IEEE TVCG, VOL. ?,NO. ?, AUGUST 200? 1

Hexagonal Global Parameterization ofArbitrary Surfaces

Matthias Nieser, Jonathan Palacios, Konrad Polthier, and Eugene Zhang

✦

Abstract—In this paper we introduce hexagonal global parameter-izations, a new type of parameterization in which parameter linesrespect six-fold rotational symmetries (6-RoSy). Such parameteri-zations enable the tiling of surfaces with regular hexagonal textureand geometry patterns and can be used to generate high-qualitytriangular remeshing. To construct a hexagonal parameterizationgiven a surface, we provide an automatic technique to generatea 6-RoSy field that respects directional and singularity features inthe surface. This is achieved by applying the trace-and-deviatordecomposition to the curvature tensor, which allows us to identifyregions of appropriate directional constraints. We also introduce atechnique for automatically merging and cancelling singularities. Thisfield will then be used to generate a hexagonal global parameteri-zation by adapting the framework of QuadCover parameterization.In particular, we formulate the energy terms needed to solve fora hexagonal parameterization. We demonstrate the usefulness ofour geometry-aware global parameterization with applications suchas surface tiling with regular textures and geometry patterns andtriangular remeshing.

Index Terms—Surface parameterization, rotational symmetry,hexagonal tiling, triangular remeshing, surface sampling.

1 INTRODUCTION

I N this paper we introducehexagonal global parameteri-zations, a new type of periodic global parameterizations

that are ideal for tiling surfaces with patterns of six-foldrotational symmetries, i.e., 6-RoSys [1]. Such hexagonalpatterns possess some rather interesting properties. First,they are one of the only two classes of rotationally-invariantpatterns that can seamlessly tile a plane (the other beingquadrangular patterns). Second, hexagonal tilings provideoptimal approximation to circle packing. These propertieshave been linked to the wide appearances of hexagonalpatterns in nature, such as honeycombs, insect eyes, fisheggs, and snow and water crystals, as well as in man-made

M. Nieser is with the Institute of Mathematics, Freie Universitat Berlin,Arnimallee 6, D-14195 Berlin, Germany. Email: Matthias.Nieser@fu-berlin.de.J. Palacios is with the School of Electrical Engineering andComputer Sci-ence, Oregon State University, 1148 Kelley Engineering Center, Corvallis,OR 97331. Email: palacijo@eecs.oregonstate.edu.K. Polthier is MATHEON-Professor with the Institute of Mathematics,Freie Universitat Berlin, Arnimallee 6, D-14195 Berlin, Germany. Email:Konrad.Polthier@fu-berlin.de.E. Zhang is an Associate Professor with the School of Electrical Engineer-ing and Computer Science, Oregon State University, 2111 Kelley Engineer-ing Center, Corvallis, OR 97331. Email: zhange@eecs.oregonstate.edu.

objects such as floor tiling, carpet patterns, and architecturaldecorations. Figure 1 shows some examples of hexagonalpatterns in nature and man-made objects.

Tiling a surface with regular texture and geometry patternsis an important yet challenging problem in pattern syn-thesis. Methods based on some local parameterization ofthe surface often lead to visible breakup of the patternsalong seams, i.e., where the surface is cut open duringparameterization. Periodic global parameterizations canal-leviate this problem when the translational and rotationaldiscontinuity in the parameterization is compatible with thetiling pattern in the input texture and geometry. For exam-ple, a quadrangular global parameterization is designed tobe compatible with an input texture of 4-RoSy (Figure 2(a)). On the other hand, it is incompatible with 6-RoSypatterns (hexagonal) (Figure 2 (b)). In contrast, a hexagonalglobal parameterization is compatible with 6-RoSy patterns(Figure 2 (c)). The hexagonal global parameterizations weintroduce in this paper are designed to facilitate the tilingof surfaces with hexagonal texture and geometry patterns.

Another motivation of our work is triangular remeshing,in which an input triangular mesh is used to generate anew triangular mesh that approximates the same underlyingsurface with improved quality. In triangular remeshing, itis often desirable to have all the triangles in the meshbeing equilateral and of uniform sizes. Furthermore, theirregular vertices in the mesh can lead to difficulties insubsequently applications. With these conditions, we canreformulate the problem of triangular remeshing into theproblem of triangular tiling of the underlying surface, withthe irregular vertices in the mesh corresponding to thesingularities in the tiling. Note that the dual of a trian-gular tiling is a hexagonal tiling. Consequently, triangularremeshing can be accomplished by computing a hexagonalglobal parameterization of the input mesh. This provides analternative to existing approaches that perform Delaunaytriangulation or centroidal Voronoi tessellation. A benefitof using hexagonal parameterizations in remeshing is theability to explicitly control the number, type, and locationof irregular vertices in the new mesh.

The automatic generation of a hexagonal parameterizationfrom an input surface has a number of challenges. First,unlike quadrangular parameterizations whose parameterlines are parallel to either the major or the minor principal

IEEE TVCG, VOL. ?,NO. ?, AUGUST 200? 2

(a) (b) (c)

(d) (e) ( f )

Fig. 1. Hexagonal patterns can be widely observedin nature: (a) a honeycomb, (b) eyes of a fly, and(c) a snowflake. Such patterns have also been usedfor architectural decorations: (d) star of David, (e) anIslamic pattern, and (f) a floor tile pattern.

curvature directions, in hexagonal parameterizations onlyone of the two directions can be used at each point on thesurface. One must decide which direction to choose, andhow to propagate such choices from a relatively small setof points to the whole surface to maintain the smoothness ofthe resulting parameterization. Second, existing techniquesto explicitly control the singularities in the parameterizationare user-driven (including quadrangular parameterizations),and it is not an easy task to provide automatic control overthe number and location of such singularities. Third, theenergy terms developed for quadrangular parameterizationsare not appropriate for hexagonal paarameterizations, andto the best of our knowledge no such energy terms havebeen formulated.

To address these challenges, we present a two-step pipeline.First, a 6-RoSy field on the surface is automatically gen-erated based on the curvature tensor of the surface. Ourfield generation technique applies thetrace-and-deviatordecompositionto the curvature tensor, which allows us toselect appropriate sparse directional constraints from eitherthe major or minor eigenvectors in the curvature tensor.This results in remeshings whose edges are aligned withthe bending of the surface, so that the shape can be moreaptly represented with relatively few triangles or hexagons.Further, we introduce a method for automatically clusteringsingularities, inspired by mesh decimation techniques. Thesmoothness of the field, along with the reduced numberof singularities lead to highly regular meshes with a rela-tively small number of irregular vertices. Such meshes aredesirable for subdivision surface applications [2]. In thesecond step of our pipeline, we solve a sparse linear systembased on an energy term that we have developed specificallyfor hexagonal parameterizations. The resulting parameterscan then be used to genearte triangular meshes free of T-junctions. We also point out that the singularities in the

(a) (b) (c)

Fig. 2. While a quadrangular global parameterizationcan lead to perfect tiling of surfaces with quadrangu-lar patterns (4-RoSys) (a), it will lead to discontinuityacross seams when the input texture is a hexagonalpattern (i.e., 6-RoSy) (b). In this case, a hexagonalglobal parameterization is more appropriate (c).

hexagonal parameterization need to occur on the set ofEisenstein integers, which is different than the quadrangularcase, where the singularities occur on theGauss integers.This leads to the parameterization method that we refer toas HexCover.

In summary, our contributions in this paper are as follows:

1) We introduce the hexagonal global parameterizationand demonstrate its uses with applications such astriangular remeshing and surface tiling with regulartexture and geometry patterns.

2) We present the first technique to construct a hexago-nal global parameterization given an input surface.

3) We point out the importance of geometry-aware 6-RoSy fields for hexagonal global parameterization.

4) We propose an automated pipeline for generat-ing geometry-aware 6-RoSy fields. As part of thepipeline, we develop a metric for locating parabolicfeatures on surfaces based on tensor decompositionas well as a way of automatically clustering singu-larities.

5) We formulate the energy term for generating a hexag-onal global parameterization from a 6-RoSy field.

The remainder of this paper is organized as follows. We firstcover related work in relevant research areas in Section 2.Next, we describe our pipeline for generating a geometry-aware 6-RoSy field given an input surface in Section 3, andour parameterization technique in Section 4. In Section 5,we demonstrate the usefulness of our techniques withapplications in triangular remeshing and surface tiling withregular texture and geometry patterns. Lastly, we close inSection 6 with our conclusions.

2 RELATED WORK

Surface Parameterization: Surface parameterization is awell-explored research area. We will not attempt a completereview of the literature but instead refer the reader to thesurveys by Floater and Hormann [3] and Hormann et al. [4].

A very early surface parameterization method is the Tutte’sbarycentric graph embedding [5]. Tutte’s embeddings arecombinatorial in nature and do not capture the geometry of

IEEE TVCG, VOL. ?,NO. ?, AUGUST 200? 3

Fig. 3. We introduce hexagonal global parameterizations (an example on the dragon is shown in the upper-left),which can be used to perform regular texture and geometry pattern synthesis with patterns of six-fold symmetries(upper-right and lower-left) as well as geometry-aware triangular remeshing (lower-right).

the surface. Early global parameterization methods focus onconformal parameterizations [6], [7], [8], which are aimedat angle preservation at the cost of length distortion.

To reduce length distortion, Kharevych et al. [9] use conesingularities for conformal parameterization, which relaxthe constraint of a flat parameter domain at few isolatedpoints. Such singularities have proven essential for highquality parameterizations and have been used in otherparameterization schemes as well.

Tong et al. [10] use singularities at the vertices of a hand-picked quadrilateral meta layout on the surface. The patchesof the meta layout are then parameterized by solving for aglobal harmonic one-form. Dong et al. [11] use a similaridea for parameterization but create the quadrilateral metalayout automatically from the Morse-Smale complex ofEigenfunctions of the mesh Laplacian.

Ray et al. [12] parameterize surfaces of arbitrary genuswith periodic potential functions guided by two orthogonal

input vector fields, or a 4-RoSy field. This leads to a contin-uous parameterization except in the vicinity of singularitieson the surface. These singular regions are detected andreparameterized afterwards. The QuadCover algorithm [13]builds upon this idea by using the input 4-RoSy field toconstruct a four-fold covering space on which it becomesa vector field. This covering space and vector field isthen used to generate a global parameterization with noT-junctions. Our algorithm to generate a parameterizationfrom a 6-RoSy field is an adaptation of the QuadCovermethod, with a different energy formulation specified de-signed for hexagonal global parameterization.

Bommes et al. [14] propose a method similar to the afore-mentioned techniques, but formulate the parameterizationas the solution to a mixed-integer system of equations. Theyalso add constraints that force parameter lines to capturesharp edges.

Field Processing: Much work has been done on thesubject of vector and tensor (1- and 2-RoSy) field anal-

IEEE TVCG, VOL. ?,NO. ?, AUGUST 200? 4

ysis. To review all of this work is beyond the scopeof this paper; here we refer to only the most relevantwork. Helman and Hesselink [15] propose a method ofvector field visualization based on topological analysis andprovide methods of extracting vector field singularities andseparatrices. Topological analysis techniques for symmetricsecond-order tensor fields are later introduced in [16]. In thecontext of vector field design, numerous systems have beendeveloped for the purpose of vector field , most of whichhave been for specific graphics applications such as texturesynthesis [17], [18], [19], fluid simulation [20], and vectorfield visualization [21], [22]. Fisher et al. [23] propose avector field design system based on discrete one-forms.Note that the above systems do not employ any methodsof topological analysis, and do not extract singularitiesand separatrices. Systems providing topological analysisinclude [24], [25] and [26]. The last has also been extendedto design tensor fields [27], [28]. In contrast, relativelylittle work has been done onN-RoSy fields whenN > 2.Hertzmann and Zorin [29] utilizecrossor 4-RoSy fields intheir work on non-photorealistic pen-and-ink sketching, andprovide a method for smoothing such fields. Ray et al. [30]extend the surface vector field representation proposedin [25] into a design system forN-RoSy fields of arbitraryN. Palacios and Zhang [1] propose anN-RoSy designsystem that allows initialization using design elements aswell as topological editing of existing fields. They alsoprovide analysis techniques for the purpose of locating bothsingularities and separatrices, and a visualization techniquein [31]. Lai et al. [32] propose a design method based ona Riemannian metric, that gives the user control over thenumber and locations of singularities. Their system alsoallows for mixedN-RoSy fields, with different values ofN in different regions of the mesh. However, this methodis based on user design while we focus on automatic andgeometry-aware generation. Bommes et al. [14] offer amethod of producing a smooth 4-RoSy field from sparseconstraints, formulated as a mixed-integer problem.

Ray et al. [33] propose a framework to generate anN-RoSyfield that follows the natural directions in the surface andhas a reduced number of singularities which tends to fallinto natural locations. In this paper, we make use of thisframework but automatically generate the input constraints,which relieves the user from labor-intensive manual design.

3 GEOMETRY-AWARE 6-ROSY FIELD GEN-ERATION

In this section, we describe our pipeline for generating ageometry-aware 6-RoSy fieldF given an input surfaceS.This field will then be used to guide the parameterizationstage of our algorithm (Section 4).

We first review some relevant properties of 6-RoSyfields [1], [30]. AnN-RoSy fieldF has a set ofN directionsat each pointp in the domain of the field:

F(p) = {RiNv(p)}, i ∈ {0, . . . ,N−1}.

where the vectorv(p) = ρ(p)(

cosθ(p)sinθ(p)

)

is one of the

N directions, andRiN is the linear operator that rotates a

given vector by 2π i/N in the corresponding tangent plane.A singularity is a pointp0 such thatρ(p0) = 0 andθ(p0) isundefined;p0 is isolatedif the value ofρ 6= 0 for all pointsin a sufficiently small neighborhood ofp0, except atp0. Anisolated N-RoSy singularity can be measured by itsindex,which is defined in terms of the Gauss map [1] and has anindex of I

N , whereI ∈ Z. A singularity p0 is of first-orderif I =±1. When|I |> 1, p0 is referred to as ahigher-ordersingularity. A higher-order singularity with an index ofINcan be realized by mergingI first-order singularities.

Requirements and Pipeline:There are a number of goalsthat we wish to achieve with our automatic field generation.

First, we wish to control the number, location, and typeof the singularises in the field. When performing quad-rangular and triangular remeshing using periodic globalparameterizations, the singularities in the guiding 4- or 6-RoSy field correspond to the irregular vertices in the mesh.Such singularities can also lead to the breakup of textureand geometry patterns during pattern synthesis on surfaces.Consequently, the ability to control the number, location,and type of singularities in the field can improve quality ofthe remeshes and surface tilings.

Second, the field needs to be smooth, or distortion can occurin the resulting parameterizations that have undesirableeffects for triangular remeshing and surface tiling.

Third, we need the parameter lines in the parameterizationto be aligned with the feature lines in the surfaces, such asridge and valley lines (see Figure 4). In addition, it has beendocumented that having texture directions aligned with thefeature lines in the mesh can improve the visual perceptionof the texture synthesis [34].

Note that these requirements may conflict with each other.For example, excessive reduction of singularities can lead

Fig. 4. In this figure we demonstrate the importanceof having the parameter lines following the principalcurvature directions for remeshing: On the left, theedges do not follow the principal directions, resulting ina ”twisting” artifact on the ears. This problem does notoccur in the case on the right, in which the parameterlines follow the principal directions.

IEEE TVCG, VOL. ?,NO. ?, AUGUST 200? 5

to high distortion in the field, and an overly-smoothed fieldmay deviate from feature lines. To deal with this we adoptthe framework of Ray et al. [33]. Given a set of user-specified constraints and a field curvature, the frameworkgenerates a sparse linear system whose solution is the RoSyfield that matches the constraints and field curvature in theleast square sense. Each constraint represents a desiredN-RoSy value, i.e.,N directions, at a given point. In our casewe wish to have our field aligned with principal curvaturedirections. The user-specified field curvature is a vertex-based function defined on the mesh, whose value at avertex represents the desired discrete Gauss curvature at thisvertex. The field curvature must sum up to 2πχ(S) whereχ(S) is theEuler characteristicof the surfaceS. It allowsthe user to specify the location and type of singularitiesin the field. For example, a vertex with a field curvature of2kπN , after the system is solved, is likely to have a singularity

of index kN there.

Given a surface with complex geometry and topology, it canbe labor intensive to provide all necessary constraints andthe field curvature through a lengthy trial-and-error process.Consequently, we automatically generate the directionalconstraints as well as the field curvature, which is at thecore of our algorithm for field generation. Our algorithmconsists of two stages. First, we identify a set of directionalconstraints based on the curvature and solve for an initial 6-RoSy field using these constraints only. Second, we extractall the singularities in the initial field and perform iterativesingularity pair clustering until the distance between anysingularity pair is above a given threshold. The remainingsingularities will be used to create the field curvature,which, along with the initial directional constraints, willbe used to generate the final RoSy field. We now describeeach of these stages in more detail next.

Automatic Constraint Identification: To automaticallyidentify directional constraints, we need to answer thequestions of where to place constraints and what directionis assigned to each constraint.

Recall that we wish to align the parameter lines with featurelines such as ridges and valleys, i.e., the principal directionin which the least bending occurs. Note that the directionsin the 6-RoSy field is the gradient of the pameterization(Section 4). Consequently, we will choose the principaldirection that has the most bending, i.e., maximum absoluteprincipal curvature, as one of the directions in the 6-RoSy.

Principal curvature directions are most meaningful inparabolic and hyperbolic regions due to the stronganisotropy there. However, while purely hyperbolic regionspossess strong anisotropy, the absolute principal curvaturesare nearly indistinguishable, thus making both principalcurvature directions an candidate. Moreover, the two bi-sectors between the major and minor principal curvaturedirections also provide viable choices for the edge direc-tions in hyperbolic regions. Due to the excessive choices ofdirections in hyperbolic regions and insufficient choice of

Fig. 5. A summary of our surface classification. Onthe left, we show φ ∈ [−π/2,π/2] color mapped tothe [BLUE,RED] arc of the HSV color wheel on theStanford Bunny; the top shows a continuous mappingand the bottom, a binned classification based on thelegend shown on the right. Also shown on the rightare the surfaces to which points with given values ofφ are locally similar and the relationships between theprincipal curvatures for each of these surfaces. Themaximum curvature directions are shown in red, andthe minimum in blue.

directions in planar and spherical regions, we only generatedirectional constraints in parabolic regions.

We make use of a representation of the curvature tensor thatreadily exposes where on this spectrum of classificationany point on a given surface falls. Using the trace-and-deviator decomposition similar to those employed in [27],we find that the curvature tensorT at any pointp ∈ S canbe rewritten in the form:

T =

(

κ1−κ2

2

[

cos2θ sin2θsin2θ −cos2θ

]

+κ1+κ2

2

[

1 00 1

])

=ρ√2

(

cosφ[

cos2θ sin2θsin2θ −cos2θ

]

+sinφ[

1 00 1

])

(1)

where κ1 and κ2 are the principal curvatures atp, ρ =√

κ21 +κ2

2 , φ ∈ [−π/2,π/2] = arctan(κ1+κ2κ1−κ2

), and θ ∈[0,2π) is the angular component of the maximum principaldirection measured in the local frame atp. Note that thefirst component in the sum is traceless and symmetric,while the second is a multiple of the identity matrix.T(p)can now be classified using(ρ(p),φ(p)), which spans ahalf plane. There are six special configurations on thishalf plane, the first satisfyingρ(p) = 0, i.e., the localgeometry nearp is planar. For the remaining five config-urations we haveρ(p) > 0. Respectively, they correspondto φ(p) = π

2 (spherical),φ(p) = π4 (cylindrical), φ(p) = 0

(purely hyperbolic),φ(p) =−π4 (inverted cylindrical), and

φ(p) = −π2 (inverted spherical). With this representation,

we can classify any pointρ(p) as being planar ifφ(p) issmaller than a given thresholdδ , elliptical if φ(p)≥ δ and|φ(p)| > 3π

8 , hyperbolic if φ(p) ≥ δ and |φ(p)| < π8 , and

IEEE TVCG, VOL. ?,NO. ?, AUGUST 200? 6

Fig. 6. Here we show a summary of our constraintselection. On the right, we show the Buddha with colormapped to ρ . We then select the 35 percent of thesurface with the highest values of ρ (middle); here wehave colored the constraints based on φ using thelegend in Figure 5. Finally, we select the maximumcurvature directions as constraints in the regions whereφ > 0 (yellow) and the minimum where φ < 0 (cyan). Inthe close-up on the left, we can see that constraintsselected in this fashion are always orthogonal to thedirection in which the surface is bending the most. Also,notice that the nearly adjacent yellow and cyan regionsdo not conflict via a π/2 rotation, as they would if wehad selected the only one of the curvature directionseverywhere.

parabolic otherwise, i.e.,φ(p) ≥ δ and π8 ≤ |φ(p)| ≤ 3π

8 .Note that this tensor representation is a direct applicationof the eigenvalue manifold for asymmetric tensor fields [35]to the principal curvature tensor.

Given the classification, we propagate the directions in theparabolic regions into non-parabolic regions (planar, spher-ical, hyperbolic) using energy minimization, an approachtaken in [14]. To accomplish this, we pick the points whereρ (the tensor magnitude) is above certain a thresholdtρ , andlabel these points as having “strong” curvature (in all of ourexamples, we have chosentρ so that 35 percent of the areaof S is so-labeled). From this set of points, we use onlythe directions of the parabolic points as constraints; thatis, the points for whichφ ∈ [−3π/8,−π/8]∪ [π/8,3π/8](Figure 6). Finally, we select the maximum directionθas the constraint direction at points whereφ > 0 and theminimum directionθ +π/2 whereφ < 0. Recall that thedirections in the output field specify the gradients in ourresulting parameterizations, and we wish one the isolines ofthe parameters to be orthogonal to the direction in which thesurface is bending the most. Clearly, the above directionssatisfy this requirement (See the shapes on the right sideof the right image in Figure 5). Finally, the constraints areused to set up a linear system [33] whose solution givesrise to our initial RoSy field.

For our solver, we use the geometry-awareN-RoSy fieldgeneration technique proposed by [33], as it allows us tocontrol the level of geometric detail that is reflected bysingularities, and also plays a role in the implementationof our singularity clustering technique. This system, based

on Discrete Exterior Calculus (DEC) [36], filters (locallyaverages) the Gauss CurvatureK of Sto produceK and thencomputes a target field curvatureCt using the differencebetweenK and K. Ct is then used to modify the anglesby which directions rotate when parallel transported alongmesh edges. This compensates for the actual curvatureof S, and directional fields computed onS under theseconditions behave as thoughS has a Gauss curvature ofK. SinceK is smoother thanK, such fields have reducedtopological noise, which makes them more suitable for ourparameterization algorithm.

Automatic Singularity Clustering: Our initial field wasobtained from directional constraints only. Consequently,it typically consists of only first-order singularities. Givena surface with rather complex geometry and topology, thenumber of singularities can be rather large. Furthermore,while the location of the singularities tend to be appropriate(in high curvature regions), many of them form denseclusters. Having singularities in closer proximity can leadto difficulties in the resulting parameterization. This isbecause the singularities will be constrained to be on alattice in the parameter space as typically required by mostglobal parameterization methods [13], [14]. Consequently,the smallest distance between any singularity pair will bemapped to a unit in the parameter space. If the smallestdistance is too small, the two involved singularities maybe mapped to the same point on the lattice, leading to alocally infinite stretching in the parameterization.

To address this, many field generation techniques constrainthe number of singularities to be as few as possible [30], butthis represents another extreme, where the field directionscan become highly distorted in some regions. Furthermore,many of the aforementioned approaches require much userinteraction [1], [30], [33], which can be time-consumingfor models with complex geometry and topology.

Our goal is to automatically reduce the number of sin-gularities in the field while retaining the locations of theremaining singularities inside high curvature regions. Toachieve this we employ the following process.

First, we extract the singularities in the initial RoSyfield [30] which we use to build a graph embedded in thesurface. The nodes of this graph are the singularities inthe field, and the edges representing proximity informationbetween singularity pairs. We refer to this graph as thesingularity graph G. To constructG, we compute a Voronoidiagram with the singularities as sites. The dual graph givesrise to the singularity graph [37].

Second, we iteratively perform edge collapse to this graph,which is equivalent to performing singularity pair cluster-ing (merging or cancellation), until the minimal surfacedistance between any singularity pair is above a giventhreshold. Every time a singularity pair is clustered, wecompute the sum of the singularity indexes and place asingular constraint with the sum as its desired index. Notethat we do this even if the sum is zero, i.e., singularity

IEEE TVCG, VOL. ?,NO. ?, AUGUST 200? 7

(a) (b) (c) (d)

Fig. 7. A visual summary of our method for automatically clustering singularities. Our initial field has a largenumber of singularities (a). In (b) we show the singularity graph G constructed from the field in (a: ear, nose). Wethen perform a series of edge-collapses to get the reduced graph (c); the region R is shown in green. Finally, weresolve the field in Rusing the nodes of G as singular constraints and the boundary of Ras directional constraints.This gives us a reduced field (d) which has fewer singularities. Note the singularities are colored based on thesign of their indexes, not the actual value. In fact, there are higher-order singularities (d) but none in (a).

pair cancellation. The singularity constraint is placed onthe path between the two original singularities, closest tothe one with the Gaussian curvature of highest magnitude.This is an attempt to keep singularities near the features thatcaused them to originally appear during initialization andis accomplished by interpolating along the geodesic fromp0 to p1 using the value|K(p1)|/(|K(p0)+K(p1)|, whereK(p) is the Gaussian curvature atp ∈ S. We continue tocollapse edges in the order of increasing edge-length onGuntil no edge of length less thandsing remains. At the endof this process, we will have generated a set of singularityconstraints, i.e., the remaining vertices in the graph, whichis then used to update the field in the vicinity of thesesingularities. In the case of fields generated for remeshing,dsing can be selected based on the edge-length of the outputmesh. We choosedsing to be 0.1B whereB is the size ofthe bounding box for the model. For a visual summary ofthe algorithm, see Figure 7.

Third, we modify the field curvature based on the singular-ity constraints. Recall that the field curvature is simply asmoothed version of the discrete Gauss curvature during thegeneration of the initial field. The singularity constraints,produced in the previous step, consists of a set of verticesin the mesh and a desired singularity indext(p)for eachsuch singularity constraintp. We modify the field curvaturesuch that it is zero everywhere on the surface except atsingularity constraints where the value of the field curvatureis 2π

N t(p). Notice that such assignment satisfies that the totaldesired field curvature is equal to the Euler characteristicofthe underlying surface. We now modify the 6-RoSy fieldby solving the same system used to generate the initialfield, with two difference. We do not solve the systemeverywhere on the surface. Instead, we generate a regionRp0 = {p|d(p,p0)< dsing} for each singularity constraintp0

and solve the system in the union of these disks. That is,the field values are fixed for vertices outside these regionsand will serve as the boundary condition when solving thesystem. The field values for vertices inside the disks arechanged. We have found this to be fast and efficient in

Fig. 8. A geometry-aware 4-RoSy field generatedusing our automatic technique (Section 3) (top) andtexture tiling of the surface based on this field (bottom).

controlling the location, type, and number of singularities.

We wish to point out that our automatic field generationmethod can be applied toN-RoSy field generation forany N, in particular 4-RoSy fields. Figure 8 shows anexample generated using our method. The only change inthe whole field generation pipeline occurs during automatic

IEEE TVCG, VOL. ?,NO. ?, AUGUST 200? 8

identification of directional constraints. Instead of choosingθ or θ + π

2 as one of the six directions for 6-RoSy fields,we choose both for 4-RoSy fields.

4 HEXCOVER PARAMETERIZATION

We now describe the algorithm behind the second stage ofour pipeline. To do so we first define some concepts.

Given a triangular manifold mesh surfaceS, a periodicglobal parameterizationof S with respect to anN-RoSyfield is a collection of linear mapsαi = (ui ,vi) that mapeach triangleti ∈SontoR2 with the following property. Forall pairs of adjacent trianglesti , t j , the transition functionbetween them must satisfy:

α j(p) = Rr i jN αi(p)+ ti j , ∀p∈ ti ∩ t j , (2)

where r i j ∈ {0,1, ...,N− 1} and ti j are the rotational andtranslational discontinuities, respectively. Recall that Rk

N isthe linear operator that rotates a vector by2kπ

N .

For quadrangular global parameterizationsti j are requiredto lie on the set ofGauss integers, defined as{

(

a b)

|a,b∈Z}. This constraint eliminates seams in surface tiling andT-junctions in remeshing. Recent approaches generate aglobal parameterization by solving an energy minimizationproblem. Kalberer et al. [13] define the energy formulation,which is quadratic, based on the concept ofcovering spacesfrom algebraic topology. They solve the system using aconjugate gradient solver. Bommes et al. [14] employ asimilar energy term but solve it using a different solver.

In this paper we wish to compute a hexagonal globalparameterization from a 6-RoSy field using the approachof [13], [14], i.e., turning the parameterization problem intoan energy minimization problem. To do we need to answertwo questions. First, what is the energy like for a hexagonalglobal parameterization? Second, what is the set on whichti j and the singularities are situated? Both questions cananswered using the idea of covering spaces.

Energy for Hexagonal Parameterizations:In the interestof saving space we do not review the definition of coveringspaces. Instead, we provide some practical descriptionswhich are needed to understand our algorithm.

Given a mesh surfaceS and a 6-RoSy fieldF defined onS, we construct a six-fold coveringS′ of (S,F) along withtwo vector fieldsF ′

u andF ′v. Every trianglet in S will have

six corresponding triangles inS′: t0, ..., t5 (Figure 9).F ′u is

a vector field onS′ that distributes the six vectors of the6-RoSy field onto the six copies, i.e.,F ′

u(ti) =Ri6 f (t) where

f (t) is one of the six directions in the 6-RoSy int. BesideF ′

u, we also defineF ′v = R1

4F ′u in S′. The two vector fields

are mutually perpendicular, and they are the gradients of theparametersu andv, respectively, i.e.,∇u= F ′

u and∇v= F ′v.

Together they provide the energy term required for energyminimization:

E(u′,v′) :=∫

M′

(‖∇u′−F ′u‖2+‖∇v′−F ′

v‖)2dA (3)

Given two adjacent triangless and t in S, their corre-sponding layers inS′ are combinatorially glued together bysome matching dst ∈ {0,1,2,3,4,5}, which connects thesheetssi with ti+dst mod 6. The matching describes whichof the 6-RoSy vectors ins and t are paired. While anymatching assignmentdst is valid combinatorially, in ourimplementation we computedst that matches the vectorsfrom neighboring triangles to be most aligned. This isachieved by settingi + dst = argmink F ′

u(s0) ·F ′u(tk). Once

it is established thatsi and t j are neighbors inS′, we havesi+p and t j+p (1≤ p≤ 5) are neighbors inS′ as well.

ti

F ′u

F ′vt ′i,0

t ′i,1t ′i,2t ′i,3t ′i,4t ′i,5

Fig. 9. Left: Triangle ti with 6-rosy field. Right: 6-foldcovering of ti . The 6-rosy field leads to two vector fields,Fu and Fv on the covering.

Notice that to computeu in S requires to choose a vectorfrom the 6-RoSy inside each triangle. This can lead tonot only a discontinuous vector field inS but also anambiguous one as there are six choices in every triangle.The construction of a covering surface replaces the needto handle the matchingsr i j in Eqn. (2). They are fullydetermined byS′, i.e. the matchings are implicitly givenby the choice of how the six copies of adjacent triangleswere identified inS′. The problem is reduced to finding aparameterization(u′,v′) on S′ which is continuous up to atranslational offset byti j in adjacent triangles.

Lattice of Hexagonal Parameterizations:Due to the sym-metry of the covering surface and the symmetric behaviorof the algorithm, the resulting texture images on differentcopies of each triangle are congruent, i.e. for two pointsp0, p1 in the same fiber but on different layers ofS′, theparameter values are related by

(u′(p1),v′(p1)) ∈ RZ

6 (u′(p0),v

′(p0))⊕G6. (4)

where G6 is the set of possible values for translationaldiscontinuities along seams and singularities. Because ofthis property, we can now take the parameterization ofS′

and project it back ontoS by just taking the function valueat any of the six layers. The choice of layer is unimportantbecause of Equation 4 and the symmetry of the textureimage (see Figure 9).

Note thatS′ is a Riemannian surface with branch pointsat those positions where the original 6-RoSy field has

IEEE TVCG, VOL. ?,NO. ?, AUGUST 200? 9

1√

31

√30

v

u

Fig. 10. Left: texture image, which can be used forseamless hex texturing of any surface. Right: gridpoints of G6.

singularities. All six copies of a triangle are geometricallyidentical, so there is not necessarily an embedding withoutself-intersections. This does not present any difficulty forus, however, since the algorithm does not rely on an explicitembedding ofS′.

To understandG6 it is sufficient to inspect the diagramshown in Figure 10. The texture image of a hexagonalglobal parameterization is constrained to be a regular hextiling of the plane (Figure 10, left). We use a rectangularimage of width

√3 and height 1, which contains a part of

the hex grid and is tiled over the whole plane. The centerof the hexagons are given byG6 (Figure 10, right), whichhas the form of

{(

a− b2

√3

2 b)

|a,b∈ Z} (5)

Thus, the texture image is invariant under translations byany offset vectort ∈ G6 and by rotation of about multiplesof π

3 around any point inG6. While a hex parameterizationis a discontinuous map, the discontinuities are not visiblebecause of the repeating structure of the texture image.Note thatG6 is exactly theEisenstein integerlattice, whichis different from theGauss integerlattice that applies toquadrangular global parameterizations.

Parameterization Generation: The energy minimizationproblem in Equation 3 is solved using a conjugate gradientsolver. The surface is then cut open to a topological diskand the resulting vector fields are integrated yielding theparameter functionsu′ andv′. This parameterization is con-tinuous everywhere (ti j = 0 at all edges) except along thecuttings, where the parameter lines might not fit together.

To cut the surface open, we compute the shortest homotopygenerators as done in [38]. In order to keep the coveringsymmetric, it is important that the cuts are identical in alllayers, i.e. we first cut the original surfaceS and then liftthese cuts onto the coveringS′. We also need to connect allsingularities with the cut graph, since they can be seen asinfinitesimal small holes in the surface. For this purpose,the method was adapted allowing also surface boundaryand singularities by [39].

Up to this point the parameterization that we have obtaineddo not necessarily satisfy thatti j ∈ G6 along the cut curves,which can lead to mismatches between parameter linesalong these curves and thus cracks in surface tiling andT-junctions in remeshing. We correct this with a postpro-cessing step that is similar to the approach of Bommes

Fig. 11. Parameterized minimal surfaces. Left:Schwarz surface with 8 singularities of index −1/2.Right: Neovius surface with 8 index −1/2 and 6 index−1 singularities.

et al. [14] which seeks to reduce this mismatch alongthe cut curves. Note that the problem of finding theL2-smallest functionsu′∂ , v′∂ , such that allti j ’s of the finalparameterization are inG6 is NP hard, since it is equivalentto minimizing a quadratic function on a given lattice (alsocalled theclosest vector problem). If the closest vectorproblem could be solved exactly in polynomial time, thenthe result would be independent of the chosen cut graph.Figure 11 shows the hexagonal parameterization of twominimal surfaces using our technique.

Note, that this is only a heuristic for the problem of findingthe L2-smallest functionsu′∂ , v′∂ , such that all gaps of thefinal parameterization are inG6. In general, this problemis NP hard, since it is equivalent to minimizing a quadraticfunction on a given lattice (also called theclosest vectorproblem). In all our tests, the parameterization generatedwith the heuristic do not differ very much from the op-timal solution. Furthermore, the heuristic seems to givemore accurate results if the geometric length of the cutpaths are smaller. This is the reason, why the shortestpossible cutgraph is a good choice. If we would solve theclosest vector problem exactly, then the result would beindependent of the chosen cut graph. Figure 11 shows thehexagonal parameterization of two minimal surfaces usingour technique.

5 RESULTS AND APPLICATIONS

In this section, we apply our hexagonal parameterizationto two graphics applications: pattern synthesis on surfaces,and triangular remeshing.

5.1 Pattern Synthesis on Surfaces

Example-based texture and geometry synthesis on surfaceshas received much attention from the Graphics communityin recent years. We refer to [40] for a complete survey.Here we will refer to most relevant work.

Wei and Levoy [19] are the first to point out that sometextures exhibit symmetries that makeN-RoSy fields ofN > 1 more suitable for the specification of orientation.Liu et al. [41] propose techniques for the analysis, ma-nipulation, and synthesis ofnear-regulartextures (i.e. very

IEEE TVCG, VOL. ?,NO. ?, AUGUST 200? 10

Fig. 12. Here we show some regular texture and geometry patterns tiled seamlessly on some surfaces usingour global parameterization. Note how the pattern follows the features on the feline (left), highlighting the shape.The mostly hexagonal hollow structure is used to represent the bunny surface (right).

structured textures with repeating patterns) in the plane.Kaplan and Salesin [42] address the design of Islamic starpatterns in the plane. There has been some recent workin constructing circle patterns from a triangular mesh forarchitectural models [43].

Generating regular patterns on a surface can be greatlyfacilitated given an appropriate global parameterization.Given a regular hexagonal texture or geometry pattern,it is simply tiled in the parameter-space of the meshand the texture should stitch (relatively) seamlessly every-where (Figure 12). For example, to achieve circle packingfor architectural patterns, our hexagonal parameterizationallows nice hexagonal patterns to be generated from asurface, which can be used as input to such algorithms asshown in Figure 12 (right). Our method provides necessarysmoothness and feature alignment, thus leading to a high-quality model, even in the case of relatively high geometricand topological complexity. Figure 3 (b, c) provides someadditional examples in which regular hexagonal texture andgeometry patterns are placed on the dragon.

We also comment that quadrangular global parameteriza-tions can be used to synthesis patterns of 4-RoSys onsurfaces. Figure 8 provides one such example. To the bestof our knowledge this realization has not been published.Notice that our field generation algorithm can also auto-matically generate geometry-aware 4-RoSy fields, whichlead to coherent synthesized patterns that align with surfacefeatures.

5.2 Triangular Remeshing

There has been much work in triangular remeshing. Toreview all past work is beyond the scope of this paper. Werefer the reader to [46] for a complete survey of triangularremeshing literature, and review only the most relevant

work here. Common methods of mesh triangulation aretypically based on either a parameterization [47], [44], [48],[49], local optimization methods [50], [51], [52], or Delau-nay triangulations and centroidal Voronoi tessellations [53],[45].

The focus of triangular remeshing is on shape preservation,good triangle aspect ratio, feature-aware triangle sizing, andcontrol of irregular vertices (valence not equal to six). Theseobject functions are often conflicting with one another,and the output mesh is a result of compromise amongthese factors. For example, many parameterization-basedmethods suffer from artifacts in the triangulation at thelocations of the chart boundaries (though this problem canbe alleviated by using a global parameterization as in [49]).Direct and local optimization methods suffer from a lack ofglobal control over the structure of the triangulation suchas the location and number of irregular vertices.

In this paper, we perform triangular remeshing using ahexagonal global parameterization derived a shape-aware6-RoSy field. There are a number of benefits to this. First,such an approach can lead to overall better aspect ratio fortriangles in the remesh (equilateral). Second, the numberof irregular vertices can be reduced and their locations canbe controlled as these vertices correspond exactly to theset of singularities in the 6-RoSy field. Third, we haveincorporated the ability to match the orientations of theRoSy field based on natural anisotropy in the surfaces.Fourth, the size of the triangles can be controlled througha scalar sizing function (Figure 14).

Figure 13 compares the results of the foot and Venusmodels using our method with that of [44], which is basedon Delaunay triangulations, and [45], which is based onrestricted Voronoi diagrams. Table 1 provides the qualitystatistics of all the test models used in this paper as well asthe comparison to [44], [45]. Notice that all three methods

IEEE TVCG, VOL. ?,NO. ?, AUGUST 200? 11

Fig. 13. We compare the outputs of our method (right) to those of Alliez et al. [44] (left) and Yan et al. [45](middle) with a Foot model and Venus. Note the reduced number of irregular vertices in flat regions of the Foot,and increased feature alignment (around the nose and brow) on the Venus. The histograms show the occuringinner angles (on the X-axis from 0 to 120degrees).

Fig. 14. We demonstrate our capability to have vari-able sizes of triangles (right) given a scalar sizingfunction (left).

capture the underlying geometry well (comparable Haus-dorff distances to the original input mesh) but our methodtends to have the fewest irregular vertices among all threemethods and smaller variations in angles of the triangles

than [44]. Note that having relatively few irregular verticesis a direct result of our automatic singularity clusteringin the field generation (Section 3) while achieving goodtriangle aspect ratios is due to the nature of the hexagonalparameterization. In addition, our method tends to produceedge directions that better align with the features in themesh, such as along Venus’ nose ridge than [44]. Additionalremeshing results can be found in Figure 3.

5.3 Performance

The amount of time to automatically generate a geometry-aware 6-RoSy field is on average 40 seconds for a modelof 40K triangles, measured on a PC with a 2.4GHz CPU.The time to generate the parameterization is approximately120 seconds per model, measured on a PC with 2.13GHzCPU. The running time of both stages are impacted by themesh size as well as the number of singularities in the RoSyfield.

IEEE TVCG, VOL. ?,NO. ?, AUGUST 200? 12

model name Hausdorff min max SD # irregulardistance angle angle angle vertices

foot [44] 0.3373 2.82 173.88 11.92 146foot [45] 0.0094 26.92 115.85 7.40 3287foot 0.0129 22.65 125.09 5.11 13Venus [44] 0.1005 0.42 178.99 17.48 38Venus [45] 0.0439 19.89 121.13 10.37 1449Venus 0.0543 24.87 114.80 6.84 36Max Planck 0.00263 12.44 145.79 5.00 44bunny 0.00751 18.03 138.54 7.54 65feline 0.02695 5.75 167.34 11.09 121dragon 0.00762 4.80 151.88 9.10 181Buddha 0.02338 1.89 175.11 12.12 123

TABLE 1Mesh quality of all test models used in this paper. We

have also included comparisons with two models(foot, Venus) remeshed using the methods of Alliez et

al. [44] and Yan et al. [45], respectively. The meshquality is measured in terms of the Hausdorff distance(in % of the bounding box), the minimum, maximum,

and standard deviation (SD) of the angles in the meshmeasured in degrees, as well as the number irregular

vertices. Notice that our method produces similarapproximation errors (Hausdorff distance), but leadsto better overall aspect ratios of triangles in the mesh(larger minimum angle, smaller maximum angle, andsmall standard deviation of angles) than [44] as wellas fewest irregular vertices among all three methods.

6 CONCLUSIONS

In this paper we have introduced a novel type of globalparameterizations which we refer to as hexagonal globalparameterizations. We demonstrate the uses of such param-eterization with applications in pattern synthesis on surfacesand triangular remeshing.

We provide a two-step pipeline to generate a shape-awarehexagonal global parameterization. First, a 6-RoSy fieldis automatically generated based on the surface geometry.Second, the 6-RoSy field is used to generate the hexagonalglobal parameterization based on the idea of coveringspaces. As part of our field generation method, we allowautomatic constraint selection by applying the trace-and-deviator decomposition to the curvature tensor. We havealso provided the capability to automatically cluster nearbysingularities into higher-order ones, thus reducing topolog-ical noise. While these techniques are demonstrated for the6-RoSy fields, they are generic and can apply to 4-RoSyfields with relatively few modifications.

There are a number of possible future research directions.First, we wish to study objects that are close toN-RoSy,which we refer to as near-regular RoSys. In these objectsthe N member vectors do not have identical magnitudenor even angular spacings. Such objects can allow moreflexibility in both quadrangular and triangular remeshing.Second, five-fold symmetry appears in many natural objectssuch as flowers. We wish to pursue graphics applicationsthat deal with 5-RoSys. It is also our intent to investigate

appropriate mathematical representations that handle moresophisticated types of wallpaper textures which may containreflections and gliding reflections. We will also investigatemeans to improve the quality of parameterization by betterbalancing between the topological complexity (number ofsingularities) and distortion in the parameterization.

REFERENCES

[1] J. Palacios and E. Zhang, “Rotational symmetry field designonsurfaces,” inSIGGRAPH ’07. ACM, 2007, p. 55.

[2] Z. J. Wood, P. Schroder, D. Breen, and M. Desbrun, “Semi-regularmesh extraction from volumes,” inVIS ’00, 2000, pp. 275–282.

[3] M. S. Floater and K. Hormann, “Surface parameterization: atutorialand survey,” inAdvances in multiresolution for geometric modelling,N. A. Dodgson, M. S. Floater, and M. A. Sabin, Eds. SpringerVerlag, 2005, pp. 157–186.

[4] K. Hormann, K. Polthier, and A. Sheffer, “Mesh parameterization:Theory and practice,” inSIGGRAPH Asia 2008, Course Notes,no. 11. ACM, Dec 2008.

[5] W. T. Tutte, “How to draw a graph,”Proc. London Math. Soc., vol.s3-13, no. 1, pp. 743–767, 1963.

[6] S. Haker, S. Angenent, A. Tannenbaum, R. Kikinis, G. Sapiro, andM. Halle, “Conformal surface parameterization for texture mapping,”IEEE Trans. on Visualization and Computer Graphics, vol. 6, no. 2,pp. 181–189, 2000.

[7] X. Gu and S.-T. Yau, “Global conformal surface parameterization,”in Eurographics Symposium on Geometry Processing, 2003, pp. 127–137.

[8] M. Jin, Y. Wang, S.-T. Yau, and X. Gu, “Optimal global conformalsurface parameterization,” inIEEE Visualization, 2004, pp. 267–274.

[9] L. Kharevych, B. Springborn, and P. Schroder, “Discrete conformalmappings via circle patterns,”ACM Trans. on Graphics, vol. 25,no. 2, 2006.

[10] Y. Tong, P. Alliez, D. Cohen-Steiner, and M. Desbrun, “Designingquadrangulations with discrete harmonic forms,”ACM/EurogrpahicsSymposium on Geometry Processing, pp. 201–210, 2006.

[11] S. Dong, P.-T. Bremer, M. Garland, V. Pascucci, and J. C. Hart,“Spectral surface quadrangulation,” inSIGGRAPH ’06. ACM, 2006,pp. 1057–1066.

[12] N. Ray, W. C. Li, B. Levy, A. Sheffer, and P. Alliez, “Periodic globalparameterization,”ACM Transactions on Graphics, vol. 25, no. 4, pp.1460–1485, 2006.

[13] F. Kalberer, M. Nieser, and K. Polthier, “Quadcover - surfaceparameterization using branched coverings,”Comput. Graph. Forum,vol. 26, no. 3, pp. 375–384, 2007.

[14] D. Bommes, H. Zimmer, and L. Kobbelt, “Mixed-integer quadran-gulation,” ACM Trans. Graph., vol. 28, no. 3, pp. 1–10, 2009.

[15] J. L. Helman and L. Hesselink, “Visualizing vector field topologyin fluid flows,” IEEE Computer Graphics and Applications, vol. 11,pp. 36–46, May 1991.

[16] T. Delmarcelle and L. Hesselink, “The topology of symmetric,second-order tensor fields,”IEEE Computer Graphics and Appli-cations, pp. 140–147, 1994.

[17] E. Praun, A. Finkelstein, and H. Hoppe, “Lapped textures,” SIG-GRAPH 2000, pp. 465–470, Aug 2000.

[18] G. Turk, “Texture synthesis on surfaces,”SIGGRAPH 2001, pp. 347–354, 2001.

[19] L. Y. Wei and M. Levoy, “Texture synthesis over arbitrary manifoldsurfaces,”SIGGRAPH 2001, pp. 355–360, 2001.

IEEE TVCG, VOL. ?,NO. ?, AUGUST 200? 13

[20] J. Stam, “Flows on surfaces of arbitrary topology,”SIGGRAPH 2003,vol. 22, no. 3, pp. 724–731, Jul 2003.

[21] J. J. van Wijk, “Image based flow visualization,”SIGGRAPH ’02,vol. 21, no. 3, pp. 745–754, Jul 2002.

[22] ——, “Image based flow visualization for curved surfaces,” In: G.Turk, J. van Wijk, R. Moorhead (eds.), Proceedings IEEE Visualiza-tion, pp. 123–130, Oct 2003.

[23] M. Fisher, P. Schroder, M. Desbrun, and H. Hoppe, “Design oftangent vector fields,” inSIGGRAPH ’07. ACM, 2007, p. 56.

[24] H. Theisel, “Designing 2d vector fields of arbitrary topology,”in Computer Graphics Forum (Proceedings Eurographics 2002),vol. 21, 2002, pp. 595–604.

[25] W.-C. Li, B. Vallet, N. Ray, and B. Levy, “Representing higher-ordersingularities in vector fields on piecewise linear surfaces,” in IEEETransactions on Visualization and Computer Graphics (ProceedingsVisualization ’06), 2006.

[26] E. Zhang, K. Mischaikow, and G. Turk, “Vector field design onsurfaces,”ACM Transactions on Graphics, vol. 25, no. 4, pp. 1294–1326, 2006.

[27] E. Zhang, J. Hays, and G. Turk, “Interactive tensor fielddesign andvisualization on surfaces,”IEEE Transactions on Visualization andComputer Graphics, vol. 13, no. 1, pp. 94–107, 2007.

[28] G. Chen, G. Esch, P. Wonka, P. Muller, and E. Zhang, “Interactiveprocedural street modeling,”ACM Trans. Graph., vol. 27, no. 3, pp.1–10, 2008.

[29] A. Hertzmann and D. Zorin, “Illustrating smooth surfaces,” SIG-GRAPH 2000, pp. 517–526, Aug 2000.

[30] N. Ray, B. Vallet, W. C. Li, and B. Levy, “N-symmetry directionfield design,”ACM Trans. Graph., vol. 27, no. 2, pp. 10:1–10:13,2008.

[31] J. Palacios and E. Zhang, “Interactive visualization of rotationalsymmetry fields on surfaces,” inIEEE Transactions on Visualizationand Computer Graphics, to appear, 2011.

[32] Y.-K. Lai, M. Jin, X. Xie, Y. He, J. Palacios, E. Zhang, S.-M. Hu,and X. Gu, “Metric-driven rosy field design and remeshing,”IEEETransactions on Visualization and Computer Graphics, vol. 16, no. 1,pp. 95–108, 2010.

[33] N. Ray, B. Vallet, L. Alonso, and B. Lvy, “Geometry aware directionfield design,”ACM Transactions on Graphics, 2009.

[34] K. Xu, D. Cohen-Or, T. Ju, L. Liu, H. Zhang, S. Zhou, and Y.Xiong,“Feature-aligned shape texturing,”ACM Trans. Graph., vol. 28,no. 5, pp. 1–7, 2009.

[35] E. Zhang, H. Yeh, Z. Lin, and R. S. Laramee, “Asymmetric tensoranalysis for flow visualization,”IEEE Transactions on Visualizationand Computer Graphics, vol. 15, pp. 106–122, 2009.

[36] A. N. Hirani, “Discrete exterior calculus,” Ph.D. dissertation, 2003,adviser-Marsden, Jerrold E.

[37] M. Eck, T. DeRose, T. Duchamp, H. Hoppe, M. Lounsbery, andW. Stuetzle, “Multiresolution analysis of arbitrary meshes,” in SIG-GRAPH ’95. ACM, 1995, pp. 173–182.

[38] J. Erickson and K. Whittlesey, “Greedy optimal homotopy andhomology generators,” inSODA ’05: Proceedings of the sixteenthannual ACM-SIAM symposium on Discrete algorithms. Society forIndustrial and Applied Mathematics, 2005, pp. 1038–1046.

[39] F. Kalberer, M. Nieser, and K. Polthier, “Stripe parameterization oftubular surfaces,” inTopological Data Analysis and Visualization:Theory, Algorithms and Applications. Springer Verlag, 2009.

[40] L.-Y. Wei, S. Lefebvre, V. Kwatra, , and G. Turk, “State of the artin example-based texture synthesis,”Eurographics, vol. 2, 2009.

[41] Y. Liu, W.-C. Lin, and J. Hays, “Near-regular texture analysis andmanipulation,” inSIGGRAPH ’04. ACM, 2004, pp. 368–376.

[42] C. S. Kaplan and D. H. Salesin, “Islamic star patterns in absolutegeometry,”ACM Trans. Graph., vol. 23, no. 2, pp. 97–119, 2004.

[43] A. Schiftner, M. Hobinger, J. Wallner, and H. Pottmann, “Packingcircles and spheres on surfaces,” inSIGGRAPH Asia ’09. ACM,2009, pp. 1–8.

[44] P. Alliez, M. Meyer, and M. Desbrun, “Interactive geometry remesh-ing,” in SIGGRAPH ’02. ACM, 2002, pp. 347–354.

[45] D.-M. Yan, B. Levy, Y. Liu, F. Sun, and W. Wang, “Isotropicremeshing with fast and exact computation of restricted voronoidiagram,” inSGP ’09. Eurographics Association, 2009, pp. 1445–1454.

[46] P. Alliez, G. Ucelli, C. Gotsman, and M. Attene, “Recent advancesin remeshing of surfaces,”Research Report, AIMShape Network ofExcellence, 2005.

[47] X. Gu, S. J. Gortler, and H. Hoppe, “Geometry images,” vol.21,no. 3, 2002, pp. 355–361.

[48] P. Alliez, E. C. de Verdiere, O. Devillers, and M. Isenburg, “Isotropicsurface remeshing,” inSMI ’03. IEEE Computer Society, 2003,p. 49.

[49] A. Khodakovsky, N. Litke, and P. Schroder, “Globally smoothparameterizations with low distortion,” inSIGGRAPH ’03. ACM,2003, pp. 350–357.

[50] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle,“Mesh optimization,” inSIGGRAPH ’93. ACM, 1993, pp. 19–26.

[51] P. Lindstrom and G. Turk, “Image-driven simplification,”vol. 19,no. 3, 2000, pp. 204–241.

[52] V. Surazhsky and C. Gotsman, “Explicit surface remeshing,” inProceedings of Eurographics Symposium on Geometry Processing,Jun 2003, pp. 17–28.

[53] G. Turk, “Re-tiling polygonal surfaces,” vol. 26, no. 2. ACM, 1992,pp. 55–64.