Power Coordinates: A Geometric Construction of Barycentric Coordinates on Convex Polytopes

Transcription

1 Power Coordnates: A Geometrc Constructon of Barycentrc Coordnates on Convex Polytopes Max Budnnsky Caltech Bebe Lu Caltech Yyng Tong MSU Matheu Desbrun Caltech Abstract We present a full geometrc parameterzaton of generalzed barycentrc coordnates on convex polytopes. We show that these contnuous and non-negatve coeffcents ensurng lnear precson can be effcently and exactly computed through a power dagram of the polytope s vertces and the evaluaton pont. In partcular, we pont out that well-known explct coordnates such as Wachspress, Dscrete Harmonc, Vorono, or Mean Value correspond to smple choces of power weghts. We also present examples of new barycentrc coordnates, and dscuss possble extensons such as power coordnates for non-convex polygons and smooth shapes. Keywords: Generalzed barycentrc coordnates, Wachspress and mean-value coordnates, power dagrams, polytopal fnte elements. Concepts: Computng methodologes Shape modelng; Theory of computaton Computatonal geometry; 1 Introducton Generalzed barycentrc coordnates extend the canoncal case of Möbus barycentrc coordnates on smplces: they offer a smple and powerful way to nterpolate values on a polygonal (2D) or polytopal (3D) convex doman through a weghted combnaton of values assocated wth the polytope vertces. From values of a gven functon Φ at vertces v, one obtans a straghtforward nterpolatng functon φ va: n φ(x) = λ (x) Φ(v ). (1) =1 where the weghts {λ } =1..n are functons of x that are often referred to as generalzed barycentrc coordnates. Whle nterpolaton over smplcal (va lnear bass functons) or quadrlateral/hexahedral (va multlnear bass functons) elements has been used for decades n fnte element computatons and graphcs, a strong nterest n the desgn of more general nterpolants recently surfaced n varous dscplnes such as geometrc modelng (for Bézer patches, mesh parameterzaton, and volumetrc deformaton), mage processng (for mage deformaton) and computatonal physcs (to defne bass functons on polytopal elements). Among the many possble choces of coordnates that are known today, Wachspress coordnates [Wachspress 1975; Meyer et al. 2002; Warren et al. 2006] have been wdely adopted n 2D and 3D as they Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. Copyrghts for components of ths work owned by others than the author(s) must be honored. Abstractng wth credt s permtted. To copy otherwse, or republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. Request permssons from permssons@acm.org. c 2016 Copyrght held by the owner/author(s). Publcaton rghts lcensed to ACM. SIGGRAPH Asa 2016, December 5-8, 2016, Macau ISBN: /16/12 DOI: possess smple closed forms and produce natural-lookng nterpolatons n arbtrary convex domans due to beng ratonal polynomals of mnmal degree [Warren 2003]. Mean value coordnates [Floater 2003; Floater et al. 2005] are also very useful as they behave ncely even on non-convex polytopes [Hormann and Floater 2006], makng them partcularly convenent for cage-based shape deformaton [Ju et al. 2005b]. These two most-common coordnates have recently been extended to sphercal domans [Langer et al. 2006], to smooth shapes [Schaefer et al. 2007], and even to the Hermte settng [Dyken and Floater 2009]. Yet, the exstence and constructon of other generalzed coordnates on convex polytopes reman relevant n geometrc modellng [Varady et al. 2016] or even n fnte element computatons [Gllette et al. 2016], where convex elements are stll strongly preferred to ensure dscrete maxmum propertes. The goal of ths paper s to propose a prncpled geometrc approach to characterze, desgn and compute coordnates on convex polytopes that provably satsfy the key propertes expected from barycentrc coordnates. 1.1 Problem statement Ths paper focuses manly on polytopal barycentrc coordnates n R d (d = 2, 3). Let P be a non-degenerate convex polytope wth n vertces V = {v 1,...,v n}. We further assume that these vertces are all extreme ponts of the polytope, that s, no vertex s a convex combnaton of the other vertces. Note that ths mples that the dscrete Gaussan curvature of each vertex s strctly postve. Functons {λ : R d R} =1..n are called barycentrc coordnates f they satsfy the followng propertes: Partton of unty: =1..n Lnear precson: =1..n λ (x) = 1 x P (a) λ (x) v = x x P (b) Non-negatvty nsde the convex polytope P: [1..n], λ (x) 0 x P (c) Smoothness away from vertces: [1..n], λ ( ) C k( P \ j{v j} ). (d) In prevous work, addtonal propertes are often requred of barycentrc coordnates as well, ncludng: Exact nterpolaton of nodal data (Lagrange property): Restrcton on facets of P:, j [1..n], λ (v j) = δ j (e) For a facet F = Conv ( v 1,..., v m ) P, j / { 1,..., m}, λ j(x) = 0 x F. (f) However, the Lagrange property (e) and the property (f) of reducton to lnearly-precse coordnates on the boundary faces of P are

2 mere consequences of the requrements (a) (d) for convex polytopes (see App. A). Generalzed barycentrc coordnates satsfyng all these requrements are partcularly desrable n a wde varety of applcatons: they enforce exact nterpolaton at vertces, offer a dscrete maxmum prncple for nterpolants, guarantee convergence of the Galerkn method for second-order PDEs, and so on [Floater 2015]. Despte ther apparent smplcty, enforcng all these condtons s remarkably dffcult as we now brefly revew. 1.2 Prevous work Whle a thorough survey on generalzed coordnates can be found n [Floater 2015], we brefly revew prevous approaches that have been proposed over the years to better poston our contrbutons. Early developments for convex polytopes. Motvated by the desgn of bass functons over convex polygons, Wachspress [1975] proposed a general constructon of ratonal polynomal bass functons usng projectve geometry. Ths generalzaton of barycentrc coordnates was further analyzed and extended to hgher dmensons by Warren [Warren 2003], and ts use n graphcs (along wth smple 2D and 3D evaluatons) was advocated n [Meyer et al. 2002; Warren et al. 2006]. Weghted averages of smplcal barycentrc coordnates were also ntroduced by Floater [1997] for parameterzaton purposes, but these coordnates were only C 0. The cotangent weghts used for the Laplacan operator [Meyer et al. 2002] (often referred to as dscrete harmonc weghts [Eck et al. 1995]) were also noted as valuable, but fal to satsfy postvty (c). In the context of fnte elements, the work of Malsch and Dasgupta [2004; 2005] extended Wachspress approach to weakly convex polygons and convex polygons wth nteror nodes. Extensons to arbtrary polytopes. Recognzng that classcal forms of coordnates often lose smoothness or are not well-defned when appled to a non-convex polytope, Floater ntroduced Mean Value Coordnates (MVC [Floater 2003]). These coordnates stll have a smple expresson that s easly evaluated, and whle they fal to offer postvty on concave polytopes, they reman well defned everywhere n the plane gven arbtrary polygonal domans [Hormann and Floater 2006]. Floater et al. [2006] showed that whle the three most common exstng generalzed coordnates are part of a larger famly, only two of them, namely MVC and Wachspress, satsfy requrements (c) (f) for convex polytopes. A number of approaches have also explored the desgn of coordnates on nonconvex polytopes by relaxng some of the barycentrc requrements. When coordnates are used for deformaton for nstance, removng the Lagrange property and extendng the coordnates to be complex valued (or, equvalently, matrx valued) allow for a mnmzaton of angle dstorton [Weber et al. 2009; Weber and Gotsman 2010; Lpman et al. 2008]. Mean value coordnates can also be seen as a case where postvty s no longer requred for non-convex polytopes, a goal targeted n [Manson and Schaefer 2010] as well. Instead, Postve Gordon-Wxom coordnates [Manson et al. 2011] do acheve postvty for arbtrary polygons, but are only C 0. Non-analytcal coordnates. Several coordnates have been ntroduced lately to acheve stronger propertes. Harmonc coordnates [Josh et al. 2007], for nstance, proposed the use of harmonc weghts (satsfyng λ =0 nsde the doman) to enforce postvty for arbtrary polytopes. Postve Mean Value coordnates [Lpman et al. 2007] also stay, by constructon, postve on non-convex polytopes. Barycentrc coordnates dervng from numercal optmzaton have also been proposed, such as the nterpolatory maxmum entropy coordnates [Hormann and Sukumar 2008]. However, these specfc methods nvolve nexact evaluaton as they requre quadratures, optmzaton, or large memory footprnt to store PDE solutons. Ther analyss s often more dffcult as well. Further extensons. The dea of generalzed coordnates for polytopes has been extended to smooth closed domans, see [Warren et al. 2006; Belyaev 2006; Schaefer et al. 2007]. Multple methods also extended mean-value coordnates to the Hermte settng [Manson and Schaefer 2010; Weber et al. 2012; L et al. 2013] when both values and gradents (.e., normal dervatves) on the boundary need to be nterpolated. It was also recognzed n [Zhang et al. 2014] that localzed nfluence of a gven vertex of the polytope on the nterpolaton may be benefcal n some applcatons, brngng forward the dea of local coordnates. Related works. Fnally, there are a few applcatons that share commonaltes wth our work due to the generalty of the concept of barycentrc coordnates. For nstance, methods for scattered data nterpolaton such as Shepard [Shepard 1968] and natural neghbors [Sbson 1981] bear sgnfcant smlartes wth barycentrc coordnates: e.g., the Laplace nterpolant varant of natural neghbors s lnked to the dscrete harmonc barycentrc coordnates [Hyosh and Sughara 1999; Cueto et al. 2003]. The fnte element lterature s also brmmng wth dscussons of varous propertes of barycentrc coordnates on polytopes [Mlbradt and Pck 2008]. 1.3 Contrbutons Despte sgnfcant advances n extendng barycentrc coordnates, only a few coordnates verfyng condtons (a) (f) on convex polytopes are known. In ths paper, we explot an equvalence between generalzed coordnates and orthogonal dual dagrams to formulate a full geometrc parameterzaton of barycentrc coordnates va power dagrams. We also show how our formulaton nduces a smple, exact and effcent evaluaton of coordnates that leverages robust computatonal geometry tools. Furthermore, we present how to desgn customzed coordnates through smple weght functons per vertex of the polytope. Fnally, we dscuss how to extend our approach to non-convex polytopes and arbtrary shapes. 2 Power Coordnates for Convex Polytopes We begn our exposton by establshng a relatonshp between generalzed barycentrc coordnates and power dagrams, provdng a geometrc (as opposed to analytcal [Floater et al. 2006]) characterzaton of arbtrary coordnates on convex polytopes. 2.1 Background Gven a set of stes {x k } k=1..n n R d and assocated scalar weghts {w k } k=1..n, the power dagram of these weghted ponts s a tessellaton n whch each convex cell D s defned as: D = {x R d x x 2 w x x k 2 w k k}, where ndcates the Eucldean norm. Note that when weghts are all equal, power dagrams reduce to Vorono dagrams. Propertes. Cells are guaranteed to be convex and have straght edges. Note that some cells may be empty as well. Two stes x and x j are sad to be neghbors f the ntersecton D Dj s a non-empty power facet f j of codmenson 1, correspondng to a planar polygon n 3D (see nset) and a lne segment n 2D. We use l j

3 to ndcate the dstance x x j, and denote d j the sgned dstance from x to f j so that d j+d j =l j. The expresson of the dstance from a ste to a power facet s actually known analytcally: d j = x xj 2 + (w w j). (2) 2 x x j Fnally, f j denotes the volume of f j (area n 3D, and length n 2D). Akn to the famous Vorono-Delaunay dualty, a power dagram also defnes by dualty a trangulaton of the stes, known as the regular (or weghted Delaunay) trangulaton, n whch each neghborng par of stes forms an edge that s orthogonal to ts assocated power facet. In fact, all possble orthogonal prmal-dual pars correspondng to ponts {x k } k=1..n are completely parameterzed by weghts {w k } k=1..n ; see, e.g., [Memar et al. 2012]. Furthermore, the weghted crcumcenters of the tetrahedra of ths regular trangulaton n 3D (resp., trangles n 2D, see Fg. 1(left)) correspond to dual vertces of the power dagram,.e., they are at the ntersecton of four power cells n 3D (resp., three n 2D). The closed-form expresson of the weghted crcumcenter c w of a p- dmensonal smplex σ p of T (p=1... d) s: c w 1 ( (σ p) = x + x x j 2 ) + w w j N ĵ σp, (3) 2 p σ p x j σ p where Nĵσ p s the nward-pontng normal of the face of σ p opposte to x j, weghted by the volume of that face. Gauge and translaton. As Eqs. (2) and (3) ndcate, addng a constant shft to all the weghts does not nfluence the dagram, so the weghts are defned up to a constant. Moreover, addng a lnear functon t x to each weght w wll only nduce a global translaton t of the power dagram [Memar et al. 2012]. Thus, the geometry of the power dagram has a (d+1) parameter gauge. vectors n (x) and the correspondng face areas are h (x) v x. By constructon, such a polytope s orthogonally dual to the trangulaton T of ponts {x} {v } =1..n wth x and v connected ff h (x) > 0. Ths argument was already made n [Langer et al. 2006], but was not exploted to parameterze extended coordnates through power dagrams. We also note the smlarty of ths argument wth the Maxwell-Cremona equvalence of self-supportng structures wth 2D power dagrams used n [de Goes et al. 2013]. Conversely, gven any convex polytope D(x) such that each facet f (wth normal n and area f ) s orthogonal to (v x), we have: 0 = n(s)ds = n f = f v (v x). x D(x) =1..n =1..n Therefore, based on Eq. (4), the n functons h (x)= f / v x (6) defne homogeneous, non-negatve coordnates. Ther sum s also guaranteed not to vansh provded that D(x) s not degenerate (.e., not reduced to a pont). The geometry of D(x) then defnes generalzed barycentrc coordnates. Geometrc equvalence. We conclude that there s a one-to-one correspondence between non-negatve homogeneous coordnates at any nteror pont x P and orthogonal dual cells. Ths mples that we can parameterze the set of all generalzed barycentrc coordnates satsfyng condtons (a) (c) over a convex polytope P by the power weghts as functons of x. One can see ths parameterzaton as a geometrc counterpart to the analytcal parameterzaton of [Floater et al. 2006] (we wll make ths lnk more formal n Sec. 3.4), whch now generalzes to arbtrary dmensons. 2.2 From coordnates to power dagrams Power dagrams have found a varety of applcatons n geometry processng [Mullen et al. 2011; de Goes et al. 2012; de Goes et al. 2013; Lu et al. 2013; Budnnsky et al. 2016] and anmaton [Busaryev et al. 2012; de Goes et al. 2015]. As we now revew, they are also ntmately lnked to barycentrc coordnates. Even f our exposton focuses on 2D and 3D, the results of ths secton are general and hold n arbtrary dmenson. Homogeneous coordnates. Gven a convex polytope P wth vertces {v } =1..n, the homogeneous coordnates of P [Floater 2015] are a set of functons {h : R d R} =1..n that satsfy the followng property: h (x)(v x) = 0 x P, [1..n]. (4) =1..n Provded that all h (x) are C k, non-negatve, and wth a nonvanshng sum nsde P, barycentrc coordnates satsfyng condtons (a) (d) can be constructed va drect normalzaton: λ (x) = h (x)/ h k (x). (5) k=1..n Orthogonal duals. Now let x be an arbtrary pont nsde P. Defne a set of vectors n (x) = ( v x ) / v x. For a set of homogeneous coordnates, these vectors satsfy, by constructon, Eq. (4) h (x) v x n (x) = 0. =1..n Ths relaton drectly mples, through Mnkowsk s theorem [Klan 2004], the exstence of a unque (up to translaton) convex polytope D(x) n R d, of whch the face normals are gven by the unt Fgure 1: Power coordnates. From a set of weghts on the vertces v of a polygon (left) or polytope (rght) P, the power cell of a pont x (wth weght 0) offers a smple geometrc nterpretaton of contnuous, non-negatve lnearly-precse barycentrc coordnates. 2.3 From power dagrams to coordnates Our geometrc characterzaton of generalzed barycentrc coordnates suggests a smple approach to both formulate and evaluate barycentrc coordnates, whch turns out to be lnked to the noton of dscrete Hodge stars on meshes. Power coordnates evaluaton. We can now explctly descrbe how a convex polytope P and a gven choce of weghts w (x) can be used to compute exactly the correspondng coordnates. For a pont x nsde P, we proceed as follows to evaluate the power barycentrc coordnates λ (x): 1. From the n vertces v 1,..., v n of polytope P, create (n+1) weghted ponts: (v 1, w 1(x)),..., (v n, w n(x)), to whch we add (x, 0),.e., the evaluaton pont wth a weght of zero. 2. Compute the power dagram of these (n+1) weghted ponts, whch generates a power cell D assocated to pont x. 3. Compute the homogenous coordnates h as follows: f the edge from x to v s present n the regular trangulaton dual

4 to the power dagram, set: h = f / x v, where f s the facet of D dual to the edge; otherwse, set h =0. 4. Normalze the homogeneous coordnates va Eq. (5) to obtan the power coordnates λ at x. Note that we assgned a weght of 0 to the evaluaton pont x for smplcty: weghts beng defned up to a constant (see Sec. 2.1), ths choce has no bearng on the generalty of our approach. Implementaton. The power dagram constructon and the areas and lengths nvolved n the evaluaton of the coordnates are trvally handled va the use of a computatonal geometry lbrary such as CGAL [CGAL 2016]: one smply needs to use the class Regular trangulaton 3, and nsert possbly n parallel for effcency the n+1 weghted ponts (see Supplemental Materal for code), before readng off the geometrc quanttes from the resultng power dagram. For an n-vertex polytope, constructng the power dagram costs O(n log n) n 2D, and O(n d/2 ) n dd for d > 2. Power coordnates are thus very easy and effcent to compute for the typcally small values of n used n practce. Fnally, ths constructon stll holds on facets of P, where the dual polytope D s now unbounded n the normal drecton of the facet but all prevous expressons reman vald n ths lmt case. Ths mples that evaluatons of barycentrc coordnates near the boundary are very stable f a robust lbrary s used to compute power dagrams. Otherwse, a smple dstance thresholdng approach can be mplemented to avod numercal degeneraces. Coordnates as Hodge stars. Suppose we pck a pont x P, for whch we construct the dual orthogonal cell D(x) correspondng to a gven choce of weghts (we wll provde concrete examples of weghts shortly). Note that the homogeneous coordnates and thus, the barycentrc coordnates up to normalzaton are expressed as a rato of volume of dual facets of the power cell D(x) and volume of ther assocated prmal edges, see Eq. (6). In Dscrete Exteror Calculus (DEC [Desbrun et al. 2008]), ths expresson corresponds to the edge-based Hodge star 1 for 1-forms, as 1 = dag(h ). Ths connecton becomes obvous f one notces that Eq. (4) can be reexpressed n the DEC formalsm through: d t 0 1 d 0 X = 0, where X s seen as a 0-form denotng the coordnates of the vertces and the evaluaton pont, and d 0 s the exteror dervatve correspondng to a sgned adjacency matrx. Lnear precson of coordnates thus amounts to what was referred n [Glckensten 2005; de Goes et al. 2014b] as the lnear precson of the generalzed Laplacan operator when usng weghted dagonal Hodge stars. 2.4 Exstence condton Due to gauge nvarance, we saw n Sec. 2.3 that we can always fx the weght for the evaluaton pont x to 0 wthout loss of generalty. However, arbtrary assgnment of other weghts may not always lead to useful coordnates: gven an arbtrary set of weghted ponts, the power cell assocated wth the evaluaton locaton x may become empty. Weghts must thus be chosen such that the power cell D(x) correspondng to the weghted pont (x, 0) s non-degenerate for any x P. Thankfully, ths s a smple condton to enforce. Theorem. For a pont x, the power cell D(x) s not degenerate f and only f there s a vector t R d, such that v x 2 w (x) + t (v x) 0 =1... n, (7) wth the nequalty beng strct for at least one of them. Proof: By addng a lnear functon to the weght (see Sec. 2.1), we can always translate the dual polytope D(x) by a vector t(x) n order for pont x to be nsde D(x). Then the cell s not degenerate (.e., not empty) f and only f the sgned dstances from x to the faces of ths shfted dual along the correspondng prmal edges are all non-negatve, wth at least one of them beng strctly postve: ths results from Eq. (2), whch s now rewrtten usng the dstance d between x and the power cell facet dual to edge v x through d = v x 2 w (x). (8) 2 v x As a corollary of Eq. (7), a trval suffcent condton for nondegeneracy of the power cell s thus w (x) v x 2. (9) Note that ths happens to be the same condton as n [de Goes et al. 2014b] n the very dfferent context of generalzng orthogonal dual cells on non-flat trangle meshes through weghts. 2.5 Propertes If the non-degeneracy condton s enforced, power coordnates have or can easly be made to have valuable propertes beyond lnear precson: Non-negatvty. Snce the homogeneous power coordnates are non-negatve by constructon, the condton of non-degeneracy dscussed above also enforces that the denomnator of Eq. (5) can never be zero. As a result, power barycentrc coordnates are well-defned and non-negatve anywhere n P. Localty. The cells of some vertces of P may not share a dual facet wth the cell of x, thus resultng n zero coordnates. Our constructon can thus seamlessly offer local coordnates as proposed n [Zhang et al. 2014]. Indeed, dependng on the choce of weght functons, the evaluaton pont x can be far enough away from a gven vertex v that they no longer share a dual facet n the power dagram. The vertex v then loses ts nfluence on x snce the correspondng λ vanshes. Smlarty nvarance. Invarance of coordnates to rotaton, translaton, and scalng of the polytope P s often desrable. Here agan, we can easly accommodate these requrements. Power barycentrc coordnates are rotaton and translaton nvarant f the weghts themselves are an example beng weghts that are functons of geometrc measures lke dstances or volumes. They wll also be nvarant under scalng of P by a factor of s ff the pontwse duals D(x) get unformly scaled as well. Ths can be easly acheved by makng sure that the nput weghts are of SI unt [m 2 ] through normalzaton by the proper power of the volume of the polytope (or any other geometrc measure). We wll provde entre famles of such coordnates n Sec. 4. Contnuty. As can be seen from Eq. (3), a power cell D(x) (and ts face areas n partcular) are smoothly dependent on the weghts. Therefore, by choosng contnuous weght functons w (x) we automatcally nhert contnuty for the correspondng power coordnates. 2.6 Smoothness Smoothness of our coordnates deserves a closer nspecton as t s the least trval property to obtan n our framework. We frst defne a reference connectvty whch wll allow us to express a suffcent condton of smoothness and a systematc approach to render C the power coordnates derved from an arbtrary set of weghts.

5 Fgure 2: Conventons for Eq. (10) and Eq. (13), respectvely. Full-connectvty homogenous coordnates. Untl now, we always derved the connectvty of the power dagram (and of ts dual trangulaton) based on the weghts at vertces of P and on the poston of the evaluaton pont x. However, t s common practce to defne an mposed connectvty for the trangulaton, and express a weghted crcumcentrc dual W(x) for whch weghts are used to derve a weghted crcumcenter c w σ per smplex σ of ths trangulaton that are then connected nto the dual W based on the adjacency between smplces: ths s what was proposed n [de Goes et al. 2014b] for nstance, for a gven trangle mesh wth a weght per vertex. The weghted crcumcentrc dual W(x) s not necessarly embedded: flps of edges and faces are possble (see nset: red face f s flpped; for these weghts, the power cell D would be the black convex cell, but the weghted crcumcentrc dual W s, nstead, the polygon ncludng the fsh tal shape). In our context, we defne a full-connectvty trangulaton of P x to be the trangulaton formed by P (after ts facets are trangulated) and all the edges lnkng x to the vertces v of P. Ths trangulaton, along wth the weghts w (x) (and 0 for x), nduces a Hodge star H w (x) per edge xv, for whch an explct formulaton n 2D reads [Mullen et al. 2011] H w (x) = 1 (cot α,+1,x + cot αx, 1,) (10) 2 cot α+1,x, cot αx,,+1 + (w(x) w+1(x)) 2 x v 2 2 x v w+1(x) 2 + cot α,x, 1 cot α 1,,x (w(x) w 1(x)) 2 x v 2 2 x v w 1(x), 2 where α,x, 1 s the angle at x n the trangle (x, v 1, v ) as descrbed n Fg. 2(left). The 3D verson H w ( ) can be smlarly assembled from the weghted crcumcenters computed from Eq. (3). These Hodge star values H w stll form homogeneous coordnates n the sense of Eq. (4), but they are not necessarly satsfyng postvty. We refer to these star values as the full-connectvty homogeneous coordnates. Smoothness condton. Now that we have ntroduced the Hodge star edge values computed from the trangulaton wth full connectvty, we can express a suffcent condton for C of our power coordnates λ (x). Lemma. The power coordnates derved from weghts w (x) are C f the weght functons w (x) are also C, and f H w (x) 0 =1..n. (11) Proof: Postvty of the full-connectvty homogeneous coordnates H w for all x readly mples that the edge xv s part of the regular trangulaton nduced by the power dagram of the weghted vertces (v,w ) of P and (x, 0) [Glckensten 2005] Now, f every w ( ) s C and snce the regular trangulaton dual to the Fgure 3: Increasng smoothness. From an orgnal C 0 coordnates (top) wth dscontnuous dervatves (nset: y-component of gradent n orange hues), we can turn them nto C coordnates by computng the weght shft M descrbed n Sec. 2.6, or usng an even larger shft (bottom), whch smooths the bass functons. power dagram keeps the same connectvty for all x, the power coordnates are C because the sgned dual facet volumes are C functons of x: only a change of connectvty n the power dagram could nduce a loss of smoothness. Smoothness correcton. Whle the condton above s only suffcent, we can derve from t a procedure to render any set of weghts C : we can always shft them by a constant to meet the condton H w (x) 0. Indeed, for a gven d-dmensonal polytope P and a set of weghts, defne: { } H w (x) x v d M = max 0, nf, x, s (x) where s (x) equals the volume of the (d 1) dmensonal face assocated wth v of the Wachspress dual of P tmes x v d 1. M s fnte because the untless denomnator s s bounded away from zero: t s the volume (length when d =2, area when d =3) of the stereographc projecton along xv of the sphercal (d 1) polygon formed by the unt normals of the faces ncdent to v whch has a mnmum of κ, the Gaussan curvature at vertex, assumed to be strctly postve (see Sec. 1.1). In practce, we can fnd an evaluaton of M by ether samplng the nteror of P n a preprocessng phase, or usng a less tght but safe upper bound, such as nf (H w /κ ) dam(p) d, where dam s the dameter operator,.e., the maxmum dstance between pars of vertces. Then, assumng smlarty nvarance s enforced (see Sec. 2.5), the SI unt of M s [m 2d 2 ]; consequently, the use of altered weghts w defned as: w (x)=w (x) M 1/(d 1), s guaranteed to lead to smooth coordnates as ther correspondng full-connectvty homogeneous coordnates H w are non-negatve. Note that the weght at x s kept fxed at 0, and M depends on

6 the polygon shape; but the resultng λ s smlarty nvarant. Ths shft by M can be understood as a blend of the (non-smooth) power coordnates wth the always-smooth Wachspress coordnates. Whle hghly applcaton-dependent, the desgn of smooth, postve, and lnear-precse coordnates s thus rather smple, as t only requres a few condtons on weghts to guarantee relevance of the resultng power coordnates. We provde a number of examples n Sec. 4, after dscussng the geometrc nterpretaton of exstng barycentrc coordnates n our framework. 3 Exstng Power Coordnates Before formulatng new weghts, we revew exstng coordnates for convex polytopes to show that they correspond n our power coordnates framework to very smple choces of weghts, provdng straghtforward extensons to arbtrary dmensons. 3.1 Wachspress coordnates Ju et al. [2005a] notced that Wachspress coordnates can be expressed n terms of the polar dual of the nput polytope P,.e., usng a dual cell D(x) for whch the dstance from x to a dual face f s d = 1/ v x. Ths s, n fact, a partcular case of our generalzed noton of dual, and usng Eq. (2) we drectly conclude that Wachspress coordnates are expressed n arbtrary dmensons as: w Wach (x) = x v 2 2. Note that the constant 2 can be changed to any strctly postve constant: normalzaton (Eq. (5)) wll lead to the same coordnates. The ntal dervaton of mean value coordnates by Floater [2003] reles on the local ntegraton of the normal of a unt crcle centered at x. Whle ths seems unrelated to the polytopal dual cell that our constructon reles on, one realzes that n fact, creatng a cell D(x) such that each dstance d from x to a facet f of D(x) s unt (or, more generally, constant) leads to the exact same normal ntegral as n the orgnal dervaton. Ths partcular dual (see nset) s thus crcumscrbng the unt crcle, and usng Eq. (2) once agan, we fnd that the weghts needed to exactly reproduce mean value coordnates are: w MVC (x) = v x 2 2 v x. (12) Agan, the factor 2 can be replaced by any strctly postve constant wthout affectng the normalzed coordnates. Somehow surprsngly, ths orthogonal dual constructon wth unt dstances d leads n 3D to a very dfferent set of coordnates: whle the exstng 3D extenson of MVC [Ju et al. 2005b; Floater et al. 2005] reles on barycentrc coordnates over sphercal trangles, our homogeneous coordnates are stll ratos of volumes of a straght-edge dual cell that crcumscrbes the unt sphere. As a consequence, the power varant happens to be sgnfcantly less dstorted as demonstrated n Fgs. 5 and 6. Moreover, our constructon generalzes to hgher dmensons n a straghtforward manner snce the expresson of w MVC above stll holds as s. Fgure 4: Dscrete harmonc coordnates. Whle the orgnal dscrete harmonc coordnates (left) can go negatve (grey colors) n convex polygons, our power varant (rght) reman postve (ranbow colors) everywhere by constructon, but may only be C Power dscrete harmonc Dscrete Harmonc coordnates [Meyer et al. 2002] are also trvally related to a noton of dual, as they are the coordnates correspondng to the canoncal, crcumcenter-based Hodge star. Thus, they correspond to the case where D(x) s the classcal Vorono cell of x, hence to zero weghts. In 3D, these null weghts correspond to the coordnates dubbed Vorono n [Ju et al. 2007]. However, our power dagram constructon guarantees non-negatvty, they are thus more drectly related to the Laplace coordnates defned n [Hyosh and Sughara 1999]. The power form of dscrete harmonc coordnates are n general not smooth, only contnuous, see Fg. 4. However, note that ths varant s qute dfferent from a smple thresholdng of the dscrete harmonc homogeneous coordnates: nstead, our power coordnates are lnear precse, a property that smply dscardng negatve homogeneous coordnates (.e., thresholdng them to zero) would not enforce. 3.3 Mean value coordnates: old and new Fgure 5: Orgnal vs. Power Mean Value Coordnates. Even on a smple parallelepped (left), the orgnal MVC [Ju et al. 2005b] (center) for the bottom-rght corner marked wth a red dot exhbts a more asymmetrc behavor than our power varant (rght) pont and 5-pont coordnates n 2D Floater et al. [Floater et al. 2006] ntroduced a general parameterzaton of barycentrc coordnates by smooth functons c through the followng homogenous expressons: c+1a 1 cb + c 1A h =, (13) 4A A 1 where A and B are the sgned areas of trangles (x, v, v +1) and (x, v 1, v +1) respectvely, see Fg. 2(rght). Assume that for a gven x, the regular trangulaton dervng from our power coordnates nclude the edges xv ; then startng from the expresson of the canoncal weghted Hodge star gven n Eq. (10), basc trgonometry leads to weghts of the form: w (x) = x v 2 c (x), (14) provdng a smple geometrc nterpretaton of c whch extends to arbtrary dmensons. Note that ths s not an exact equvalence: our correspondng power coordnates are guaranteed to be postve, whle the orgnal parameterzaton of [Floater et al. 2006] does not: as n the dscrete harmonc case, our parameterzaton s not just a thresholdng of the orgnal homogeneous coordnates snce t always enforces lnear precson, whch nave thresholdng loses. Three-pont coordnates. Based on the relatonshp between ther parameterzaton and ours, the explct weght expressons of

7 4.2 Mean-Wachspress coordnates A 2-parameter famly of weghts that blends Wachspress and mean value coordnates can be trvally formulated as well: for any strctly postve values a and b, weght functons defned as: w MW (x) = v x 2 a b x v / P 1/d, (15) Fgure 6: 3D mean values. For a 3D polytope wth 7 faces (left), the mean value coordnates defned n [Ju et al. 2005b] (top) are sgnfcantly more dstorted than our 3D mean value coordnates (bottom) for the pont ndcated by the red sphere. Three slces are shown wth the order ndcated: one horzontal, and two vertcals. wll nduce power coordnates, where the volume P of P s rased to the nverse of the dmenson to enforce smlarty nvarance. These weghts always produce C coordnates: snce they are formed as a combnaton of Wachspress and mean-value homogeneous coordnates that are strctly postve nsde P: thus, the suffcent condton of smoothness dscussed n Sec. 2.6 s guaranteed to hold. Fg. 7 llustrates a few of these coordnates n 2D. ther three-pont famly are smply, for µ [0, 1]: w µ,1 (x) = x v 2 x v µ. Only two cases n ths famly (µ = 0 and µ = 1) were producng non-negatve homogenous coordnates n ther formulaton; ours enforce non-negatvty for all µ nstead, at the cost of smoothness: for µ ]0,1[, the resultng coordnates are only C 0. Fve-pont coordnates. Smlarly, the expressons for the 5-pont Wachspress famly and the two types of 5-pont mean value famly are (for µ [0, 1] and where angles are denoted usng the conventons depcted n Fg. 2): w µ,4 w µ,3 w µ,2 (x) = x v 2 A 1 + A B (1 + µ ), A 1 + A (x) = x v 2 (1 + µ)(sn α,x, 1 + sn α +1,x,) x v sn α,x, 1 + sn α +1,x, + µ sn(α, +1,x, 1) cos(µ(α,x, 1 α+1,x,/4)) (x) = x v 2 x v cos(µ(α,x, 1 + α. +1,x,/4)) 4 Other Examples of Power Coordnates We provde next a seres of new coordnates wth varyng degrees of smoothness, to llustrate the generalty of power coordnates. 4.1 Smple power coordnates One can ntroduce smple coordnates by smply choosng weghts functons w that are negatve (to trvally enforce the suffcent condton n Eq. (9)) and translaton nvarant homogeneous functons of x {v } =1..n of degree 2 (to enforce smlarty nvarance): they wll drectly enforce all the condtons (a) (f), although wth C 0 contnuty n general. One can also use weghts of the form w (x)= v x 2 ŵ (x) for any ŵ s that are non-negatve translaton nvarant homogeneous functons of x {v } =1..n of arbtrary degree. Scale nvarance s easy to enforce through the proper choce of exponent or through normalzaton by the volume, surface area, or mean-wdth of the polytope P, as dscussed n Sec Moreover, weghts that are functons of edge dstances such as the ones used n [Malsch and Dasgupta 2004] and ther trval extenson usng facet dstances can also be employed, offerng a drect generalzaton of ther results to 3D. Furthermore, should such power coordnates need to be C, the procedure descrbed n Sec. 2.6 can be drectly used to ensure smoothness on any gven polytope P. Fgure 7: Mean-Wachspress. Usng the unfed expresson n Eq. (15), we recover mean value coordnates (left), Wachspress coordnates (rght), or a blend between the two (mddle). 4.3 Ansotropc power coordnates Whle all the choces of weghts presented so far used Eucldean geometrc measures such as dstances or areas, they can be easly extended to ncorporate ansotropy through a norm defned by v 2 g = v T Gv for a constant, symmetrc postve defnte matrx G. Ths ansotropc norm allows us to talor our barycentrc coordnates qute drectly by substtutng the Eucldean norm by g n prevous formulas. In partcular, weghts of the form w (x) = v x 2 ŵ (x) nduce ansotropc barycentrc coordnates whle stayng smlarty nvarant f the dstances used n ŵ (x) s are modfed to be ansotropc. Note that ths treatment s remnscent of the creaton of ansotropc Hodge stars (.e., [de Goes et al. 2014a]) due to the nterpretaton of coordnates as Hodge star values mentoned n Sec Ansotropc mean value coordnates. The ansotropc varant of mean value coordnates s partcularly nterestng, as t has a nce geometrc nterpretaton. Indeed, we defne ts weghts as: w AMVC (x) = v x 2 2 v x g. (16) (Notce that unlke Eq. (15), there s no need to dvde by the volume of P: smlarty nvarance s automatcally enforced by the normalzaton step.) Based on Eq. (2), the dstance from the evaluaton pont x to the facet of D(x) that s dual to edge xv s: d = v x g v x We clam that ths partcular set of dstances d can be geometrcally constructed as follows. For a gven evaluaton pont x, construct a secondary polytope Z wth vertces z that are at the ntersecton of the rays xv and the ellpsod defned by the set of ponts y such that (y x) t G(y x) = 1. The new polytope s trvally convex, and the new vertces z are expressed as z x = v x. v x g

8 4.4 Iterated power coordnates Fgure 8: Ansotropc 2D mean value coordnates. Computng the mean value weghts usng a constant metrc G = dag(15, 1) (top) or G = dag(1, 15) (bottom) generates ansotropc varants of the orgnal mean value coordnates, stll satsfyng the same propertes but wth deformed shape functons (see Fg. 7(rght)). Fnally, we note that one can even defne power coordnates mplctly through repeated teratons. For example, for a gven evaluaton pont x and a convex polytope P, one can compute a seres of weghts w (k) (startng wth w (0) w (k+1) = w Wach for nstance) usng: (x) = v x 2 v x λ(k) (x), (18) where λ (k) (x) denotes the power barycentrc coordnates derved from weghts w (k). Ths procedure converged (n only 3 to 5 teratons) to the same barycentrc coordnates for all w (0) we chose. The resultng barycentrc coordnates (see Fgs. 10 and 12(rght)) are vsually and numercally smooth. Whle we do not know how to analyze or explctly formulate the resultng coordnates, note that they come from a repeated barycentrc averagng of weghts, whch may allow us to algorthmcally defne weghts n such a way that the resultng coordnates satsfy possbly-ntrcate mplct relatons maybe even dfferental relatons. Now consder the polar dual of Z w.r.t. a unt sphere centered at x and denote the vertces of that polar dual by p. One realzes that p x = v x g (v x), v x 2 and thus p x = d. We conclude two thngs from ths observaton: frst, the dual cell D(x) s always n-faceted snce t corresponds to Wachspress homogeneous coordnates for the polytope Z; second, snce Z depends smoothly on x, these ansotropc mean value coordnates are always C, see Fg. 8. The SPD matrx G defnng the norm. g can also be spatally varyng,.e., a dfferent matrx for the evaluaton of w AMVC at a dfferent x: the constructon and resultng propertes are unchanged, offerng even more degrees of freedom to derve addtonal coordnates, see Fg. 12 (left). Ansotropc mean-wachspress coordnates. By the same reasonng, and because Wachspress coordnates are such that w Wach x v 2 s constant, the ansotropc varant of mean-wachspress coordnates are also C, and expressed wth weghts w AMW (x) = v x 2 a b v x g/ P 1/d. (17) Fgure 9: Ansotropc 3D mean value coordnates. Usng the same polytope and cuts as n Fg. 6, our 3D mean value coordnates for G = dag(10, 1, 1) (top) and G = dag(1, 10, 1) (bottom). Fgure 10: Iterated power coordnates. From arbtrary C 0 coordnates (top), 3 teratons of Eq. (18) result n smooth coordnates. 5 Extensons to Non-Convex Polytopes The lnk between homogeneous coordnates and orthogonal duals partally extends to the case of non-convex polygons and polytopes, n partcular f we remove the postvty constrant (c). 5.1 Sgned coordnates on non-convex 2D polygons In R 2, there s stll a 1-to-1 correspondence between homogeneous coordnates and weghted crcumcentrc dual cells W(x) descrbed n Sec. 2.6, created as the dual of x for the smplcal d-manfold wth a full connectvty lnkng x to the vertces of P. Indeed, for any set of functons h (x) satsfyng Eq. (4), the 2D vectors f (x)=h (x)(v x) where denotes a counterclockwse π/2 rotaton, form a closed loop, thus defnng the boundary of an orthogonal dual cell W(x) wth possble self-ntersectons (see nset n Sec. 2.6). The converse s also true: gven any orthogonal dual W(x), h (x) defned through the sgned dagonal Hodge star would satsfy Eq. (4). Thus, through the weghted crcumcentrc dual, all homogeneous coordnates h have a one-to-one correspondence to an assgnment of w up to a constant gauge. Note that the homogeneous coordnates can now be negatve as a consequence, thus volatng condton (c). Replacng the power cell D(x) by the weghted crcumcentrc dual W(x) n our approach recovers the tradtonal 2D mean value coordnates, whch are vald (but often wth negatve coordnates) on arbtrary polygons. 5.2 Sgned coordnates on non-convex 3D polytopes The 3D case s more nvolved for two reasons. Frst, even though there s stll a 1-to-1 correspondence between power cells and

9 non-negatve homogeneous coordnates as mpled by the work of Memar et al. [2012], the resultng coordnates are not guaranteed to be contnuous for contnuous weghts anymore, thus volatng condton (d): the power cell tself can exhbt dscontnuous behavor because the vsblty vectors connectng x to vertces v change ther relatve confguraton as the evaluaton pont moves around a non-convex polytope. Second, f we remove the postvty constrant, the resultng arbtrary homogeneous coordnates are not fully characterzed by our weghted crcumcentrc constructon: whle Mnkowsk s theorem guarantees the exstence of a convex orthogonally dual polytope from a sequence of (possbly flpped) normals, such a polytope s no longer necessarly a weghted crcumcentrc dual. As a consequence, a truly general extenson of power coordnates for non-convex polytopes remans open. Nevertheless, our weghted crcumcentrc approach stll spans vald homogeneous coordnates that are potentally negatve, leadng to sgned barycentrc coordnates f ther sums h are bounded away from zero; e.g., we can recover well-known coordnates such as the Vorono coordnates n 3D mentoned n [Ju et al. 2005a]. 5.3 Postve-power mean value coordnates. If one stll requres true generalzed barycentrc coordnates satsfyng all the condtons (a)-(d), our geometrc characterzaton of coordnates over convex polytopes n partcular, our use of a secondary polytope n Sec. 4.3 suggests a smple way to extend power coordnates to non-convex polytopes: one can construct a secondary polytope by smoothly dsplacng the orgnal vertces dependng on the evaluaton pont. We can ndeed alter the poston of the vertces to make the modfed polytope star-shaped wth respect to the evaluaton pont; usng the mean value weghts gven n Eq. (12), we can stll construct a non-degenerate power cell whch wll be, by defnton, stll crcumscrbed to the unt sphere, but that may nvolve only a subset of the vertces of the non-convex polytopes. For nstance, n order to evaluate postve generalzed barycentrc coordnates for the evaluaton pont x nsde the 7-vertex non-convex polygon dsplayed n the nset, the pnk polygon s used. Ths star-shaped polygon s found by gnorng the obstructed vertex v 5 blocked by the concave vertex v 6, and by dsplacng vertex v 4 to a modfed poston ṽ 4 along the edge v 3v 4. (Note that t corresponds to the vsble porton of the polygon used n the postve MVC coordnates [Lpman et al. 2007].) Ths constructon, once the barycentrc coeffcents are redstrbuted to the dsplaced vertces, results n true coordnates, wthout dscontnuty due to vsblty as demonstrated n Fg. 11. The ensung coordnates are thus both local (n the sense advocated n [Zhang et al. 2014]), and postve [Lpman et al. 2007], but requre no quadrature to evaluate makng them partcularly well suted to cage-based deformaton of meshes. We leave mplementaton detals and analyss of ths constructon n 2D and 3D to a forthcomng paper. 6 Future Work Our characterzaton of coordnates based on power duals opens a number of avenues for future work. We menton a few n leu of a concluson to llustrate both the current lmtatons of power coordnates and the opportuntes that our work offers. Theoretcal developments. Gven the generalty of power coordnates, fndng necessary condtons on the weghts for smoothness, dervng smple and effcent evaluatons of ther dervatves, or provng contracton for terated power coordnates are examples Fgure 11: Postve Power Mean Value Coordnates. Even on nonconvex polytopes, one can use the power mean value coordnates on a secondary star-shaped polytope to force postvty. Here, four bass functons are dsplayed usng a ranbow color ramp. of results that could have very practcal consequences. The use of power coordnates to defne hgher-order Whtney bass functons usng the work of Gllette et al. [2016] s also of great nterest. Harmonc coordnates. Coordnates ssued from partal dfferental equatons have not found wde acceptance due to the memory footprnt requred by the storage of solutons, despte the clear advantage of beng postve on arbtrary polygons. A natural queston s whether the addtonal nsght afforded by power coordnates can result n an explct or terated formulaton (see Sec. 4.4) whch could be evaluated wthout the need for precomputed solutons. Bézer patches. Non-negatve barycentrc coordnates over polytopes have been shown crucal to offer multsded Bézer patches n [Loop and DeRose 1989], and recently n [Varady et al. 2016]. Rewrtng ther constructon n the context of power coordnates may help n provdng fast evaluatons of these general patches. Coordnates on non-convex polytopes. Although we showed how our constructon extends naturally to non-convex polytopes, we have only scratched the surface of what can be accomplshed for ths case. Understandng a general constructon wth the same effcency of evaluaton s an obvous future work. Transfnte nterpolants. Fnally, the noton of power dagram for pontsets extends to contnuous power dagrams of curves, whch can be computed rather effcently [Hoff et al. 1999]. Consequently, our constructon wll converge as a polytope s refned to capture a smooth shape (nset: example of power Vorono dual for 3D bean shape). Ths extenson s smlar to the transfnte form of Sbsons nterpolant [Gross and Farn 1999], and closedform solutons of specfc choces of weght functons are possble and could offer valuable new transfnte nterpolants as well.

10 Acknowledgements We are ndebted to Fernando de Goes and the revewers for ther feedback. Ths work was partally supported through NSF grants CCF , IIS , CMMI and III MD gratefully acknowledges the Inra Internatonal Char program and all the members of the TITANE team for ther support as well. References ARROYO, M., AND ORTIZ, M Local maxmum-entropy approxmaton schemes: a seamless brdge between fnte elements and meshfree methods. Int. J. Num. Meth. Eng. 65, 13, BELYAEV, A On transfnte barycentrc coordnates. In Symp. Geo. Processng, BEN-CHEN, M., WEBER, O., AND GOTSMAN, C Varatonal harmonc maps for space deformaton. ACM Trans. Graph. 28, 3 (July), Art. 34. BRUVOLL, S., AND FLOATER, M. S Transfnte mean value nterpolaton n general dmenson. Journal of Computatonal and Appled Mathematcs 233, 7, BUDNINSKIY, M., LIU, B., DE GOES, F., TONG, Y., ALLIEZ, P., AND DESBRUN, M Optmal Vorono tessellatons wth Hessan-based ansotropy. ACM Trans. Graph. 36, 6. BUSARYEV, O., DEY, T., WANG, H., AND REN, Z Anmatng bubble nteractons n a lqud foam. ACM Trans. Graph. 31, 4, Art. 63. CGAL CGAL 4.8 User and Reference Manual. CGAL Edtoral Board, CUETO, E., SUKUMAR, N., CALVO, B., MARTÍNEZ, M. A., CEGOÑINO, J., AND DOBLARÉ, M Overvew and recent advances n natural neghbour Galerkn methods. Archves of Computatonal Methods n Engneerng 10, 4, DASGUPTA, G., AND WACHSPRESS, E. L Bass functons for concave polygons. Computers & Mathematcs wth Applcatons 56, 2, DE GOES, F., BREEDEN, K., OSTROMOUKHOV, V., AND DES- BRUN, M Blue nose through optmal transport. ACM Trans. Graph. 31, 6, Art DE GOES, F., ALLIEZ, P., OWHADI, H., AND DESBRUN, M On the equlbrum of smplcal masonry structures. ACM Trans. Graph. 32, 4, Art. 93. DE GOES, F., LIU, B., BUDNINSKIY, M., TONG, Y., AND DES- BRUN, M Dscrete 2-tensor felds on trangulatons. Comput. Graph. Forum 33, 5, DE GOES, F., MEMARI, P., MULLEN, P., AND DESBRUN, M Weghted trangulatons for geometry processng. ACM Trans. Graph. 33, 3, Art. 28. DE GOES, F., WALLEZ, C., HUANG, J., PAVLOV, D., AND DES- BRUN, M Power partcles: An ncompressble flud solver based on power dagrams. ACM Trans. Graph. 34, 4, Art. 50. DESBRUN, M., KANSO, E., AND TONG, Y Dscrete dfferental forms for computatonal modelng. In Dscrete dfferental geometry. Brkhäuser Basel, DYKEN, C., AND FLOATER, M. S Transfnte mean value nterpolaton. CAGD 26, ECK, M., DEROSE, T., DUCHAMP, T., HOPPE, H., LOUNSBERY, M., AND STUETZLE, W Multresoluton analyss of arbtrary meshes. In Proceedngs of ACM SIGGRAPH, FLOATER, M. S., KÓS, G., AND REIMERS, M Mean value coordnates n 3D. CAGD 22, 7, FLOATER, M. S., HORMANN, K., AND KÓS, G A general constructon of barycentrc coordnates over convex polygons. Advances n Computatonal Mathematcs 24, 1, FLOATER, M. S Parametrzaton and smooth approxmaton of surface trangulatons. CAGD 14, 3, FLOATER, M. S Mean value coordnates. CAGD 20, 1, FLOATER, M. S Generalzed barycentrc coordnates and applcatons. Acta Numerca 24 (5), GILLETTE, A., RAND, A., AND BAJAJ, C Constructon of scalar and vector fnte element famles on polygonal and polyhedral meshes. Computatonal Methods n Appled Math 19. GLICKENSTEIN, D., Geometrc trangulatons and dscrete Laplacans on manfolds. arxv.org:math/ GROSS, L., AND FARIN, G A transfnte form of Sbson s nterpolant. Dscrete Appled Mathematcs 93, 1, HIYOSHI, H., AND SUGIHARA, K Two generalzatons of an nterpolant based on Vorono dagrams. Internatonal Journal of Shape Modelng 05, 02, HOFF, K. E., KEYSER, J., LIN, M., MANOCHA, D., AND CUL- VER, T Fast computaton of generalzed Vorono dagrams usng graphcs hardware. In Proceedngs of ACM SIG- GRAPH, HORMANN, K., AND FLOATER, M. S Mean value coordnates for arbtrary planar polygons. ACM Trans. Graph. 25, 4, HORMANN, K., AND SUKUMAR, N Maxmum entropy coordnates for arbtrary polytopes. Comp. Graph. Forum 27, 5, JACOBSON, A., BARAN, I., POPOVIĆ, J., AND SORKINE, O Bounded bharmonc weghts for real-tme deformaton. ACM Trans. Graph. 30, 4 (July), Art. 78. JOSHI, P., MEYER, M., DEROSE, T., GREEN, B., AND SANOCKI, T Harmonc coordnates for character artculaton. ACM Trans. Graph. 26, 3, Art. 71. JU, T., SCHAEFER, S., WARREN, J., AND DESBRUN, M A geometrc constructon of coordnates for convex polyhedra usng polar duals. In Symp. Geo. Processng, JU, T., SCHAEFER, S., AND WARREN, J Mean value coordnates for closed trangular meshes. ACM Trans. Graph. 24, 3, JU, T., LIEPA, P., AND WARREN, J A general geometrc constructon of coordnates n a convex smplcal polytope. CAGD 24, 3, KLAIN, D. A The Mnkowsk problem for polytopes. Advances n Mathematcs 185, 2, LANGER, T., AND SEIDEL, H.-P Hgher order barycentrc coordnates. Comp. Graph. Forum 27, 2,

11 LANGER, T., BELYAEV, A., AND SEIDEL, H.-P Sphercal barycentrc coordnates. In Symp. Geo. Processng, LI, X.-Y., JU, T., AND HU, S.-M Cubc mean value coordnates. ACM Trans. Graph. 32, 4 (July), Art LIPMAN, Y., KOPF, J., COHEN-OR, D., AND LEVIN, D GPU-asssted postve mean value coordnates for mesh deformatons. In Symp. Geo. Processng, LIPMAN, Y., LEVIN, D., AND COHEN-OR, D Green coordnates. ACM Trans. Graph. 27, 3, Art. 78. LIU, Y., HAO, P., SNYDER, J., WANG, W., AND GUO, B Computng self-supportng surfaces by regular trangulaton. ACM Trans. Graph. 32, 4. LOOP, C. T., AND DEROSE, T. D A multsded generalzaton of Bézer surfaces. ACM Trans. Graph. 8, 3, MALSCH, E. A., AND DASGUPTA, G Shape functons for polygonal domans wth nteror nodes. Internatonal Journal for Numercal Methods n Engneerng 61, 8, MALSCH, E. A., AND DASGUPTA, G Algebrac constructon of smooth nterpolants on polygonal domans. The Mathematca Journal 9, 3, MALSCH, E. A., LIN, J. J., AND DASGUPTA, G Smooth two-dmensonal nterpolatons: A recpe for all polygons. Journal of Graphcs, GPU, and Game Tools 10, 2, MANSON, J., AND SCHAEFER, S Movng least squares coordnates. Comp. Graph. Forum 29, 5, MANSON, J., LI, K., AND SCHAEFER, S Postve Gordon- Wxom coordnates. CAD 43, 11, MEMARI, P., MULLEN, P., AND DESBRUN, M Parametrzaton of generalzed prmal-dual trangulatons. In Internatonal Meshng Roundtable MEYER, M., BARR, A., LEE, H., AND DESBRUN, M Generalzed barycentrc coordnates on rregular polygons. J. Graph. Tools 7, 1, MILBRADT, P., AND PICK, T Polytope fnte elements. Int. J. Num. Meth. Eng. 73, 12, MULLEN, P., MEMARI, P., DE GOES, F., AND DESBRUN, M HOT: Hodge-optmzed trangulatons. ACM Trans. Graph. 30, 4. SCHAEFER, S., JU, T., AND WARREN, J A unfed, ntegral constructon for coordnates over closed curves. CAGD 24, 8-9, SHEPARD, D A two-dmensonal nterpolaton functon for rregularly-spaced data. In ACM Natonal Conference, SIBSON, R A bref descrpton of natural neghbor nterpolaton. In Interpolatng Multvarate Data. John Wley & Sons, ch. 2, SUKUMAR, N., AND BOLANDER, J Vorono-based nterpolants for fracture modellng. Tessellatons n the Scences. SUKUMAR, N Constructon of polygonal nterpolants: a maxmum entropy approach. Int. J. Num. Meth. Eng. 61, 12, VARADY, T., SALVI, P., AND KARIKO, G A mult-sded Bézer patch wth a smple control structure. Comp. Graph. Forum 35, 2, Fgure 12: More 3D coordnates. Left: ansotropc mean value coordnates for G = dag(5 cos 2 (10x)+1, 1, 1). Rght: terated power coordnates from Eq. (18) n 3D. WACHSPRESS, E A Ratonal Fnte Element Bass. Academc Press. WARREN, J., SCHAEFER, S., HIRANI, A. N., AND DESBRUN, M Barycentrc coordnates for convex sets. Advances n Computatonal Mathematcs 27, 3, WARREN, J On the unqueness of barycentrc coordnates. Contemporary Mathematcs, WEBER, O., AND GOTSMAN, C Controllable conformal maps for shape deformaton and nterpolaton. ACM Trans. Graph. 29, 4, Art. 78. WEBER, O., BEN-CHEN, M., AND GOTSMAN, C Complex barycentrc coordnates wth applcatons to planar shape deformaton. Comp. Graph. Forum 28, 2. WEBER, O., BEN-CHEN, M., GOTSMAN, C., AND HORMANN, K A complex vew of barycentrc mappngs. Comp. Graph. Forum 30, 5, WEBER, O., PORANNE, R., AND GOTSMAN, C Bharmonc coordnates. Comp. Graph. Forum 31, 8, ZHANG, J., DENG, B., LIU, Z., PATANÈ, G., BOUAZIZ, S., HOR- MANN, K., AND LIU, L Local barycentrc coordnates. ACM Trans. Graph. 33, 6, Art A Lagrange & Facet Restrcton Propertes We here show that condtons (a)-(d) mply propertes (e) and (f) for the case of convex polytope P wth vertces {v } =1...n n arbtrary dmenson. As a consequence of (a) and (b), we have λ(x)(v x) = 0. When evaluatng at a vertex vj, one gets: λ (v j)(v v j) = 0 =1...n, j Snce all vertces are extreme ponts of the polytope, a convex combnaton of vectors (v v j) j s zero f and only f all the coeffcents are zeros,.e., λ (v j) = 0 for = 1... n, j: ndeed, there exsts a drecton at v j along whch all vectors (v v j) j have postve components. Usng condton (a), we conclude that the Lagrange property (e) holds,.e., λ (v j)=δ j. As for condton (f), we can construct a coordnate system such that a gven boundary facet contans vertces wth ther frst coordnates at 0, and such that all other vertces have postve frst coordnates due to the convexty of P. Thus, smlar to the prevous argument, property (f) reflectng the restrcton of coordnates to the boundary facets follows from condtons (a)-(d) snce a postve lnear combnaton of postve numbers must be a postve number.