Computational Geometry: Theory and Applications

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1 JID:COMGEO AID:1238 /FLA [m3g; v 1.87; Prn:22/11/2012; 15:16] P.1 (1-25) Computatonal Geometry ( ) Contents lsts avalable at ScVerse ScenceDrect Computatonal Geometry: Theory and Applcatons Geometrc and combnatoral propertes of well-centered trangulatons n three and hgher dmensons Evan VanderZee a, Anl N. Hran b,,damrongguoy c, Vadm Zharntsky d,edgara.ramos e a Argonne Natonal Laboratory, Lemont, IL, USA b Department of Computer Scence, Unversty of Illnos at Urbana-Champagn, 201 N. Goodwn Avenue, Urbana, IL 61801, USA c Synopsys Inc., Mountan Vew, CA, USA d Department of Mathematcs, Unversty of Illnos at Urbana-Champagn, Urbana, IL, USA e Escuela de Matemátcas, Unversdad Naconal de Colomba, Colomba artcle nfo abstract Artcle hstory: Receved 21 January 2010 Accepted 7 November 2012 Avalable onlne xxxx Communcated by R. Sedel Keywords: Mesh generaton Acute trangulatons Fnte element method Crcumcentrc dual Dscrete exteror calculus An n-smplex s sad to be n-well-centered f ts crcumcenter les n ts nteror. We ntroduce several other geometrc condtons and an algebrac condton that can be used to determne whether a smplex s n-well-centered. These condtons, together wth some other observatons, are used to descrbe restrctons on the local combnatoral structure of smplcal meshes n whch every smplex s well-centered. In partcular, t s shown that n a 3-well-centered (2-well-centered) tetrahedral mesh there are at least 7 (9) edges ncdent to each nteror vertex, and these bounds are sharp. Moreover, t s shown that, n stark contrast to the 2-dmensonal analog, where there are exactly two vertex lnks that prevent a well-centered trangle mesh n R 2, there are nfntely many vertex lnks that prohbt a well-centered tetrahedral mesh n R Elsever B.V. All rghts reserved. 1. Introducton An n-dmensonal smplex s n-well-centered f ts crcumcenter les n ts nteror. More generally, t s k-well-centered f each of ts k-dmensonal faces s k-well-centered. It s completely well-centered f t s k-well-centered for each k, 1 k n [20]. It can be shown that an n-dmensonal trangulaton consstng of n-well-centered smplces s a Delaunay trangulaton, however n-well-centeredness s stronger than Delaunay and s dfferent from other famlar propertes of smplces [20,3]. Several authors have noted the possble applcaton of well-centered meshes to partcular problems. Among these are Ncolades [15] and Sazonov et al. [16,17], who have dscussed the covolume method and ts applcaton n electromagnetcs smulatons. Also, Kmmel and Sethan [9] descrbed an algorthm for numercally solvng the Ekonal equaton on trangulated domans. Ther algorthm, whch can be used to compute geodesc paths on trangulated surfaces, s descrbed frst for acute trangulatons (.e., 2-well-centered trangulatons) and requres addtonal work for trangulatons that are nonacute. Üngör and Sheffer [18] used acute planar trangulatons when they ntroduced the tent-ptchng algorthm for space tme meshng. For lnear ellptc problems, planar acute trangulatons result n a dscrete maxmum prncple as well as dagonal domnance of the stffness matrx [5,12]. Well-centered meshes also fnd applcaton wthn Dscrete Exteror Calculus (DEC), a framework for desgnng numercal methods for partal dfferental equatons [7,6]. In DEC, a suffcent condton for dscretzng the Hodge star operator so that t s represented by a dagonal matrx wth postve dagonal entres s to use a well-centered mesh (recent work shows that the well-centeredness condton can be relaxed [8]). A dagonal Hodge star matrx can lead to effcent numercal soluton. * Correspondng author. E-mal address: hran@llnos.edu (A.N. Hran) /$ see front matter 2012 Elsever B.V. All rghts reserved.

2 JID:COMGEO AID:1238 /FLA [m3g; v 1.87; Prn:22/11/2012; 15:16] P.2 (1-25) 2 E. VanderZee et al. / Computatonal Geometry ( ) Constructng well-centered meshes s a nontrval task, so researchers have put effort nto fndng ways to work around an algorthmc requrement for well-centered meshes. Recently, however, there has been progress towards makng wellcentered meshes through mesh optmzaton. In [21] we descrbed a heurstc for obtanng well-centered meshes of planar domans by startng wth a gven trangle mesh of the doman and relocatng the nteror vertces of the mesh to mnmze a cost functon defned on the coordnates of those vertces. We later generalzed the cost functon, makng t possble to optmze meshes of any dmenson [22]. In[20] we gve a varety of examples of tetrahedra that are and are not wellcentered, and we show that t s possble to mesh many smple shapes n R 3 wth completely well-centered tetrahedra. In many cases, the completely well-centered tetrahedral meshes of [20] were obtaned by creatng ntal relatvely hgh-qualty meshes by hand and applyng the optmzaton method to the ntal meshes. Not all smplcal meshes can be made well-centered by optmzng the cost functons of [21] and [22]. In some cases the combnatoral propertes of the mesh prevent the mesh from becomng well-centered. In such cases the mesh wll not be well-centered for any choce of vertex coordnates. Ths paper develops theory and ntuton related to the geometry of well-centered smplces and apples the geometrc condtons to nvestgate combnatoral propertes of well-centered meshes. Before dscussng the specfc results of ths paper, a few comments about termnology are n order. The term wellcentered may be used wthout a qualfyng dmenson to refer to the general concept, or may refer to one of the more precse terms f the context makes clear whch more precse term s approprate. When speakng of a trangle, for example, well-centered smultaneously means 2-well-centered and completely well-centered, snce every smplex s trvally 1-wellcentered. All of these defntons can be appled to smplcal meshes or smplcal complexes embedded n Eucldean space. Thus f a smplcal mesh s sad to be k-well-centered, ths means that every k-dmensonal smplex appearng n the mesh properly contans ts crcumcenter. 2. Results After gvng defntons and notaton n Secton 3, we ntroduce n Secton 4 several geometrc and algebrac condtons for an n-smplex to be well-centered. Sectons 5 and 6 apply the theory developed n Secton 4 to establsh condtons on the combnatoral structure of the neghborhood of a vertex n a well-centered tetrahedral mesh. Fnally, Secton 7 records some observatons specfc to constructng well-centered meshes of the cube, and Secton 8 offers some concludng thoughts. We enumerate the contrbutons of ths paper n more detal n the followng paragraphs. In Secton 4 we prove three new statements on n-well-centeredness of an n-smplex, each phrased n terms of the locaton of a vertex v wth respect to the facet opposte v : (a) a necessary geometrc condton (Proposton 3 the Cylnder Condton), (b) a suffcent geometrc condton (Proposton 8 the Prsm Condton), and (c) a both necessary and suffcent algebrac condton expressed n terms of cubc polynomal nequaltes (Proposton 11). The two geometrc condtons are generalzatons to hgher dmensons of condtons n R 2. Secton 5 nvestgates combnatoral propertes that follow from the results n Secton 4. (d) We prove a new combnatoral condton that must be satsfed by the lnk of an nteror vertex n an n-well-centered mesh n R n (Theorem 13). (See Secton 3 for the defnton of the lnk of a vertex.) (e) As an easy corollary we show that n a 3-well-centered tetrahedral mesh n R 3, every nteror vertex has at least seven ncdent edges (Corollary 14). (f) We show that, n stark contrast to the analogous case n R 2, where there are only two vertex lnks that cannot appear n a 2-well-centered mesh, there are nfntely many vertex lnks that cannot appear n a 3-well-centered mesh n R 3 (Corollary 15). (g) We also construct an nfnte famly of vertex lnks that can appear n a completely well-centered tetrahedral mesh n R 3. (h) The secton closes by showng that f a vertex lnk can appear n a 3-well-centered mesh and the vertex lnk satsfes some mnor addtonal condtons, then degree three vertces can be successvely nserted nto the vertex lnk to create an nfnte famly of vertex lnks that can appear n a 3-well-centered mesh (Proposton 19). Secton 6 develops combnatoral condtons that 2-well-centered tetrahedral meshes n R 3 must satsfy. () We prove n Theorem 23 that no trangulaton of S 2 on m vertces wth a vertex of degree at least m 3 can appear n a 2-wellcentered tetrahedral mesh n R 3. (j) It follows that n a 2-well-centered tetrahedral mesh n R 3, every nteror vertex has at least nne ncdent edges (Corollary 24). (k) We show that vertces of degree three can be nserted nto or deleted from trangulatons that permt 2-well-centered neghborhoods to create other trangulatons that permt 2-well-centered neghborhoods (Proposton 25). (l) Vertces of degree four can also be added to such trangulatons (Proposton 26). At several ponts n the paper, we make clams about addtonal results beyond what s actually proved n ths paper. Further detals about some of these clams appear n the frst author s dssertaton [19]. 3. Defntons and notaton We begn by ntroducng some defntons and notaton that wll be used throughout the paper. A smplex s referred to wth a Greek letter, usually σ or τ. A superscrpt for a smplex ndcates the dmenson, so, for example, σ n s an n-smplex. The notaton σ n =[v 0 v 1...v n ] s used to ndcate that σ n s the convex hull of the n + 1vertcesv 0, v 1,...,v n. It s assumed that the vertces of a smplex are n general poston,.e., that the vertces are affnely ndependent, so σ n s fully n-dmensonal. An (n 1)-dmensonal face of an n-dmensonal smplex σ n s called a facet of σ n.thecrcumcenter of a smplex σ s denoted c(σ ). Forann-smplex σ n embedded n R n, c(σ n ) s the unque pont whch has the same

3 JID:COMGEO AID:1238 /FLA [m3g; v 1.87; Prn:22/11/2012; 15:16] P.3 (1-25) E. VanderZee et al. / Computatonal Geometry ( ) 3 Fg. 1. Four vews of the same 3-well-centered tetrahedron σ 3 n the same orentaton. From left to rght the vews show σ 3 wth the equatoral balls of ts bottom, rght, left, and rear facets. For each facet τ 2 of σ 3, the crcumsphere (.e., crcumcrcle) of τ 2, whch s an equator of the equatoral ball B(τ 2 ), s shown. Because the tetrahedron s 3-well-centered, the vertex v opposte facet τ 2 les outsde of B(τ 2 ). (See Theorem 1.) For the bottom facet and rear facet vews n the fgure, the reader may need to look closely to see that the edges ncdent to v do perce B(τ 2 ), andv les outsde B(τ 2 ). dstance from every vertex of σ n.whenσ n s embedded n R m for m > n, c(σ n ) s the unque pont that among all ponts equdstant from the vertces of σ n mnmzes the dstance to the vertces of σ n.thecrcumradus of a smplex σ,.e., the dstance from c(σ ) to the vertces of σ,sdenotedr(σ). We also use the cone operaton of algebrac topology [14], wrtngu σ n to ndcate the smplex formed by takng the convex hull of a vertex u together wth the n-dmensonal smplex σ n to form a smplex of dmenson n + 1. Ths notaton may also be used for a set K of smplces; u K s the set of smplces {u σ : σ K }. Theaffne hull of a set S R m, whch we denote aff(s), s the smallest affne space that contans S. Forasmplexσ n =[v 0...v n ] we can defne t as ( aff [v 0...v n ] ) { n } n = λ v : λ = 1, <λ < for = 0,...,n. =0 =0 The affne hull of a smplex σ may also be called the plane of σ. When referrng to a smplex σ,theboundary of the smplex, denoted Bd(σ ), s the unon of the set of proper faces of σ,.e., the set of all faces of σ other than σ tself. The nteror of the smplex, denoted by Int(σ ), sdefnedasσ Bd(σ ). More generally, for a set S, weuseint(s) to refer to the nteror of S takenwthrespecttotheusualtopologyofaff(s). For the closure of a set S we use the notaton Cl(S). For a vertex u of a smplcal complex we defne St u, thestar of the vertex, to be the unon of the nterors of all the smplces for whch u s a vertex. The closure of the star, or the closed star, Cl(St u), s the unon of all smplces ncdent to u. Thelnk of a vertex u s defned by Lk u = Cl(St u) St u. Many of the terms brefly defned here are defned and dscussed at more length n [14]. We wsh to avod any ambguty about the dmenson of the crcumsphere or crcumball of a smplex σ. We adopt the notaton S k for the k-dmensonal sphere. Throughout ths paper the objects crcumsphere and crcumball always lve n aff(σ ). Thus the crcumsphere of a trangle s always a copy of S 1, even when the trangle s embedded n R 3 as the facet of a tetrahedron. The equatoral ball of a smplex σ,sometmesdenotedb(σ), s a ball of radus R(σ ) centered at c(σ ), but dstngushed from the crcumball of σ by the fact that the equatoral ball s consdered n a hgher-dmensonal space. For example, the equatoral ball of a trangle τ consdered n R 3 s the unque 3-dmensonal ball that has the crcumcrcle of τ as an equator (see Fg. 1). The ambent hgher-dmensonal space that contans the equatoral ball should be made clear wherever the term s used. In ths paper, wherever the term equatoral ball s used and a hgher-dmensonal space s not explctly specfed, the term appears n the context of a smplex σ n = u τ n 1, and B(τ n 1 ) s an n-dmensonal subset of aff(σ n ). We frequently dscuss the facets of a smplex σ n =[v 0...v n ]. As a matter of conventon, the facets usually are denoted τ n 1 wth the understandng that the facet τ n 1 0,...,τn n 1 = 0, 1,...,n. Fnally, the word mesh wll refer to a manfold smplcal complex. 4. Characterzng the well-centered smplex s the facet opposte vertex v.thusσ n = v τ n 1 for each We now nvestgate geometrc propertes of an n-well-centered n-smplex. The context for ths dscusson s a smplex σ n = u τ n 1 wth facet τ n 1 gven. The vertex u s free to move, and we wsh to determne whether σ n s n-well-centered based on the poston of u relatve to the fxed vertces of τ n 1. We frst recall an alternate geometrc characterzaton of the n-well-centered n-smplex and state ts consequences n ths context. The alternate characterzaton s stated n terms of equatoral balls. Usng the result, whch adopts the notatonal conventon that σ n = v τ n 1 for each = 0, 1,...,n, one can determne whether an n-smplex s n-well-centered wthout explctly computng c(σ n ).

4 JID:COMGEO AID:1238 /FLA [m3g; v 1.87; Prn:22/11/2012; 15:16] P.4 (1-25) 4 E. VanderZee et al. / Computatonal Geometry ( ) Fg. 2. Afacetτ of a tetrahedron σ (not shown n the fgure) and the reflecton τ of τ through c(σ ). Inthecaseontheleftτ s not 2-well-centered; on the rght τ s 2-well-centered. The tetrahedron σ s 3-well-centered f and only f ts vertex u opposte τ les n the sphercal trangle determned by the ntersecton of the crcumsphere of σ and a cone on τ wth apex c(σ ). (The sphercal trangle s shown n the case on the rght, but not on the left.) Theorem 1 (Equatoral Balls Condton). The smplex σ n s n-well-centered f and only f vertex v les strctly outsde B(τ n 1 ) for each = 0, 1,...,n. Proof. See [22]. Fg. 1 llustrates Theorem 1 as t apples to a tetrahedron. For each vertex v of the tetrahedron, Fg. 1 shows the equatoral ball B(τ ) of the facet τ opposte v, emphaszng n a darker color the crcumcrcle of τ (whch s an equator of B(τ )). The fgure shows that n each case v s outsde the equatoral ball of τ, so we can conclude that the tetrahedron s 3-well-centered. In the context of an n-smplex σ n wth a free vertex u opposte a fxed facet τ n 1, Theorem 1 becomes a necessary condton that vertex u must satsfy f σ n s to be n-well-centered. Corollary 2 (One-Facet Equatoral Ball Condton). Let σ n = u τ n 1. If the smplex σ n s n-well-centered, then u les strctly outsde of B(τ n 1 ). To ntroduce the remanng results of ths secton and get a somewhat dfferent perspectve on Corollary 2, we consder the sketch n Fg. 2. Trangle τ represents a facet of a tetrahedron σ whose fourth vertex u has not yet been determned. (Thus tetrahedron σ does not appear n Fg. 2.) We suppose, however, that the crcumcenter c(σ ) of the tetrahedron s known, so u s constraned to le on the crcumsphere of σ.thetwosdesoffg. 2 show two dfferent cases; on the left τ s not 2-well-centered, and on the rght τ s 2-well-centered. Now τ les n the plane aff(τ ), and the reflectons of τ and aff(τ ) through c(σ ) are τ and aff(τ ) respectvely. The plane aff(τ ) ntersects the crcumsphere to determne a lower sphercal cup C, andaff(τ ) determnes an upper sphercal cup C. The necessary condton of Corollary 2 says that when σ s 3-well-centered u does not le n the lower sphercal cup C, because C s contaned n B(τ ). From the geometry one can see that, n fact, f σ s to be 3-well-centered (that s, f c(σ ) s to fall strctly nsde σ) thenu must le strctly nsde the sphercal trangle determned by the ntersecton of C, the upper cup of the crcumsphere of σ, wth the geometrc cone on τ wth apex c(σ ). (Ths sphercal trangle s drawn n the case on the rght, but not n the case on the left.) In partcular, there s a necessary condton that u must le strctly n the upper cup C of the sphere. Thus f we project u orthogonally nto aff(τ ) (movng u vertcally n the fgure), the projecton of u les n the nteror of the crcumdsk of τ. Moreover, f τ s 2-well-centered, as on the rght, then the projecton of the sphercal trangle nto aff(τ ) contans the projecton of τ nto aff(τ ). Thus f the projecton of u nto aff(τ ) les nsde the projecton of τ nto aff(τ ), thss suffcent to establsh that σ s 3-well-centered. These condtons and ther generalzatons nto hgher dmensons are the frst two condtons dscussed n ths secton. The geometrc ntuton developed here s formalzed and proved algebracally n Propostons 3 and 8. Fnally, we consder varyng the poston of c(σ ). Notce that as c(σ ) moves n Fg. 2 from the crcumcenter of τ upward along a lne orthogonal to aff(τ ), the sphercal trangle of u-postons that produce a 3-well-centered tetrahedron wth crcumcenter c(σ ) sweeps out a sold 3-dmensonal regon. Tetrahedron σ wll be 3-well-centered f and only f u les n ths regon. The secton closes by descrbng ths regon for arbtrary dmensons n terms of polynomal nequaltes. (See Fg. 9.) Now we state a proposton that gves a necessary condton for an n-smplex σ n to be n-well-centered. See Fg. 3. Proposton 3 (Cylnder Condton). Let σ n be an n-well-centered n-smplex n R n wth u a vertex of σ n and τ n 1 the facet of σ n opposte u. That s, let σ n = u τ n 1. Let P be the orthogonal projecton P : R n aff(τ n 1 ).Then P(u) c(τ n 1 ) < R(τ n 1 ),.e., u projects to the nteror of the crcumball of τ n 1.

5 JID:COMGEO AID:1238 /FLA [m3g; v 1.87; Prn:22/11/2012; 15:16] P.5 (1-25) E. VanderZee et al. / Computatonal Geometry ( ) 5 Fg. 3. Because the tetrahedron σ n = u τ n 1 s 3-well-centered, P(u) les nsde the crcumball of τ n 1. Proof. Consder the coordnate system on R n such that c(σ n ) s the orgn, and aff(τ n 1 ) ={x R n : x n = k} for some constant k 0. In ths coordnate system, P s the projecton P : (x 1,...,x n 1, x n ) (x 1,...,x n 1,k). Let u = (x 1,...,x n ) n ths coordnate system. We have assumed that σ n s n-well-centered, so c(σ n ) (the orgn) s strctly nteror to σ n. It follows that k < 0 and x n > 0. Consder the lne segment l from u to u. Observe that l s a dameter of the crcumsphere of σ n. Moreover,l Int(τ n 1 ). Ths follows from the fact that σ n s n-well-centered; we have σ n = u τ n 1 and c(σ n ) Int(σ n ),sothere must be some pont w Int(τ n 1 ) such that c(σ n ) les on uw l. We notce, then, that the pont u les below aff(τ n 1 ) and conclude that x n > k. By the Pythagorean theorem, R(τ n 1 ) 2 + k 2 = R(σ n ) 2.Wealsohave n x 2 = ( R σ n) 2, =1 snce u les on the crcumsphere of σ n. It follows that ( ) P(u) c τ n 1 2 n 1 = x 2 = ( R σ n) 2 x 2 n < ( R σ n) 2 k 2 = ( R τ n 1) 2. =1 The statement s not lmted to σ n R n, of course. For a smplex σ n R m wth m > n, there exsts a coordnate system such that aff ( σ n) = { x R m : x = 0for = n + 1,...,m }, and the same proof apples. Remark 4. Gven a partcular smplex τ n 1 R n, Proposton 3 provdes a geometrc necessary condton on the locaton of vertex u to create an n-well-centered smplex σ n = u τ n 1.Vertexu must le wthn a sold rght sphercal cylnder over the crcumsphere of τ n 1 f σ n s to be n-well-centered. Fg. 4 llustrates the condton n 2D and 3D, makng t clear how ths condton generalzes from the famlar 2D case nto hgher dmensons. In each case the vertces of the base smplex τ n 1, as well as the crcumcenter c(τ n 1 ), are marked by small dark-colored balls. If u τ n 1 s n-wellcentered, then the vertex u must le nsde the gray cylnder over the crcumsphere of τ n 1. In the notaton of Fg. 2, where the crcumcenter of σ s known, the Cylnder Condton says that u must le ether n the upper cup C or the lower cup C. Remark 5. The One-Facet Equatoral Ball Condton (Corollary 2) s also a necessary condton on the locaton of vertex u. In R 2, the combnaton of Corollary 2 and the Cylnder Condton s suffcent to guarantee that a trangle (a 2-smplex) s acute (s 2-well-centered). In R n for n 3, however, an n-smplex u τ n 1 for whch u satsfes both of these necessary condtons mght not be n-well-centered. Example 6. For example, consder the tetrahedron σ = σ 3 wth vertces ( 0.152, 0.864, 0.48), ( 0.64, 0.6, 0.48), (0.6, 0.64, 0.48), and ( 0.192, 0.64, 0.744), whose crcumcenter les at the orgn. For three of the four facets τ 2 of σ 3,vertexv satsfes both necessary condtons wth respect to τ 2, and for the fourth facet v satsfes the Cylnder Condton, but not the One-Facet Equatoral Ball Condton. The tetrahedron σ 3 s not 3-well-centered.

6 JID:COMGEO AID:1238 /FLA [m3g; v 1.87; Prn:22/11/2012; 15:16] P.6 (1-25) 6 E. VanderZee et al. / Computatonal Geometry ( ) Fg. 4. If a smplex u τ n 1 s n-well-centered, then u s nteror to a sold rght sphercal cylnder over the crcumsphere of τ n 1. Fg. 5. A tetrahedron that s not 3-well-centered, even though every vertex satsfes the necessary condton of Proposton 3. Three of the vertces also le outsde the equatoral balls of ther respectve opposte facets. Fg. 5 shows several dfferent vews of σ 3. The large vew at left shows that σ 3 s not 3-well-centered; the crcumcenter of σ 3, marked by a small axes ndcator, les outsde the tetrahedron. The four small vews on the rght sde of Fg. 5 are vews drectly down onto the facets τ 2 of σ 3. In each case the crcumcrcle of τ 2 s rendered n a darker color, and one can see that the vertex above the facet projects to the nteror of the crcumball of the facet,.e., that vertex v satsfes the Cylnder Condton wth respect to τ 2. In three of the four cases all except the case at lower left the vertex also satsfes the One-Facet Equatoral Ball Condton. The partcular example n Fg. 5 s also mentoned n [20], whch gves some addtonal statstcs on the tetrahedron. Example 7. The tetrahedron wth vertces at ( 0.01, 0.01, 0.01), (1, 0, 0), (0, 1, 0), and (0, 0, 1) s another tetrahedron that s not 3-well-centered. It also has three vertces that satsfy the One-Facet Equatoral Ball Condton and four vertces that satsfy the Cylnder Condton. Ths example s dhedral acute, n contrast to the prevous example. The above examples llustrate that the One-Facet Equatoral Ball Condton and the Cylnder Condton are not enough to establsh that the n-smplex u τ n 1 s n-well-centered. However, the followng proposton does provde suffcent condtons that u τ n 1 s n-well-centered. See also Fg. 6. Proposton 8 (Prsm Condton). Let τ n 1 be an (n 1)-well-centered smplex n R n and σ n = u τ n 1. If u les outsde the equatoral ball B(τ n 1 ) and the reflecton of P(u) through c(τ n 1 ) s nteror to τ n 1,thenσ n s n-well-centered.

7 JID:COMGEO AID:1238 /FLA [m3g; v 1.87; Prn:22/11/2012; 15:16] P.7 (1-25) E. VanderZee et al. / Computatonal Geometry ( ) 7 Fg. 6. Because P( u) and c(τ n 1 ) = P(c(σ n )) are both nteror to τ n 1 and c(σ n ) s above aff(τ n 1 ),weknowthatthetetrahedronσ n = u τ n 1 s 3-well-centered. Fg. 7. A tetrahedron for whch the top vertex and bottom facet satsfy all of the suffcent condtons for beng 3-well-centered except that the bottom facet s not 2-well-centered. Proof. We assume the stated hypothess and take the same coordnate system that was used n the proof of Proposton 3. Observe that f u were on the equatoral ball of τ n 1,thenc(σ n ) would le n τ n 1, concdng wth c(τ n 1 ).Becauseu les outsde the equatoral ball of τ n 1, c(σ n ) les nteror to the same halfspace as u wth respect to aff(τ n 1 ). It follows that k < 0 and x n > k. Observe that, as shown n Fg. 6, the reflecton of P(u) through c(τ n 1 ) s P( u). Bythehypothess,P( u) s nteror to τ n 1.ThusP( u) s nteror to the crcumball of σ n and n 1 P(u) 2 = P( u) 2 = k 2 + x 2 < ( R σ n) n 2 = x 2. =1 =1 It follows that x n > k = k. Snce we know that x n > k, we conclude that x n > k > 0. Let l bethelnesegmentfromu to u. We wll show that l ntersects the nteror of τ n 1. Then, because σ n = u τ n 1 and k < 0 < x n (so that c(σ n ) l s above τ n 1 and below u), we wll be able to conclude that c(σ n ) s nteror to σ n. We know that P(c(σ n )) = c(τ n 1 ) s nteror to τ n 1 because τ n 1 s (n 1)-well-centered. Snce P( u) and P(c(σ n )) are both nteror to τ n 1, the lne segment from c(σ n ) to u, whch s contaned n l, s nteror to the (convex) nfnte prsm τ n 1 R. Moreover,0> k > x n (.e., c(σ n ) s above τ n 1 and u s below τ n 1 ), so ths part of segment l ntersects the nteror of τ n 1. As was the case for Proposton 3, Proposton 8 s not lmted to σ n R n ; n hgher-dmensonal spaces R m there s a coordnate system such that aff ( σ n) = { x R m : x = 0for = n + 1,...,m }, and the same proof apples. After readng Proposton 8, one mght ask whether the requrement that the facet τ n 1 be (n 1)-well-centered can be removed from the proposton. It may already be clear from the dscusson of Fg. 2 that the answer to ths queston s no. The tetrahedron n Fg. 7 s an explct example that confrms the requrement cannot be removed. Example 9. The tetrahedron n Fg. 7 s the convex hull of vertces v 0 = (0.224, 0.768, 0.6), v 1 = (0.8, 0, 0.6), v 2 = (0.224, 0.768, 0.6), and v 3 = ( 0.28, 0, 0.96). The bottom facet n Fg. 7, whch s the trangle τ 2 3 =[v 0v 1 v 2 ],lesnthe

8 JID:COMGEO AID:1238 /FLA [m3g; v 1.87; Prn:22/11/2012; 15:16] P.8 (1-25) 8 E. VanderZee et al. / Computatonal Geometry ( ) Fg. 8. If the base smplex τ n 1 s (n 1)-well-centered and vertex u s both outsde B(τ n 1 ) and nsde the nfnte prsm over the reflecton of τ n 1 through c(τ n 1 ), then the smplex u τ n 1 s n-well-centered. plane x 3 = 0.6 and s an obtuse trangle. The obtuse angle s at vertex v 2, the rghtmost vertex n Fg. 7. Takng ths bottom facet to be τ 2 as n Proposton 8, and the top vertex to be u = v 3, we satsfy the condtons that u le outsde the equatoral ball of τ 2 and that the reflecton of P(u) through c(τ 2 ) be nteror to τ 2.Indeed,c(τ 2 ) = (0, 0, 0.6) and R(τ 2 ) = 0.8 wth u c(τ 2 ) = , so u s outsde B(τ 2 ), and P( u) = (0.28, 0, 0.6) les nsde τ 2.Thuswe satsfy all of the Prsm Condton except the requrement that τ 2 be 2-well-centered. It s clear from Fg. 7 that ths s not suffcent; the crcumcenter of tetrahedron u τ 2, marked by a small axes ndcator, les outsde the tetrahedron, so the tetrahedron s not 3-well-centered. Remark 10. Lke the condton of Proposton 3, the condton of Proposton 8 has a nce geometrc nterpretaton. Gven an (n 1)-well-centered facet τ n 1,fthevertexu opposte τ n 1 les outsde B(τ n 1 ) and wthn an nfnte prsm (a rght cylnder) over the reflecton of τ n 1 through ts crcumcenter, then σ n = u τ n 1 s n-well-centered. Fg. 8 portrays the regon defned by the Prsm Condton for specfc examples n 2 and 3 dmensons. In each case the base smplex τ n 1 s shown n dark colors and sold lnes, and ts reflecton s outlned wth lghter colors and dashed lnes. In the fgure, each τ n 1 s (n 1)-well-centered, so for a vertex u lyng nsde the prsm over the reflecton of τ n 1 through c(τ n 1 ) and outsde the equatoral ball of τ n 1,.e., for a vertex u lyng n the gray regon shown n Fg. 8, thesmplexu τ n 1 wll be n-well-centered. Note that on the left n Fg. 8 the base smplex and ts reflecton should actually le on top of each other, but are set slghtly apart n the drawng so the reader can dstngush them from each other. We have now establshed two dfferent condtons for an n-smplex to be n-well-centered. One condton s a necessary condton, and the other condton s a suffcent condton. Both condtons are stated n terms of the locaton of a vertex u relatve to the facet τ n 1 opposte u. The regons defned by the necessary condton and the suffcent condton may be qute dfferent from each other. For example, n the 3D portons of Fgs. 4 and 8 the same base smplex τ n 1 yelds rather dfferent regons for the two condtons. It s natural to seek a precse descrpton of the regon where the vertex u wll produce an n-well-centered n-smplex u τ n 1. The followng dscusson develops just such a set of condtons on the locaton of u. The condtons take the form of a system of cubc polynomal nequaltes n the coordnates of u. Thesmplex u τ n 1 wll be n-well-centered f and only f the coordnates of u satsfy the polynomal nequaltes. The nequaltes are derved from a lnear system of equatons dscussed n [2]. Ths lnear system, whch provdes one way to compute the crcumcenter of a smplex σ n embedded n R m for m n, s brefly revewed here. We may wrte the crcumcenter c of a smplex σ n =[v 0 v 1...v n ] as a lnear combnaton of the vertces v R m, c = α 0 v 0 + α 1 v 1 + +α n v n, wth the coeffcents α satsfyng n =0 α = 1. The coeffcents α are known as the barycentrc coordnates of the crcumcenter. The condton that σ n be n-well-centered s the same as the condton that 0 < α for every α,.e., the condton that the crcumcenter be a convex combnaton of the vertces of σ n wth strctly postve coeffcents. Suppose we are gven the coordnates of the vertces v of σ n. We know that c v, c v = c v 2 = R 2 for each vertex v. Introducng the varable λ = R 2 c 2, we obtan the n + 1 equatons 2 c, v +λ = v 2.Sncethe vertces v are known, each equaton s a lnear equaton n the n + 2 unknowns α 0, α 1,...,α n,λ. The fnal equaton of the

9 JID:COMGEO AID:1238 /FLA [m3g; v 1.87; Prn:22/11/2012; 15:16] P.9 (1-25) E. VanderZee et al. / Computatonal Geometry ( ) 9 system s n =0 α = 1, whch forces the α to be barycentrc coordnates. As long as ths lnear system of n + 2 equatons n n + 2 unknowns s nonsngular, we can solve for the barycentrc coordnates. If the smplex s nondegenerate,.e., f the n + 1 vertces are affnely ndependent, then the smplex has a unque, fnte crcumcenter, whch has unque barycentrc coordnates. It follows that the lnear system has a unque soluton; hence the matrx s nonsngular. Let A be the matrx of ths lnear system and b the rght-hand sde, 2 v 0, v 0 2 v 0, v 1 2 v 0, v n 1 v 0, v 0 2 v 1, v 0 2 v 1, v 1 2 v 1, v n 1 v 1, v 1 A = , b =.. 2 v n, v 0 2 v n, v 1 2 v n, v n 1 v n, v n For = 0, 1,...,n we let A be the matrx A wth column +1 replacedbyb. Cramer s rule tells us that α = det(a )/ det(a). If we consder vertces v 0,...,v n 1 to be the vertces of some gven τ n 1 and v n to be a free vertex u, then the barycentrc coordnates α are ratonal functons of the coordnates of u. Thus the condtons α > 0 become algebrac nequaltes n the coordnates of u. To smplfy matrx A a lttle, we translate each vertex of the smplex by v 0. The translaton may change the value of λ n the soluton vector n fact, λ = 0 always holds for the translated system but the barycentrc coordnates of the crcumcenter are not changed by translatng the vertces of the smplex. If m > n we make one further smplfcaton. In the translated coordnate system, we rotate the smplex about the orgn v 0 to obtan a smplex for whch vector v v 0 {x: x n+1 = = x m = 0} for each = 1,...,n. Rotaton about the orgn s an orthogonal transformaton, so t does not change any of the entres of the lnear system and does not affect the barycentrc coordnates. If we restrct our attenton to one of the open halfspaces bounded by aff(τ n 1 ),wehaveetherdet(a)>0ordet(a)<0 throughout the halfspace, because det(a) s a contnuous functon of the entres n A and A s sngular only when u aff(τ n 1 ). We wll see that, n fact, det(a) 0 holds everywhere, so det(a)<0 throughout the halfspace. The frst row and the frst column of A n the smplfed lnear system are all zeroes except for the last entry, whch s 1 n both cases. Computng the determnant of A by frst expandng across the frst row and then expandng down the frst column (one wth an odd number of entres and the other wth an even number of entres) we fnd that det(a) = det(b) where B s the submatrx of A spannng rows 2 to n + 1 and columns 2 to n + 1. The n n submatrx B has the form 2V T V, where V s the m n matrx V = ( v 1 v 0 v 2 v 0 v n v 0 ). Because of the earler rotaton of the smplex, the last m n coordnates of each vector v v 0 are zeroes, and f we take Ṽ to be the frst n rows of V,thenṼ s an n n matrx that satsfes V T V = Ṽ T Ṽ. It follows that B = 2Ṽ T Ṽ.Thus det(b) = 2 n det(ṽ ) 2 0. Observng that det(ṽ ) s the sgned volume of the parallelepped spanned by the vectors that form the columns of Ṽ, we note that det(b)>0 holds when the columns of Ṽ are lnearly ndependent,.e. when the vertces of the orgnal smplex are affnely ndependent. Thus wth the assumpton that τ n 1 s a fully (n 1)-dmensonal smplex, we know that det(a) <0 when the vertex u les n ether of the open halfspaces bounded by aff(τ n 1 ). For u outsde aff(τ n 1 ), then, we conclude that α = det(a )/ det(a) >0 f and only f det(a )<0. Hence the smplex u τ n 1 wll be n-well-centered f and only f the coordnates of u satsfy the polynomal nequalty det(a )<0. It remans to show that the equaton det(a ) = 0 s a polynomal n the coordnates of u of degree at most 3. To do ths we examne the entres of A that depend on u. All of these entres appear n row n + 1 or n column n + 1. At most two of these entres are quadratc n the coordnates of u the entry at poston (n + 1,n + 1) and the entry at (n + 1, + 1). (Only one entry s quadratc n the coordnates of u when = n.) Every other entry that depends on u s lnear n the coordnates of u. UsngS n to denote the group of permutatons on n letters, the determnant of an n n matrx M can be wrtten as det(m) = π S n sgn(π) n M jπ( j), j=1 where M jk stands for the entry n row j, column k of matrx M, and sgn(π) s the sgnum functon appled to the permutaton. Consderng the structure of matrx A, we observe that each product n ths defnton of det(a ) nvolves at most two terms that depend on u, and at most one of these the entry selected from row n + 1 s quadratc n the coordnates of u. Thus the determnant s a summaton of terms that are polynomal n the coordnates of u and have degree at most 3. We can also explan ths from the perspectve of computng the determnant by expandng t along a row or column. We wll consder a specfc example wth = 2 arsng from a tetrahedron (dmenson n = 3), but the dscusson apples to the general case. We have

10 JID:COMGEO AID:1238 /FLA [m3g; v 1.87; Prn:22/11/2012; 15:16] P.10 (1-25) 10 E. VanderZee et al. / Computatonal Geometry ( ) Fg. 9. Gven a facet τ n 1, the regon where the vertex u may le to produce an n-well-centered smplex u τ n 1 s defned by a system of polynomal nequaltes. When τ n 1 s (n 1)-well-centered, so that there are regons related to both the necessary and suffcent condtons, the actual regon where u may le s somewhere n between the regons defned by the necessary Cylnder Condton (Fg. 4) and the suffcent Prsm Condton (Fg. 8). 2 v 0, v 0 2 v 0, v 1 v 0, v 0 2 v 0, u 1 2 v 1, v 0 2 v 1, v 1 v 1, v 1 2 v 1, u 1 A 2 = 2 v 2, v 0 2 v 2, v 1 v 2, v 2 2 v 2, u 1 2 u, v 0 2 u, v 1 u, u 2 u, u for ths partcular example. If we compute det(a ) by expandng down column n + 1 (column 4, n ths case), we fnd that term n + 1 of the summaton s a quadratc functon of the coordnates of u multpled by the determnant of a submatrx that s constant wth respect to u. In our example, ths s the fourth term n the summaton, 2 v 0, v 0 2 v 0, v 1 v 0, v u, u det 2 v 1, v 0 2 v 1, v 1 v 1, v v 2, v 0 2 v 2, v 1 v 2, v The remanng n + 1 of the terms n the summaton are lnear (or constant) functons of u multpled by a determnant of some other submatrx of A that s not constant wth respect to u. For our example, the frst term of the summaton s 2 v 1, v 0 2 v 1, v 1 v 1, v v 0, u det 2 v 2, v 0 2 v 2, v 1 v 2, v u, v 0 2 u, v 1 u, u Expandng the approprate row (usually row n) of each of these submatrces n smlar fashon, we obtan a summaton of terms that are ether lnear or quadratc n the coordnates of u (at most one term s quadratc), each multpled by the determnant of a smaller submatrx that s constant wth respect to u. We state the conclusons of the foregong dscusson as a formal proposton. Proposton 11. Let σ n = u τ n 1 for a fxed facet τ n 1. The n-smplex σ n s n-well-centered f and only f the coordnates of vertex u satsfy the nequaltes det(a )<0, whch are cubc polynomal nequaltes n the coordnates of u. Fg. 9 gves a graphcal representaton of the precse regon where the vertex u may be placed to produce a 3-wellcentered tetrahedron u τ n 1.Thefacetτ n 1 used n Fg. 9 s the same facet used to llustrate the necessary condton for a tetrahedron n Fg. 4 and the suffcent condton for a tetrahedron n Fg. 8, so readers can see for ths specfc case how the full regon compares to the regons defned by the necessary condton and the suffcent condton. The facet τ n 1 along wth ts crcumcrcle and the reflecton of τ n 1 through c(τ n 1 ) are shown n Fg. 9 to ad ths comparson. It should also be noted that Fg. 9 was generated usng MATLAB s sosurface functon and evaluatons of the polynomal nequaltes on a fnte grd, so the graphcal representaton has some slght mperfectons. For nstance, the entre crcumcrcle of τ n 1 les n the boundary of the regon even though n Fg. 9 t appears that there s a small gap above and below aff(τ n 1 ). 5. Local combnatoral propertes of 3-well-centered tetrahedral meshes The geometrc propertes of the n-well-centered n-smplex dscussed n Secton 4 have mplcatons for the combnatoral propertes of well-centered meshes. As a smple motvatng example we consder the 2-dmensonal case of a trangle mesh

11 JID:COMGEO AID:1238 /FLA [m3g; v 1.87; Prn:22/11/2012; 15:16] P.11 (1-25) E. VanderZee et al. / Computatonal Geometry ( ) 11 Fg. 10. If v and v j are both contaned n the nteror of the sold rght sphercal cylnder over the crcumcrcle of τ u,thenu les outsde the smplex formed from v, v j, and the other vertces of τ u. n the plane. If v s a vertex nteror to ths mesh and there are fewer than fve edges ncdent to v, then some angle ncdent to v has measure π/2 radans or larger. Thus the mesh has a nonacute trangle. Ths geometrc observaton can be restated as a combnatoral property of 2-well-centered (.e., acute) trangle meshes. Namely, there are at least fve edges ncdent to every nteror vertex of an acute trangle mesh n R 2. Ths well-known fact s a key ngredent n the generaton of 2-well-centered trangle meshes through optmzaton of vertex coordnates; the mesh must satsfy ths combnatoral condton at every nteror vertex f optmzng the vertex coordnates s to have any hope of fndng an acute mesh. Smlarly, tetrahedral meshes n R 3 that are 2-well-centered or 3-well-centered must satsfy certan local mesh connectvty condtons. These combnatoral condtons, whch are key to creatng well-centered tetrahedral meshes, are analyzed n the next two sectons. Ths secton develops some of the combnatoral propertes of 3-well-centered tetrahedral meshes, and the next secton examnes combnatoral propertes of 2-well-centered tetrahedral meshes. The combnatoral propertes of tetrahedral meshes n R 3 are more complex than the analogous propertes for trangle meshes n R 2. In a trangle mesh n R 2, the lnk of an nteror vertex s a set of edges that form a cycle around the vertex,.e., a trangulaton of a topologcal crcle (S 1 ). The number of edges ncdent to the nteror vertex, whch s the number of vertces on the cycle, completely characterzes the neghborhood of the vertex combnatorally. In tetrahedral meshes n R 3, on the other hand, the lnk of an nteror vertex s a trangulaton of a topologcal sphere S 2. Thus the number of edges ncdent to the vertex does not completely characterze the neghborhood of the vertex. We do, however, prove necessary condtons on the number of edges that must be ncdent to an nteror vertex n a tetrahedral mesh n R 3 n order for the mesh to be 3-well-centered, 2-well-centered, or completely well-centered. We also show that there s no suffcent condton n terms of the number of edges ncdent to an nteror vertex. Much of the dscusson n Sectons 5 and 6, then, s phrased n terms of the lnk of an nteror vertex. For a tetrahedral mesh n R 3, ths s a trangulaton of S 2, whch corresponds to a planar trangulaton n a graph theoretc sense. We try to avod the term planar trangulaton to prevent possble confuson wth trangle meshes n R 2. We begn wth two results that apply n arbtrary dmenson. The frst lemma generalzes the followng statement about planar trangle meshes, usng the Cylnder Condton (Proposton 3) to relate geometry to combnatorcs. If a planar trangle σ 2 =[abc] s subdvded nto three trangles by addng a vertex u nteror to σ 2 and addng edges [ua], [ub], and [uc] to obtan u Bd(σ 2 ), then at most one of the three trangles [abu], [bcu], and [cau] n u Bd(σ 2 ) s an acute trangle. The man dea of the proof of Lemma 12 s llustrated by Fg. 10. Lemma 12. For n 2, letσ n =[v 0 v 1...v n ] have facets τ n 1 0, τ n 1 1,...,τn n 1. If u s a pont lyng n σ n, then at most one of the n-smplces of u Bd(σ n ),.e., at most one of the smplces u τ n 1 0, u τ n 1 1,...,u τn n 1, s an n-well-centered n-smplex. Proof. It suffces to prove the statement when u s n the nteror of σ n.indeed,fu s on the boundary of σ n and two or more of the smplces u τ n 1 are n-well-centered, then we can slghtly perturb u nto the nteror and obtan a pont u Int(σ n ) wth at least two n-well-centered n-smplces. Thus we assume that u Int(σ n ). Let τ n 1 and τ n 1 be two dstnct facets of σ n.thenu τ n 1 and u τ n 1 are n-smplces, and τ n 1 τ n 1 s an j j j (n 2)-dmensonal face of σ n.thefaceτ n 1 τ n 1 s ncdent to all but two of the vertces of σ n, the two vertces v j and v j. (Recall that v s opposte τ n 1 and v j s opposte τ n 1.) Notce that u τ n 1 and u τ n 1 have a common facet, j j the (n 1)-smplex τu n 1 := u (τ n 1 τ n 1 ). WeletT aff(σ n ) be the sold rght sphercal cylnder over the crcumball j of τu n 1. Assume towards contradcton that u τ n 1 and u τ n 1 are both n-well-centered. By the Cylnder Condton (Proposton 3), both v and v j le n Int(T ). NowT s a convex set, and all the vertces of σ n le n T,soσ n T. On the j other

12 JID:COMGEO AID:1238 /FLA [m3g; v 1.87; Prn:22/11/2012; 15:16] P.12 (1-25) 12 E. VanderZee et al. / Computatonal Geometry ( ) Fg. 11. In R 2 the smplcal complex K consstng of two trangles [v 0 v 1 v 2 ] and [v 1 v 3 v 2 ] and ther faces (left) satsfes the hypothess of Theorem 13 for the mesh M = u L embedded n R 2 (center), so M s not 2-well-centered. The embeddng of M n R 2 nduces a geometrc realzaton of K nto R 2 (rght). The geometrc realzaton of K s not an embeddng n ths case, snce [v 1 v 3 v 2 ] s nverted here. The partcular embeddng of M does not affect the exstence of the abstract complex K n Theorem 13; there s no embeddng of M that s 2-well-centered. hand, u les on the crcumsphere of τu n 1,sou Bd(T ). Thusu / Int(σ n ) Int(T ), contradctng the assumpton we made n the frst paragraph of the proof. We conclude that at most one of u τ n 1, u τ n 1 j s n-well-centered. The next theorem shows that Lemma 12 has mplcatons for the local combnatoral propertes of n-well-centered meshes. The theorem s stated usng the language of smplcal complexes. We say that a vertex u s an nteror vertex n an n-dmensonal smplcal complex embedded n R n f (the underlyng space of) Lk u s homeomorphc to S n 1,the sphere of dmenson n 1. Thus the closed star of u s homeomorphc to an n-dmensonal ball n R n, and the pont v les n the nteror of the ball n the standard topology on R n. When we speak of an abstract smplcal complex K we make an mportant dstncton between an embeddng of K and a geometrc realzaton of K.Anembeddng of K s an assgnment of coordnates n R n to the vertces of K such that K s a smplcal complex n R n wth vertces at the specfed locatons. By a geometrc realzaton of K we mean merely some assgnment of coordnates n R n to the vertces of K. Thus n a geometrc realzaton of K n R n, t s possble for K to have self-ntersectons. Fg. 11, whch s related to the proof of Theorem 13, llustrates the dstncton between these two terms. Theorem 13 (One-Rng Necessary Condton). Let u be an nteror vertex of an n-dmensonal smplcal complex M (e.g., a mesh) embedded n R n,andsetl= Lk u. If there exsts an abstract fnte n-dmensonal smplcal complex K such that () K s an n-manfold complex (wth boundary), () Bd(K ) s somorphc to L, and () for every n-smplex σ n K, there are at least two (n 1)-smplces n Bd(σ n ) Bd(K ), then u L s not n-well-centered. Proof. We frst observe that every vertex of K must also be a vertex of Bd(K ). By assumpton (), every smplex of K s a face of some n-smplex of K,sofK had a vertex v not n Bd(K ), then there would be some n-dmensonal smplex σ n K ncdent to v, and σ n would have only one (n 1)-dmensonal face not ncdent to v. Sncev / Bd(K ), t follows that Bd(σ n ) Bd(K ) would contan at most one (n 1)-smplex, and () would not be satsfed. The embeddng of M n R n ncludes an embeddng of u L n R n. Snce every vertex of K s a vertex of Bd(K ) and Bd(K ) = L, ths embeddng of u L n R n nduces a geometrc realzaton of K nto R n.(asshownnfg. 11, thegeometrc realzaton mght not be an embeddng.) We have an embeddng of the smplcal complex u L n R n. Snce t s an embeddng, each n-dmensonal smplex s a fully n-dmensonal geometrc object, and we have consstent orentaton. Moreover, L s star-shaped wth respect to u. We clam that by () and () ths mples that there s some smplex n the nduced geometrc realzaton of K that contans the pont u (possbly on ts boundary). We return to ths clam n a moment, but frst we show how ths completes the proof. Fx a smplex σ n K that contans u. Now consder the n-smplces of u Bd(σ n ). By assumpton () of the hypothess, at least two of these smplces have a facet n L. Eachsmplexofu Bd(σ n ) wth a facet n L s a member of u L, and by Lemma 12 at most one of these smplces s n-well-centered. We conclude that at least one of the smplces of u L M s not n-well-centered. Now we prove the clam that there s a smplex of the geometrc realzaton of K that contans the pont u. Choose a lne l through u n general poston. General poston here means that l does not ntersect any face of K of dmenson less than n 1. Such an l canbechosenunlessu tself les on a smplex ρ k of K of dmenson k < n 1, and n that case we are done, snce ρ k s a face of some σ n and that σ n contans u.

13 JID:COMGEO AID:1238 /FLA [m3g; v 1.87; Prn:22/11/2012; 15:16] P.13 (1-25) E. VanderZee et al. / Computatonal Geometry ( ) 13 Fg. 12. There s only one trangulaton of S 2 wth 5 vertces, and t has two correspondng tetrahedral complexes such that each tetrahedron has at least two facets n common wth the trangulaton. Snce u L s a smplcal complex and L s star-shaped wth respect to u, l ntersects exactly two smplces of L, eachof dmenson n 1, and the ntersecton ponts are n opposte drectons from u along l. For reference, we desgnate a + and a drecton and name facet τ+ n 1 (resp. τ n 1 )asthefacetofl ntersected by l n the + ( ) drecton from u. Startng from τ+ n 1 we descrbe a walk along l through n-smplces and (n 1)-smplces of the geometrc realzaton of K that ends at τ n 1 n 1. By contnuty of ths walk and τ+, τ n 1 n opposte drectons from u, there must be some n-smplex n the geometrc realzaton of K that contans u. The walk s as follows. Snce K s a manfold wth a boundary and τ+ n 1 s on the boundary, there s a unque σ n 1 ncdent to τ n 1 0 := τ+ n 1.Thenforagvenσn the walk s on l at τ n 1 n, and l ntersects some unque second facet of σ, whch 1 we name τ n 1.Aslongasτ n 1 τ n 1, we are not on the boundary of K,so(snceK s manfold) there are exactly two n-dmensonal smplces ncdent to τ n 1.Oneofthesesσ n, and the other we name σ n +1.SnceK s a manfold complex, the sequence τ n 1 has no repettons and must eventually end at τ n 1.(Theσn n the sequence may flp back and forth n orentaton, whch corresponds to the walk gong back and forth along l.) It s worth notng that the exstence of the abstract smplcal complex K has no dependence on the partcular embeddng of M n R n. Theorem 13 s really a combnatoral statement, and we can use t to show that a partcular abstract smplcal complex L = Bd(K ) cannot appear as the lnk of an nteror vertex n an n-well-centered mesh embedded n R n. The case n = 3 s of partcular nterest. Usng the One-Rng Necessary Condton of Theorem 13 t s farly easy to establsh a tght lower bound on the number of edges ncdent to a vertex n a 3-well-centered tetrahedral mesh embedded n R 3. Corollary 14. Let M be a 3-well-centered tetrahedral mesh embedded n R 3. For every vertex u nteror to M, at least 7 edges of M are ncdent to u. Proof. Brtton and Duntz have assembled a catalog of all polyhedra wth at most 8 vertces, whch ncludes all the trangulatons of S 2 wth at most 8 vertces [4]. ByTheorem 13 t suffces to show that each such trangulaton L of S 2 wth at most 6 vertces has a correspondng tetrahedral complex K such that each tetrahedron of K has at least two facets n common wth L. There s only one trangulaton of S 2 wth 4 vertces the boundary of a tetrahedron. The correspondng tetrahedral complex s that sngle tetrahedron. There s also only one trangulaton of S 2 wth 5 vertces. Ths trangulaton s shown n Fg. 12 along wth two correspondng tetrahedral complexes. Ether complex certfes that the trangulaton cannot be the lnk of any vertex n a 3-well-centered mesh. For sx vertces there are two nonsomorphc trangulatons of S 2.ThefrstsshownnFg. 13 along wth ts correspondng tetrahedral complex. The second s drawn n Fg. 14 along wth ts correspondng tetrahedral complex. When there are m 7 vertces, there exst trangulatons L of S 2 wth m vertces such that there s no tetrahedral complex K satsfyng both Bd(K ) = L and the condton that every tetrahedron of K have at least two facets n L. In partcular, the trangulatons of S 2 wth 7 vertces and degree lsts (5, 5, 5, 4, 4, 4, 3) and (6, 5, 5, 5, 3, 3, 3),.e., polyhedra 7 1 and 7 4 n the catalog of Brtton and Duntz, both can appear as the lnk of a vertex n a 3-well-centered mesh. The problem of decdng whether such confguraton s possble can be expressed by a frst order formula over the reals and, hence, t s decdable [1]. Fg. 15 shows an example of a 3-well-centered mesh n R 3 consstng of a sngle vertex u and ts neghborhood Cl(St u) such that Lk u s a trangulaton wth degree lst (5, 5, 5, 4, 4, 4, 3). Fg. 16 shows a smlar example for the degree lst (6, 5, 5, 5, 3, 3, 3). The boundares of these complexes are embeddngs of polyhedra 7 1 and 7 4 mentoned above. We obtaned these embeddngs by mposng symmetres and usng our optmzaton software descrbed n [22].

14 JID:COMGEO AID:1238 /FLA [m3g; v 1.87; Prn:22/11/2012; 15:16] P.14 (1-25) 14 E. VanderZee et al. / Computatonal Geometry ( ) Fg. 13. In the frst trangulaton of S 2 wth 6 vertces, each vertex has exactly four neghbors. There s a tetrahedral complex consstng of four tetrahedra such that each tetrahedron has two facets n common wth ths trangulaton of S 2. Fg. 14. In the second trangulaton of S 2 wth 6 vertces, the degree lst s (5, 5, 4, 4, 3, 3). Ths trangulaton of S 2 also has a correspondng tetrahedral complex such that each tetrahedron has at least two facets n common wth the trangulaton. Fg. 15. A 3-well-centered mesh wth an nteror vertex u such that Lk u has seven vertces and degree lst (5, 5, 5, 4, 4, 4, 3). The vertex coordnates are lsted n the table at rght; vertex u s at the orgn. Ths complex s symmetrc w.r.t. rotatons by 120 around the z-axs. Fg. 16. A 3-well-centered mesh wth an nteror vertex u such that Lk u has seven vertces and degree lst (6, 5, 5, 5, 3, 3, 3). The vertex coordnates are lsted n the table at rght; vertex u s at the orgn. Ths complex s symmetrc w.r.t. rotatons by 120 around the z-axs. There are three other trangulatons of S 2 wth 7 vertces. Each has a correspondng tetrahedral complex K satsfyng the requrements of the One-Rng Necessary Condton (Theorem 13), so none of these trangulatons can appear as the lnk of avertexna3-well-centeredmesh.

15 JID:COMGEO AID:1238 /FLA [m3g; v 1.87; Prn:22/11/2012; 15:16] P.15 (1-25) E. VanderZee et al. / Computatonal Geometry ( ) 15 There are 14 nonsomorphc trangulatons of S 2 wth 8 vertces. Of these, 5 have tetrahedral complexes K that certfy they cannot be the lnk of a vertex n a 3-well-centered tetrahedral mesh n R 3. Each of the other 9 trangulatons can appear as the lnk of a vertex n a 3-well-centered tetrahedral mesh n R 3. (We menton these results wthout proof here.) For m 8 vertces, then, the necessary condton of Theorem 13 completely characterzes whch trangulatons can and cannot be made 3-well-centered. We leave open the queston of whether the One-Rng Necessary Condton stated n Theorem 13 s a complete characterzaton for m > 8 vertces n 3 dmensons. We also have not nvestgated f the theorem provdes a complete characterzaton for n-well-centeredness n dmensons n 4. The trangulatons on 8 vertces that cannot be made 3-well-centered are polyhedra 8 4, 8 5, 8 6, 8 7, and 8 13 n the catalog [4] of Brtton and Duntz. It s nterestng to note that the degree lst of 8 7, whch cannot be made 3-well-centered, s the same as the degree lst of 8 8, whch can be made 3-well-centered [19]. Thus the degree lst of a trangulaton does not provde enough nformaton to determne whether the trangulaton can be the lnk of a vertex n a 3-well-centered tetrahedral mesh n R 3. There are 50 nonsomorphc trangulatons wth 9 vertces and an exponentally growng number of trangulatons wth more vertces [13], so although makng a catalog for 9 or 10 vertces mght be somewhat nterestng, somethng more abstract wll be necessary to defntvely characterze whch trangulatons can be made 3-well-centered. In the rest of ths secton we dscuss some more general results n the drecton of characterzng whch trangulatons of S 2 can appear as the lnk of a vertex n a 3-well-centered mesh n R 3. The results fall short of a complete characterzaton, but do show that the set of trangulatons of S 2 that cannot appear as the lnk of a vertex n a 3-well-centered mesh and the set of trangulatons that can appear as the lnk of a vertex are both nfnte. Corollary 15. For any nteger m 4 there s a trangulaton of S 2 wth m vertces that cannot appear as the lnk of a vertex n a 3-well-centered mesh. Proof. We have already proved that ths holds for 4 m 6. For m 7 we note that the tetrahedral complexes shown on the rght-hand sdes of Fgs. 12 and 13 can be generalzed. Consder a tetrahedral complex K consstng of a set of m 2 tetrahedra that close around a common edge. The complex K satsfes the condtons of Theorem 13, sobd(k ) cannot appear as the lnk of a vertex n a 3-well-centered mesh. Bd(K ) s a trangulaton of S 2 on m vertces wth degree lst (m 2,m 2, 4,...,4). We note that by removng a sngle tetrahedron from the example complex K of Corollary 15, we obtan another nfnte famly of trangulatons of S 2 that cannot appear as the lnk of a vertex n a 3-well-centered mesh. Each member of ths famly s a trangulaton on m vertces wth degree lst (m 1,m 1, 4,...,4, 3, 3). Ths famly generalzes the tetrahedral complexes shown on the left-hand sde of Fg. 12 and the rght-hand sde of Fg. 14. Remark 16. Usng an argument based on Euler s formula, t can be shown that tetrahedral complexes satsfyng the condtons of Theorem 13 are ether from the famly dscussed n Corollary 15 or from the famly of chans of tetrahedra. (A chan of p tetrahedra s a complex consstng of tetrahedra σ 1, σ 2,...,σ p such that for 2 p 1 tetrahedron σ has two boundary facets and shares one facet wth each of σ 1 and σ +1.) It follows that for each m there are, up to somorphsm, at most 3 m m-vertex trangulatons of S 2 for whch Theorem 13 apples. Corollary 15 s one nstance that shows how substantal the dfference s between tetrahedral and trangle meshes. In the case of trangle meshes n R 2, where we consder trangulatons of S 1 as the lnk of a vertex, the only two trangulatons that cannot appear as the lnk of a vertex are the 3-cycle and the 4-cycle. In contrast, there are nfntely many trangulatons of S 2 that cannot be the lnk of a vertex n a 3-well-centered mesh n R 3. One may wonder whether there are stll nfntely many trangulatons of S 2 that can appear as the lnk of a vertex n a 3-well-centered mesh n R 3. The answer s yes. One way to prove ths s to explctly construct an nfnte famly of 3-well-centered meshes wth dfferent vertex lnks. We wll do exactly that n a moment, wth the help of the followng lemma, whch we prove usng the Prsm Condton (Proposton 8). Lemma 17. Let S n 1 u be a unt (n 1)-sphere centered at a pont u. If τ n 1 s an (n 1)-well-centered (n 1)-smplex whose vertces le on S n 1 u,andthedstancefromutoaff(τ n 1 ) s greater than 1/ 2,thenσ n := u τ n 1 s an n-well-centered n-smplex. Proof. Suppose that τ n 1 s an (n 1)-well-centered smplex meetng the condtons specfed n the hypothess. Let S n 2 u be the crcumsphere of τ n 1. S n 2 u s the ntersecton of aff(τ n 1 ) wth S n 1 u,.e., an (n 2)-sphere lyng n S n 1 u.the orthogonal projecton of u nto aff(τ n 1 ), whch we denote by P(u), s the center of S n 2 u,.e., the crcumcenter c(τ n 1 ) of τ n 1. Snce τ n 1 s (n 1)-well-centered, t contans the pont c(τ n 1 ).Thusτ n 1 contans the reflecton of P(u) through c(τ n 1 ). The crcumradus of τ n 1 satsfes R(τ n 1 ) 2 + z 2 = 1, where z s the dstance from u to aff(τ n 1 ),sobecause z 2 > 1/2, we have R(τ n 1 )<1/ 2, and u les outsde the equatoral ball of τ n 1. By the Prsm Condton, σ n s n-wellcentered.

16 JID:COMGEO AID:1238 /FLA [m3g; v 1.87; Prn:22/11/2012; 15:16] P.16 (1-25) 16 E. VanderZee et al. / Computatonal Geometry ( ) Fg. 17. For k 4 we can create an acute trangulaton of the unt sphere from a set of vertces consstng of the north and south poles and two out-ofphase regular k-gons. Conng such a trangulaton to the orgn produces a completely well-centered tetrahedral mesh. The fgure shows the tetrahedral mesh obtaned for k = 7. It s relatvely straghtforward to prove the converse as well. For σ n = u τ n 1 wth the vertces of τ n 1 lyng on a sphere S n 1 u centered at u, fτ n 1 s not (n 1)-well-centered or the dstance z from u to aff(τ n 1 ) satsfes z 1/ 2, then σ n s not n-well-centered. Ths proof s left to the reader; the result s not needed n ths paper. The smplex σ n = u τ n 1 n Lemma 17 s an sosceles smplex wth all vertces of τ n 1 equdstant from the apex vertex u. Whenn = 2, Lemma 17 reduces to the statement that an sosceles trangle s acute f the apex angle s acute. In hgher dmensons Lemma 17 tells us when an sosceles smplex s n-well-centered. Note that n an sosceles smplex all of the faces ncdent to the apex vertex u are sosceles; the plane of each such face ntersects the sphere S n 1 n some lower-dmensonal sphere centered at u, and Lemma 17 can be appled to these sosceles faces. It follows that σ n wll be completely well-centered f τ n 1 s completely well-centered and z > 1/ 2. In partcular, for the case n = 3, an sosceles tetrahedron wth an acute trangle facet opposte the apex vertex s a completely well-centered tetrahedron. Thus from any trangulaton of a unt sphere S 2 wth suffcently small acute trangles we can create a completely wellcentered tetrahedral mesh n R 3 by takng the cone u τ 2 of each acute trangle τ 2 wth the center of the sphere u. Fg. 17 shows a completely well-centered tetrahedral mesh constructed n ths fashon. The boundary of the mesh n Fg. 17 s an acute trangulaton of S 2 selected from an nfnte famly of acute trangulatons of S 2. The next two paragraphs descrbe ths famly. Consder the set of vertces consstng of the north pole (0, 0, 1), thesouthpole(0, 0, 1), and the vertces of two regular k-gons, one n the plane z = and the other n the plane z = We set the polygons exactly off phase from each other. For nstance, let the coordnates of the polygon vertces be ( ( ) ( ) ) 2π 2π cos, sn, 0.352, = 0, 1,...,k 1, k k ( ( ) ( ) ) (2 + 1)π (2 + 1)π cos, sn, 0.352, = 0, 1,...,k 1. k k These coordnates were chosen to satsfy the condtons of Lemma 17 whle makng well-centeredness of the complex shown n Fg. 17 vsually persuasve. Let each pole vertex be adjacent to all of the vertces of the closer regular polygon. Ths constructs k sosceles trangles ncdent to each pole. We take each vertex of a regular polygon to be adjacent to the closer pole, the two neghbors on ts own regular polygon, and two vertces from the other regular polygon. Trangles formed entrely from vertces of the two regular polygons are also sosceles. The example n Fg. 17 uses the result of ths constructon for the case k = 7. We clam that f k 4, then each trangle τ 2 of ths constructon s acute and satsfes the condton that the dstance from the orgn to τ 2 s greater than 1/ 2. Snce k 4 t s clear that the apex angles of the sosceles trangles ncdent to the poles are acute angles. Verfyng that the other trangles are acute and that the trangles are far enough from the orgn s straghtforward and we omt the detals. Lemma 17 apples, and as an mmedate consequence we have the followng. Proposton 18. There are nfntely many trangulatons of S 2 that can appear as the lnk of a vertex n a completely well-centered mesh. For large enough k, ths constructon of completely well-centered neghborhoods of a vertex usng acute trangulatons of a unt sphere S 2 can be generalzed. One can use more than two regular k-gons, alternatng the phase between each successve k-gon. We have seen that there are nfntely many trangulatons of S 2 that cannot appear and nfntely many that can appear as the lnk of a vertex n a 3-well-centered mesh. The authors suspect that for m 8 vertces the majorty of trangulatons of S 2 on m vertces are trangulatons that can appear as a lnk of a vertex n a 3-well-centered mesh. We do not formally

17 JID:COMGEO AID:1238 /FLA [m3g; v 1.87; Prn:22/11/2012; 15:16] P.17 (1-25) E. VanderZee et al. / Computatonal Geometry ( ) 17 Fg. 18. Gven a 3-well-centered tetrahedron σ =[uv 2 v 3 v 4 ] wth acute angles uv v j, one can construct three tetrahedra [uv 2 v 3 v 1 ], [uv 3 v 4 v 1 ],and [uv 4 v 2 v 1 ] by addng a new vertex v 1 = u ε along the lne l through u and c(σ ). The crcumcenters of the constructed tetrahedra le along lnes connectng c(σ ) to the crcumcenters c(τ ) of the [uv v j ] facets of σ. As dscussed n Proposton 19, whenv 1 s close enough to u the reflecton of u through c(σ ) the constructed tetrahedra wll be 3-well-centered and the angles uv 1 v, uv v 1 wll be acute. The angles v uv j do not need to be acute for ths constructon. For example, v 2 uv 3 snotanacuteanglenthsfgure. prove that conjecture n ths paper, but n lght of the next proposton, t s hghly lkely; Proposton 19 provdes a method for constructng new trangulatons that can appear as the lnk of a vertex n a 3-well-centered tetrahedral mesh n R 3. In Proposton 19 we consder a trangulaton G of S 2 wth a vertex of degree 3. In ths context, the notaton G v 1 refers to the trangulaton of S 2 obtaned by deletng vertex v 1 and all faces ncdent to v 1, replacng them wth the face [v 2 v 3 v 4 ], where v 2, v 3, v 4 are the neghbors of v 1 n G, ordered to keep the orentaton consstent. Proposton 19. Let G be a trangulaton of S 2 wth a vertex v 1 of degree 3,andletv 2,v 3,v 4 be the neghbors of v 1 n G. Let M be a tetrahedral mesh n R 3 consstng of a vertex u and ts closed neghborhood Cl(St u),wthlk u somorphc to G v 1.If () Ms3-well-centered, () face angle uv v j s acute for each, j {2, 3, 4}, j, then there exsts a tetrahedral mesh M nr 3 and a vertex u of M suchthat () Lk u s somorphc to G, () M s3-well-centered, () face angle uv v j s acute for each, j {1, 2, 3, 4}, j. Proof. Fg. 18 accompanes ths proof and may help the reader understand the geometrc constructons dscussed n the proof. Consder a partcular tetrahedral mesh that satsfes the condtons of the hypothess. In ths mesh the tetrahedron σ = σ 3 =[uv 2 v 3 v 4 ] s 3-well-centered, so c(σ ) s nteror to σ. Let l be the lne through u and c(σ ). Lnel ntersects the crcumsphere of σ at two ponts. One of these s u, and the other we name u.wedefne u ε = (1 ε)u + εu, apontlyngonl. Becauseσ s 3-well-centered, we know that segment uu ntersects trangle [v 2 v 3 v 4 ] at some pont u ε 0, wth 1/2 > ε 0 > 0. We can cut σ nto the three tetrahedra [uv 2 v 3 u ε 0 ], [uv 3 v 4 u ε 0 ], and [uv 4 v 2 u ε 0 ]. For ε 0 > ε > 0 we consder the three tetrahedra [uv 2 v 3 u ε ], [uv 3v 4 u ε ], and [uv 4v 2 u ε ]. We clam that for suffcently small ε > 0 these three tetrahedra are 3-well-centered and the face angles uu ε v, uv u ε are acute for = 2, 3, 4. Examnng the face angles frst, we note that at ε = 0 the crcumcenters of the facets [uv u ε ] concde wth c(σ ) and wth each other. Indeed, each of these facets s a rght trangle wth ts crcumcenter lyng on the hypotenuse uu ε.asε ncreases, v uu ε does not change, uv u ε decreases, becomng smaller than π/2, and uu ε v ncreases but remans less than π/2 for suffcently small ε. Turnng to the tetrahedra, then, we wll argue that the specfc tetrahedron [uv 2 v 3 u ε ] s 3-well-centered for suffcently small ε. An argument dentcal except for changed labels apples to the other two tetrahedra, so ths wll complete the proof. We know that, regardless of the value of ε, the crcumcenter of [uv 2 v 3 u ε ] les on the lne orthogonal to aff([uv 2v 3 ]) = aff(τ 4 ) passng through c(τ 4 ); ths lne s the locus of ponts equdstant from u, v 2, and v 3. The locaton of c([uv 2 v 3 u ε ]) vares contnuously wth ε. Atε = 0, the crcumcenter of tetrahedron [uv 2 v 3 u ε ] concdes wth c(σ ), and as ε ncreases from0towardsε 0, c([uv 2 v 3 u ε ]) moves n the drecton of vector c(τ 4) c(σ ). Because uv 2 v 3 and uv 3 v 2 are acute,

18 JID:COMGEO AID:1238 /FLA [m3g; v 1.87; Prn:22/11/2012; 15:16] P.18 (1-25) 18 E. VanderZee et al. / Computatonal Geometry ( ) we know that c(τ 4 ) les n the sector of aff(τ 4 ) nteror to angle v 2 uv 3. Thus segment c(σ )c(τ 4 ) σ s contaned n [uv 2 v 3 u ε 0 ] [uv 2 v 3 u ε ], and for suffcently small ε > 0, tetrahedron [uv 2v 3 u ε ] s 3-well-centered. Because the face angles uv 1 v, uv v 1 are acute n the constructon of Proposton 19, the constructon can be terated. If a trangulaton G of S 2 satsfes the condtons of Proposton 19, then a degree 3 vertex v 1 can be nserted nto face [v 2 v 3 v 4 ]. In the new trangulaton of S 2, the three new faces ncdent to v 1 satsfy the condtons of Proposton 19, so a degree 3 vertex can be nserted nto any one of those three faces, and so on. In partcular, startng from any completely well-centered mesh constructed from an acute trangulaton of a unt sphere S 2, one can successvely nsert vertces of degree 3 to create an nfnte famly of trangulatons that can appear as the lnk of a vertex n a 3-well-centered mesh. It s also worth mentonng that each trangulaton of a topologcal S 2 wth 8 vertces v 1,...,v 8 that can appear as Lk u for a vertex u n a 3-well-centered mesh n R 3 has an embeddng nto R 3 for whch all of the face angles uv v j are acute for, j {1,...,8}, j, where v v j s an edge n the trangulaton. Recall that there are 50 nonsomorphc trangulatons of S 2 wth 9 vertces [13]. UsngProposton 19 to add vertces of degree 3 to the varous faces of trangulatons of S 2 wth 8 vertces, one can show that at least 34 of these 50 trangulatons of S 2 wth 9 vertces can appear as the lnk of a vertex n a 3-well-centered tetrahedral mesh embedded n R Local combnatoral propertes of 2-well-centered tetrahedral meshes Corollary 14 shows that n a 3-well-centered mesh there are at least 7 edges ncdent to each vertex. In the followng dscusson we wll see that the combnatoral constrants for a mesh to be 2-well-centered are qute dfferent from the constrants for a mesh to be 3-well-centered, and n terms of the mnmum number of edges ncdent to a vertex, they are more strngent. As n Secton 5, the dscusson focuses on Lk u where u s a vertex nteror to a tetrahedral mesh n R 3. Defnton. We say that a partcular trangulaton G of S 2 permts a 2-well-centered neghborhood of a vertex u f there exsts a tetrahedral mesh M n R 3 such that u s an nteror vertex of M, Lku s somorphc to G (as a smplcal complex), and all facets of M ncdent to u are 2-well-centered. (A facet means a 2-smplex n ths context.) It should be noted that ths defnton does not drectly address the queston of whether the tetrahedra ncdent to u are 2-well-centered, snce each tetrahedron ncdent to u has one facet lyng on Lk u, and that facet s not ncdent to u. We shall see, however, that for tetrahedral meshes n R 3, the smallest trangulaton that permts a 2-well-centered neghborhood n the sense of ths defnton can, n fact, appear as the lnk of a vertex n a completely well-centered mesh. Fnally, note that phrasng the problem n terms of the facets of M ncdent to u actually reduces the problem to determnng whether the face angles at u are acute, because f there s an arrangement of rays at u such that all of the face angles formed at u by these rays are acute, then we can place the neghbors of u at the ponts where these rays ntersect a unt sphere centered at u. Ths wll create a neghborhood of u n whch every 2-dmensonal face ncdent to u s an sosceles trangle wth an acute apex angle at u. The frst result of ths secton s a smple observaton that forms the foundaton for the theory developed n the rest of the secton. Lemma 20. Let u and v 1 be adjacent vertces n a tetrahedral mesh M embedded n R 3 and let v be a vertex of Lk u that s adjacent to v 1. The angle v 1 uv s acute f and only f v H 1,whereH 1 s the open halfspace that contans v 1 and s bounded by the plane through u orthogonal to the vector v 1 u. Proof. The angle v 1 uv s acute f and only f v 1 u, v u > 0, where, s the standard nner product on R 3, and ths holds f and only f v les n H 1. The next two techncal lemmas are based on Lemma 20. They lead to the proof of the man result of ths secton. In both lemmas and n the subsequent theorem we use the followng notaton. We denote by u a vertex n a tetrahedral mesh n R 3, and the m vertces of Lk u are labeled v 1,...,v m.foreachvertexv, the plane through u orthogonal to v u s denoted P, and the open halfspace bounded by P that contans v s denoted H. The other halfspace bounded by P wll be called H, and we take ths to be a closed halfspace, whch contans ts boundary P. The orthogonal projecton of a vertex v j nto P wll be denoted Pr (v j ). Lemma 21. Let v 1 and v 2 be nonadjacent vertces of Lk u, wth v 2 H 1.Ifv s a vertex adjacent to both v 1 and v 2 such that v 1 uv and v 2 uv are both acute angles, then the orthogonal projecton of v nto P 1 les n P 1 H 2. Proof. The sketch n Fg. 19 llustrates ths result. For an algebrac proof we assgn a coordnate system wth u as the orgn and v 1 lyng on the postve z-axs. Usng coordnates (x, y, z ) for vertex v, the condton v 2 H 1 means that z 2 0. Snce the angle v 1 uv s acute, Lemma 20 mples that v must le n H 1, and snce the angle v 2 uv s acute, Lemma 20

19 JID:COMGEO AID:1238 /FLA [m3g; v 1.87; Prn:22/11/2012; 15:16] P.19 (1-25) E. VanderZee et al. / Computatonal Geometry ( ) 19 Fg. 19. If a 2-well-centered mesh contans two vertces v 1 and v 2 that both le n Lk u, are not adjacent to each other, and have a common neghbor v and f v 2 les n H 1, then the orthogonal projecton of v nto P 1,.e., the pont Pr 1 (v ), must le n P 1 H 2. Fg. 20. Let u be a vertex of a tetrahedral mesh embedded n R 3,andletv 1, v 2, v, v j be vertces of Lk u wth adjacences as shown. If the face angles at u between adjacent vertces of Lk u are all acute angles, but v 1 uv 2 s nonacute, then the projecton of facet [v v 2 v j ] nto P 1 les n P 1 H 2. mples that v must le n H 2.Thusv les n H 1 H 2.SnceH 1 H 2 would be empty f v 2 had coordnates (0, 0, z 2 ),we can conclude that v 2 does not le on the z-axs. Wth the remanng freedom n defnng a coordnate system we specfy that v 2 has coordnates (x 2, 0, z 2 ) wth x 2 < 0. Now snce v H 1, we know that z 0. We also know that v, v 2 =x x 2 + z z 2 > 0, because v H 2. We have establshed that z z 2 0 and that x 2 < 0. It follows that x < 0. The projecton Pr 1 (v ) has coordnates (x, y, 0) and s nteror to P 1 H 2 ={(x, y, 0): x < 0}. Lemma 22. Let v 1 and v 2 be nonadjacent vertces of Lk u, wth v 2 H 1.If[v v 2 v j ] s a 2-smplex of Lk u, such that v,v j are both adjacent to v 1 and the face angles v 1 uv, v 1 uv j, v 2 uv, v 2 uv j, are all acute angles, then Pr 1 ([v v 2 v j ]) P 1 H 2,.e., the orthogonal projecton of the entre facet [v v 2 v j ] nto P 1 les n H 2. Proof. See the sketch n Fg. 20. From the gven hypotheses we can conclude by Lemma 21 that Pr 1 (v ) and Pr 1 (v j ) both le n P 1 H 2. Usng the same coordnate system defned n the proof of Lemma 21, thepontpr 1 (v 2 ) has coordnates (x 2, 0, 0) wth x 2 < 0, thus t les n P 1 H 2 as well. It follows that the orthogonal projecton of the facet [v v 2 v j ] nto P 1, whch s the convex hull of Pr 1 (v ), Pr 1 (v 2 ), and Pr 1 (v j ), les entrely n the convex set P 1 H 2. Applyng the above two lemmas, we obtan a combnatoral necessary condton on the neghborhood of an nteror vertex n a 2-well-centered mesh. Theorem 23. Let G be a trangulaton of S 2 wth m vertces. If G contans a vertex v 1 of degree d(v 1 ) m 3, then G does not permt a 2-well-centered neghborhood. Proof. We consder a vertex u such that Lk u s somorphc to G where G has a vertex of degree at least m 3 and consder a geometrc realzaton of Cl(St u) n R 3. Label the vertces of Lk u wth the labels v 1, v 2,...,v m such that v 1 s a vertex of maxmum degree and the (at most two) vertces not adjacent to v 1 are lsted mmedately after v 1 (e.g., labeled v 2, v 3 f there are two of them). We choose a coordnate system on R 3 such that u s at the orgn and v 1 les on the postve z-axs. Assume that all of the face angles v uv j are acute. We clam ths mples that for any facet [v v j v k ] wth at least one vertex n H 1, the orthogonal projecton of the facet nto P 1,.e., Pr 1 ([v v j v k ]), does not contan vertex u. Assumng ths clam for the moment, we see that u les outsde the (sold) polyhedron bounded by Lk u. (See Fg. 21.) Snce u s outsde ths polyhedron, some 3-smplex ncdent to u must be nverted. Thus the geometrc realzaton of Cl(St u) s not an embeddng, and the clam completes the proof. We proceed to prove the clam. Notng that v 1 H 1 by our defnton of H 1, we observe that for 4, vertex v must le n H 1 because v s adjacent to v 1. (Ths follows from Lemma 20.) Thus there are only two types of facets that may have nonempty ntersecton wth H 1.Thefrsttypes[v v 2 v j ] or [v v 3 v j ] where v and v j both are adjacent to v 1, and the

20 JID:COMGEO AID:1238 /FLA [m3g; v 1.87; Prn:22/11/2012; 15:16] P.20 (1-25) 20 E. VanderZee et al. / Computatonal Geometry ( ) Fg. 21. When Lk u has m vertces and one of the vertces v 1 has degree d(v 1 ) m 3, any geometrc realzaton of Cl(St u) n R 3 wth all face angles at u acute s not an embeddng. Theorem 23 shows that f we consder such a geometrc realzaton and project every facet that ntersects H 1 nto P 1,then the unon of the projected facets does not contan u. The sketch at left shows an example of a geometrc realzaton of a tetrahedral mesh Cl(St u) n R 3 such that every face angle at vertex u s acute. In the sketch, v 1 has degree d(v 1 ) = 6 = m 3nLku. Thesketchatrghtshowstheresultoftakngthe geometrc realzaton on the left and projectng each facet that ntersects H 1 nto P 1. Fg. 22. In the proof of Theorem 23, the most dffcult case to analyze s a facet of the second type [v 2 v 3 v j ] wth z 2 < 0andz 3 < 0. These sketches llustrate the projecton of the facet [v 2 v 3 v j ] nto P 1 for the two subcases x 3 < 0(left)andx 3 0(rght).Inbothsubcases,Pr 1 (v j ) les n P 1 H 2 H 3. When x 3 < 0, Pr 1 (v 3 ) P 1 H 2, so the projecton of facet [v 2 v 3 v j ] nto P 1 s a subset of P 1 H 2.Whenx 3 0, the projected facet can be decomposed nto two peces meetng along [p 0 Pr 1 (v j )]. OnepecelesnP 1 H 2, and the other pece les n P 1 H 3.Inbothsubcasesweseethatu / Pr 1 ([v 2 v 3 v j ]). second type s [v 2 v 3 v j ] for j 4. Consder, then, the frst type of facet, takng the specfc notaton [v v 2 v j ].(Thesame argument apples to [v v 3 v j ].) If v 2 les n H 1, we are done; the facet does not ntersect H 1.Otherwsev 2 les n H 1.Hence v 1 uv 2 s nonacute, and v 2 s not adjacent to v 1. Lemma 22 apples. The proof for facets of the second type s more complcated. If both v 2 and v 3 le n H 1, we are done. If one vertex les n H 1 P 1 and the other les n H 1, we assume wthout loss of generalty that z 2 0 and z 3 0. Then v 2 s not adjacent to v 1.Ifv 3 s adjacent to v 1,thenLemma 22 apples drectly wth v 3 functonng as v.onthe other hand, even f v 3 s not adjacent to v 1, the arguments of Lemmas 21 and 22 can be appled wth v 3 functonng as v. (In the proofs of Lemmas 21 and 22 we used v adjacent to v 1 to establsh only that z 0 and that v 2 does not le on the z-axs. The latter holds n ths case because v 2 and v 1 have common neghbor v j v 3.) Ths leaves the case z 2 < 0 and z 3 < 0. As noted above, v 2 does not le on the z-axs. We choose the coordnate system wth v 2 = (x 2, 0, z 2 ), x 2 < 0. We also assume wthout loss of generalty that y 3 0. (We can reflect through the plane y = 0 f y 3 < 0.) See Fg. 22 for sketches related to ths case. By applyng Lemma 20 three tmes, we obtan v j H 1 H 2 H 3, and by applyng Lemma 21 twceweobtanpr 1 (v j ) P 1 H 2 H 3.Ifx 3 < 0, then the whole segment Pr 1 ([v 2 v 3 ]) les n P 1 H 2, and snce Pr 1 (v j ) P 1 H 2, t follows that Pr 1 ([v 2 v 3 v j ]) P 1 H 2.ThusPr 1 ([v 2 v 3 v j ]) does not contan u. So we assume that x 3 0. Now f x 3 = 0 we know that y 3 0becausev 3,lkev 2,doesnotleonthez-axs. Moreover, x 3 > 0 also mples y 3 0, snce otherwse P 1 H 3 would be {(x, y, 0): x > 0}, yeldng P 1 H 2 H 3 = Pr 1 (v j ).Apont on Pr 1 ([v 2 v 3 ]) has the form λ Pr 1 (v 2 ) + (1 λ) Pr 1 (v 3 ) = λ(x 2, 0, 0) + (1 λ)(x 3, y 3, 0), wth 0 λ 1. Thus for a pont p = (x p, y p, 0) on Pr 1 ([v 2 v 3 ]), etherx p < 0 and the pont les n P 1 H 2 or both x p 0 and y p > 0 so that p, v 3 =x p x 3 + y p y 3 > 0 and the pont les n P 1 H 3. We conclude that Pr 1 ([v 2 v 3 ]) P 1 (H 2 H 3 ).

21 JID:COMGEO AID:1238 /FLA [m3g; v 1.87; Prn:22/11/2012; 15:16] P.21 (1-25) E. VanderZee et al. / Computatonal Geometry ( ) 21 Fg. 23. A completely well-centered mesh wth an nteror vertex u such that Lk u has nne vertces and degree lst (5, 5, 5, 5, 5, 5, 4, 4, 4). The vertex coordnates are lsted n the table at rght; vertex u s at the orgn. Fnally we note that there must exst a pont p 0 = (ε, y p0, 0) wth ε < 0 such that p 0 les on Pr 1 ([v 2 v 3 ]) and p 0 P 1 H 2 H 3. Thus we can decompose Pr 1 ([v 2 v 3 v j ]) nto the two peces p 0 Pr 1 ([v 2 v j ]) and p 0 Pr 1 ([v 3 v j ]), wththe frst pece lyng n P 1 H 2 and the second pece lyng n P 1 H 3. It follows that Pr 1 ([v 2 v 3 v j ]) does not contan u. Recall that Euler s formula specfes a relatonshp between the number of vertces, edges, and faces n a planar graph. If m, e, and f are the number of vertces, edges, and faces respectvely, then Euler s formula states that m e + f = 2 for planar graphs. In a planar trangulaton each face s ncdent to three edges and each edge s ncdent to two faces, so 2e = 3 f, and the relatonshp f = 2(m 2) can be derved. Moreover, n a planar trangulaton each face s ncdent to three vertces, and vertex v s ncdent to d(v ) faces, so d(v ) = 3 f = 6(m 2). Combnng these consequences of Euler s formula wth Theorem 23, we easly obtan a lower bound on the number of edges ncdent to an nteror vertex n a 2-well-centered tetrahedral mesh n R 3. Corollary 24. Let M be a 2-well-centered tetrahedral mesh n R 3. For every vertex u nteror to M, at least 9 edges of M are ncdent to u. Proof. Let G = Lk u for some nteror vertex u of a 2-well-centered mesh M, and let m be the number of edges ncdent to u,.e., the number of vertces of G. Consder the possblty m = 8. Euler s formula shows that for m = 8wehave d(v ) = 36 so the average vertex degree s 4.5, and there must be at least one vertex of degree at least 5 = m 3. By Theorem 23, ths cannot occur, for such a graph G would not permt a 2-well-centered neghborhood of u. Smlarly, f m = 7 the average degree s 30/7 > 4 and there must be a vertex of degree at least m 2. For m = 6 the average degree s 4 and there must be a vertex of degree at least m 2. In each of the cases m = 5 and m = 4, there s only one trangulaton, and ths trangulaton has a vertex of degree m 1. When m = 9, the average degree s 4 2 3, and there s a trangulaton of S2 wth degree lst (5, 5, 5, 5, 5, 5, 4, 4, 4) that permts a completely well-centered neghborhood. Fg. 23 shows a completely well-centered mesh that has a sngle nteror vertex u such that Lk u s a 9-vertex trangulaton of S 2 wth the specfed degree lst. We have already seen that there are nfntely many trangulatons of S 2 that can appear as the lnk of an nteror vertex n a 2-well-centered mesh (Proposton 18). In the sprt of Proposton 19, we now dscuss some ways to use an exstng trangulaton that permts a 2-well-centered neghborhood to construct new trangulatons that permt a 2-well-centered neghborhood. The next two propostons show that one can add vertces of degree 3, subtract vertces of degree 3, or add vertces of degree 4 to obtan new trangulatons that permt a 2-well-centered neghborhood. In Proposton 25 we agan use the notaton G v 1 used n Proposton 19. Proposton 25. A trangulaton G of S 2 that contans a vertex v 1 of degree three permts a 2-well-centered neghborhood f and only f the trangulaton G v 1 permts a 2-well-centered neghborhood. Proof. Frst we suppose that G permts a 2-well-centered neghborhood. Then consder some tetrahedral mesh embedded n R 3 that contans a vertex u wth Lk u somorphc to G and all face angles v uv j acute. We choose a coordnate system on R 3 such that u les at the orgn and dentfy each vertex v of Lk u wth the vector orgnatng at the orgn and termnatng at v. Now vector v 1 makes an acute face angle for each of the three facets that are ncdent to the edge [uv 1 ]. Deletng v 1 from Lk u removes the three facets that are ncdent to edge [uv 1 ], but has no effect on the other facets ncdent