decompositions using Steiner points.
Chazelle [6] has shown that any polyhedron with n vertices can be decomposed into O(n2) tetrahedra,
∗Department of Computing Science, TU Eindhoven.
P.O. Box 513, 5600 MB Eindhoven, the Netherlands. Email:
{mdberg,cgray}@win.tue.nl. This research was supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 639.023.301.
and that this is tight in the worst case: there are polyhedra with n vertices for which any decomposi- tion uses Ω(n2) tetrahedra. (In fact, the result is even stronger: any convex decomposition—a convex de- composition is a decomposition into convex pieces—
uses Ω(n2) pieces, even if one allows pieces of non- constant complexity.) Since the complexity of algo- rithms that need a decomposition depends on the number of tetrahedra in the decomposition, this is rather disappointing. Chazelle’s polyhedron is quite special, however, and one may hope that polyhedra arising in practical applications are easier to handle.
This is the topic of our paper: are there types of polyhedra that can be decomposed into fewer than a quadratic number of pieces? Erickson [9] has an- swered this question affirmatively for so-called local polyhedra by showing that any such 3-dimensional polyhedron can be decomposed intoO(nlogn) tetra- hedra. We considerfat polyhedra.
Types of fatness. Before we can state our results, we first need to give the definition of fatness that we use. In the study of realistic input models [5], many definitions for fatness have been proposed. When the input is convex many of these definitions are basically equivalent. When the input is non-convex, however, this is not the case: polyhedra that are fat under one definition may not be fat under a different definition.
Therefore we study two different definitions.
The first is a generalization of the (α, β)-covered ob- jects introduced by Efrat [8] to 3-dimensional objects.
A simply-connected object P in R3 is (α, β)-covered if the following condition is satisfied: for each point p∈∂P there is a simplexT with one vertex atpthat is fully insideP such thatT isα-fat and has diameter β·diam(P). Here a tetrahedron is called α-fat if all its solid angles are at least α, and diam(P) denotes the diameter ofP.
The second definition that we use was introduced by De Berg [2]. For an objecto and a ball B whose center lies inside o, we define B⊓o to be the con- nected component of B∩o that contains the center of B. An object o is locally-γ-fat if for every ball B that has its center inside oand which does not com- pletely contain o, we have vol(B ⊓o) ≥ γ·vol(B), where vol(·) denotes the volume of an object. Note that if we replace⊓with∩—that is, we do not restrict the intersection to the component containing the cen- ter of B—then we get the definition of fat polyhedra
