26 GD11 Abstracts

IP0

Pierre Bezier Award Lecture - On Geometric In-teroperability

Interoperability of CAD systems has been sought aftersince the earliest design systems came to be used. Buttranslating shape representations, and models, faces sub-stantial difficulties that originate in part from mathemat-ical facts. Moreover, shape translation also runs counterto the business models of the software houses. The talkwill discuss approaches to geometric interoperability thathave the potential to overcome many of these obstacles andto deliver practical interoperability that side-steps the tra-ditional problems associated with model translation andstandardization. This is joint work with Vadim Shapiro(Wisconsin) and Vijay Srinivasan (NIST).

Christoph M. HoffmannPurdue Universitycmh@cs.purdue.edu

IP1

Human-Centered CAD System

Consumers increasingly demand products that are easy touse and cause the minimum fatigue on the user side orprovide the maximum effect while in use. These types ofdesign activity that consider the effect on the human bodyin the course of design are referred to as human-centereddesign. An integrated framework that allows a simultane-ous modeling of a product and a human user, simulationof both product and human behavior, and modificationof the product based upon the simulation results is de-scribed. Case studies using the proposed approach are alsoillustrated.

Kunwoo LeeSeoul National UniversityKunwoo@snu.ac.kr

IP2

Isogeometric Analysis: Progress and Perspectives

Computational geometry has until recently had little im-pact upon the numerical solution of partial differentialequations. The purpose of this talk is to explore Isoge-ometric Analysis, in which NURBS (Non-Uniform Ratio-nal B-Splines) and T-Splines are employed to constructexact geometric models of complex domains. I will reviewrecent progress toward developing integrated ComputerAided Design (CAD)/Finite Element Analysis (FEA) pro-cedures that do not involve traditional mesh generationand geometry clean-up steps, that is, the CAD file is di-rectly utilized as the analysis input file. I will summa-rize some of the mathematical developments within Isoge-ometric Analysis that confirm the superior accuracy androbustness of spline-based approximations compared withtraditional FEA, and I will summarize the requirementsthat methods of computational geometry must satisfy tobe suitable for FEA.

Thomas HughesInstitute for Computational Engineering and SciencesThe University of Texas at Austinhughes@ices.utexas.edu

IP3

Generalized Barycentric Coordinates

In 1827 August Ferdinand Mbius published his seminalwork on the ”barycentric calcul” which provided a novelapproach to analytic geometry. One key element in hiswork is the idea of barycentric coordinates which allow towrite any point inside a triangle as a unique convex com-bination of the triangle’s vertices. Only recently, this ideahas been extended to polygons with more than three ver-tices and arbitrary polytopes in higher dimensions, yield-ing an efficient method for interpolating data given at thevertices of an arbitrary polytope. We discuss the theoreti-cal background of these generalized barycentric coordinatesand present several useful applications, e.g. in computergraphics, computer aided geometric design, and image pro-cessing.

Kai HormannUniversity of Luganokai.hormann@usi.ch

IP4

Spectral Properties of Discrete Surface Energies

Beyond the Laplacian and its associated Dirichlet energyexist several other differential operators and energies onsurfaces. The spectra encode useful information, oftencharacterizing local and global geometric topological fea-tures of a surface, many of them beyond classical curvatureproperties. The spectral analysis of surfaces has many openproblems, most famous the reconstruction of a surface fromits spectrum. In the talk we design a new family of ener-gies and operators on discrete surface meshes, and studytheir usefulness for applications including fast interactivesurface modeling, derivation of surface signatures and meshfiltering.

Konrad PolthierFreie Universitaet BerlinFB Mathematik & Informatikkonrad.polthier@fu-berlin.de

IP5

Nearest Neighbor Searching in Metric Spaces, andRelated Concepts

Metric spaces are very simple geometric structures: theycomprise a set and a distance measure that obeys the tri-angle inequality. Nearest neighbor searching is a funda-mental algorithmic problem that can be posed and solvedin the general setting of metric spaces, in particular whenthe (intrinsic) dimension of the space is small. A vari-ety of geometric constructions, such as building meshesand graded triangulations, can be usefully understood asfinding a ”well distributed” set in a metric space with anappropriate distance measure. I will lightly survey suchalgorithms and constructions.

Ken ClarksonIBM Almaden Research Centerklclarks@us.ibm.com

IP6

Computational Geometry and Topology Meet Ge-ometric Modeling

In recent years several techniques primarily developed inthe area of computational geometry and topology have

GD11 Abstracts 27

been applied to various geometric modeling problems. Asa result robust methods for various geometric modelingproblems have been developed along with theoretical guar-antees. In this talk we consider the problems of curve andsurface reconstruction, mesh generation, and feature iden-tification to illustrate how mathematics from geometry andtopology interacts with algorithms to provide such certifiedcomputational tools. We explain the theoretical results,delineate the algorithms, and present experimental results.

Tamal DeyThe Ohio State Universitytamaldey@cse.ohio-state.edu

CP0

Inference-Based Procedural Modeling of Solids

As virtual environments become larger and more complex,there is an increasing need for more automated construc-tion algorithms to support the development process. Wepresent an approach for modeling solids by combining priorexamples with a simple sketch. Our algorithm uses aninference-based approach to incrementally fit patches to-gether in a consistent fashion to define the boundary ofan object. This algorithm samples and extracts surfacepatches from input models, and develops a Petri net struc-ture that describes the relationship between patches alongan imposed parameterization. Then, given a new param-eterized line or curve, we use the Petri net to logicallyfit patches together in a manner consistent with the inputmodel. This allows us to easily construct objects of varyingsizes and configurations using arbitrary articulation, repe-tition, and interchanging parts. The result of our process isa solid model representation of the constructed object thatcan be integrated into a simulation-based environment.

Keith Biggers, John KeyserTexas A&M Universitybiggers@tamu.edu, keyser@cse.tamu.edu

CP0

Efficient and Good Delaunay Meshes From Ran-dom Points

Our Conforming Delaunay Triangulation (CDT) algorithmuses uniform random points, and achieves the same qual-ity as deterministic Delaunay refinement: angles in [30,120] degrees. Random meshes are preferred for graphics,fracture mechanics, and mesh validation. We mesh 2d non-convex domains with holes, required points, and multipleregions in contact. Point-generation takes O(n) memoryand E(n log n) time; triangulation is O(n) for both. Givenpoints located in a square background grid, each point ismeshed independently in constant time, using radial sort-ing. Our CPU and GPU implementations scale well. CPUspeed matches Triangle’s, our GPU is 2x faster.

Mohamed S. Ebeida, Scott A. MitchellSandia National Laboratoriesmsebeid@sandia.gov, samitch@sandia.gov

Andrew Davidson, Anjul PatneyUniversity of California, Davisaaldavidson@ucdavis.edu, anjul.patney@gmail.com

Patrick M. KnuppSandia National Laboratoriespknupp@sandia.gov

John OwensUniversity of California, Davisjowens@ece.ucdavis.edu

CP0

Geometric Characteristics of Conics in Bezier Form

In this paper we address the calculation of elements of conicsections (axes, asymptotes, centres, eccentricity, foci...)given in Bezier form in terms of their control polygons andweights, making use of real and complex projective andaffine geometry.

Leonardo Fernandez-JambrinaUniversidad Politecnica de Madrid, Spainleonardo.fernandez@upm.es

Alicia Canton, Maria Eugenia Rosado MariaUniversidad Politecnica de Madrid, Spainalicia.canton@upm.es, eugenia.rosado@upm.es

CP0

Extended Grassfire Transform on Medial Axes of2D Shapes

We present a global shape measure on the medial axis ofa 2D shape and a definition of a center point. We reveal anumber of properties of the shape measure and center pointthat resemble those of the boundary distance function andthe medial axis, and show that they can be generated usinga firefront propagation similar to Blum’s grassfire analogy.The shape measure and center point are compared withexisting formulations, and demonstrated in several appli-cations including pruning medial axes, aligning shapes, andshape matching.

Tao Ju, Lu LiuWashington University in Saint LouisDepartment of Computer Science & Engineeringtaoju@cse.wustl.edu, ll10@cse.wustl.edu

Erin Chambers, David LetscherSaint Louis Universityechambe5@slu.edu, letscher@slu.edu

CP0

Triangular Bubble Spline Surfaces

We present a new method for generating a Gn-surface froma triangular mesh of compatible surface strips, which is anetwork of polynomial curves with an associated implic-itly defined surface. Our method is based on a new typeof surface patches, called bubble patches, to represent theindividual surfaces. The construction of a single patch issimple, independent of the neighboring patches and worksuniformly for triangular meshes with vertices of arbitraryvalency. The resulting surface is a piecewise rational sur-face, which interpolates the given network of polynomialcurves.

Mario KaplJohannes Kepler University, Linz, Austriamario.kapl@jku.at

Marek ByrtusUniversity of West Bohemia, Plzen, Czech Republicbyrtus@kma.zcu.cz

28 GD11 Abstracts

Bert JuettlerJohannes Kepler University of LinzInstitute of Applied Geometrybert.juettler@jku.at

CP0

Gpu-Accelerated Hausdorff Distance ComputationBetween Dynamic Deformable Nurbs Surfaces

We present a parallel GPU-accelerated algorithm for com-puting the Hausdorff distance from one NURBS surfaceto another, within a bound. We make use of axis-alignedbounding-box hierarchies that bound the NURBS surfacesto accelerate the computations. We dynamically constructas well as traverse the bounding-box hierarchies for theNURBS surfaces using operations that are optimized forthe GPU. To compute the Hausdorff distance, we traversethis hierarchy after culling bounding-box pairs that do notcontribute to the Hausdorff distance. Our contribution in-cludes two-sided culling tests that can be performed inparallel using the GPU. The culling, based on the mini-mum and maximum distance ranges between the boundingboxes, eliminates bounding-box pairs from both surfacesthat do not contribute to the Hausdorff distance simulta-neously. We calculate accuracy bounds for our computedHausdorff distance based on the curvature of the surfaces.Our algorithm runs in real-time with very small guaran-teed error bounds for complex NURBS surfaces. Since wedynamically construct our bounding-box hierarchy, our al-gorithm can be used to interactively compute the Hausdorffdistance for models made of dynamic deformable surfaces.

Adarsh KrishnamurthyUniversity of California, Berkeley, CAadarsh@me.berkeley.edu

Sara McMainsUniversity of California, Berkeleytba@siam.org

Iddo HannielTechnion, Israel Institute of Technologytba@siam.org

CP0

An Iterative Algorithm for Homology Computa-tion on Simplicial Shapes

In this paper, we propose a new iterative algorithm forcomputing the homology of arbitrary shapes, discretizedthrough simplicial complexes. Here, we demonstrate howthe simplicial homology of a shape can be effectively ex-pressed in terms of the homology of its sub-components.The proposed algorithm retrieves the complete homologi-cal information including the Betti numbers, the torsioncoefficients and the representative homology generators.To the best of our knowledge, it is the first algorithmbased on the constructive Mayer-Vietoris sequence, whichrelates the homology of a topological space to the homolo-gies of its sub-spaces, i.e. the sub-components of the inputshape, and their intersection. We demonstrate the valid-ity of our approach with a particular shape decomposition,based only on topological properties, which minimizes thesize of the intersections between the sub-components andincreases the efficiency of the algorithm.

Dobrina BoltchevaGrenoble University

dobrina.boltcheva@inrialpes.fr

David CaninoUniversity of Genoa, Italycanino@disi.unige.it

Sara Aceituno, Jean-Claude LeonGrenoble Universityn/a, jean-claude.leon@inpg.fr

Leila De FlorianiUniversity of Genovadeflo@disi.unige.it

Franck HetroyGrenoble Universityfranck.hetroy@imag.fr

CP0

Isogeometric Analysis and Shape Optimization ViaBoundary Integral

In this paper, we present a boundary integral based ap-proach to isogeometric analysis and shape optimization.It uses the same basis, Non-Uniform Rational B-Spline(NURBS) basis, for both geometry representation andboundary integral based analysis. Boundary points cor-responding to Greville abscissae are used as collocationpoints. We conducted h-, p- and k-refinement study andshape optimization for linear elasticity problems. Ourstudy finds that 1) the NURBS based boundary integralmethod is computationally efficient, 2) it bypasses theneed for domain parameterization, a bottleneck in currentNURBS based volumetric isogeometric analysis; 3) it en-ables tighter integration of CAD and analysis.

Kang Li, Xiaoping QianIllinois Institute of Technologykli@iit.edu, qian@iit.edu

CP0

Estimating Effects of Removing Negative Featureson Engineering Analysis

This paper provides a general framework for the quanti-tative estimation of the effects of removing negative fea-tures on engineering analysis, or modification sensitivityfor short. There are two main applications: (i) when de-featuring models so that finite element analysis may becarried out more quickly and with lower memory require-ments, and (ii) when performing iterative design based onfinite element analysis. Our approach can handle large aswell as small features, and features with Neumann/naturalboundary conditions prescribed on them; previous meth-ods have difficulties in handling such cases. Estimationof the modification sensitivity is achieved by reformulatingit as a modeling error caused by use of different mathe-matical models to describe the same engineering analysisproblem. Results are obtained using the dual weightedresidual (DWR) method in combination with a heuristicassumption of small variation of dual solution after defea-turing. The final derived sensitivity estimator is expressedin terms of the difference of local boundary integrationsover the feature boundary, which can be explicitly evalu-ated using solutions defined on the defeatured model. Thealgorithm’s performance is demonstrated using a Poissonequation. Comparisons to results obtained by previous ap-proaches indicate it is both accurate and computationally

GD11 Abstracts 29

efficient.

Ming LiState Key Laboratory of CAD&CG, Zhejiang University,Chinaeming.li@gmail.com

Shuming GaoZhejiang Universitysmgao@cad.zju.edu.cn

Ralph MartinSchool of Computer Science& Informatics, Cardiff Universitralph@cs.cf.ac.uk

CP0

Fast and Robust Generation of City-Scale Seamless3D Urban Models

Since the introduction of the concept of “Digital Earth,”almost every major international city has been re-constructed in the virtual world. A large volume of ge-ometric models describing urban objects has become freelyavailable in the public domain via software like ArcGlobeand Google Earth. Although mostly created for visual-ization, these urban models can benefit many applicationsbeyond visualization including city scale evacuation plan-ning and earth phenomenon simulations. However, thesemodels are mostly loosely structured and implicitly definedand require tedious manual preparation that usually takesweeks if not months before they can be used. Designing al-gorithms that can robustly and efficiently handle unstruc-tured urban models at the city scale becomes a main tech-nical challenge. In this paper, we present a frameworkthat generates seamless 3D architectural models from 2Dground plans with elevation and height information. Theseoverlapping ground plans are commonly used in the currentGIS software such as ESRI ArcGIS and urban model syn-thesis methods to depict various components of buildings.Due to measurement and manual errors, these ground plansusually contain small, sharp, and various (nearly) degener-ate artifacts. In this paper, we show both theoretically andempirically that our framework is efficient and numericallystable. Based on our review of the related work, we be-lieve this is the first work that attempts to automaticallycreate 3D architectural meshes for simulation at the citylevel. With the goal of providing greater benefit beyondvisualization from this large volume of urban models, ourinitial results are encouraging.

Jyh-Ming Lien, Yanyan Lu, Evan Behar, StephenDonnelly, Fernando Camelli, David WongGeorge Mason Universityjmlien@cs.gmu.edu, tba@siam.org, tba@siam.org,tba@siam.org, tba@siam.org, tba@siam.org

CP0

Positive Gordon-Wixom Coordinates

We introduce a new construction of transfinite barycentriccoordinates for arbitrary closed sets in 2D. Our methodextends weighted Gordon-Wixom interpolation to non-convex shapes and produces coordinates that are positiveeverywhere in the interior of the domain and that aresmooth for shapes with smooth boundaries. We achievethese properties by using the distance to lines tangent tothe boundary curve to define a weight function that is pos-itive and smooth. We derive closed-form expressions for

arbitrary polygons in 2D and compare the basis functionsof our coordinates with several other types of barycentriccoordinates.

Josiah M. MansonTexas A&M Universityjosiahmanson@gmail.com

Kuiyu LiIntelkuiyuli@gmail.com

Scott SchaeferTexas A&M Universityschaefer@cs.tamu.edu

CP0

Computing the Medial Axis of 3D RegionsBounded by B-Spline Surfaces

A new approach is presented for computing the interior3D medial axis of regions bounded by parametric B-splinesurfaces. In the generic situation, the 3D medial axis con-sists of smooth surfaces along with a singular set consistingof edge curves, branch curves, fin points and six junctionpoints. The medial axis singular set is first computed us-ing a collection of robust higher order techniques. Medialaxis surfaces are computed as a time trace of the evolvingself-intersection set of the boundary under the the eikonal(grassfire) flow. The proposed algorithm presents a com-plete solution along with accurate topological structure.

Suraj Musuvathy, Elaine CohenUniversity of Utahsrm@cs.utah.edu, cohen@cs.utah.edu

James DamonUniversity of North Carolina at Chapel Hilljndamon@math.unc.edu

CP0

C2 Splines Covering Polar Configurations

A polar configuration is a triangle fan in a quad-dominantmesh; it allows for many mesh lines to join at a singlepolar vertex. This paper shows how a single tensor-productspline of degree (3, 6) can cap a polar configuration witha C2 surface. By design, this C2 polar spline joins C2

with surrounding bi-3 tensor-product splines and therebycomplements algorithms that smoothly cap star-like, multi-sided regions.

Ashish MylesNew York Universitymarcianx@gmail.com

Jorg PetersUniversity of Floridajorg@cise.ufl.edu

CP0

A Hybrid Parallel Solver for Systems of Multivari-ate Polynomials Using CPUs and GPUs

In this paper, we present a hybrid parallel algorithm forsolving systems of multivariate constraints by exploitingboth the CPU and the GPU multicore architectures. Wededicate the CPU for the traversal of the subdivision tree

30 GD11 Abstracts

and the GPU for the multivariate polynomial subdivision.By decomposing the constraint solving technique into twodifferent components, hierarchy traversal and polynomialsubdivision, each of which is more suitable to CPUs andGPUs, respectively, our solver can fully exploit the avail-ability of hybrid, multicore architectures of CPUs andGPUs. Furthermore, our GPU-based subdivision methodtakes advantage of the inherent parallelism in the multi-variate polynomial subdivision.

Cheon-Hyeon ParkNexon Corporation, Koreacheonhyeon@gmail.com

Gershon ElberTechnion, HaifaIsraelgershon@cs.technion.ac.il

Ku-Jin KimKyungpook National University, Koreakujinkim@yahoo.com

Gye-Young Kim, Joon-Kyung SeongSoongsil University, Koreagykim11@ssu.ac.kr, seong@ssu.ac.kr

CP0

Modeling with Rational Biquadratic Splines

We develop a rational bi-quadratic G1 analogue of the non-uniform polynomial C1 B-spline paradigm. The G1 splinescan exactly reproduce parts of multiple basic shapes, suchas cyclides and quadrics, and combine them into onesmoothly-connected structure.

Jorg PetersUniversity of Floridajorg@cise.ufl.edu

Kestutis KarciausKestutis.Karciauskas@mif.vu.ltvilnius university

CP0

Blends of Canal Surfaces from Polyhedral MedialTransform Representations

We present a new method for constructing G1 blending sur-faces between an arbitrary number of canal surfaces. Thetopological relation of the canal surfaces is specified via aconvex polyhedron and the design technique is based ona generalization of the medial surface transform. The re-sulting blend surface consists of trimmed envelopes of one-and two-parameter families of spheres. Blending the me-dial surface transform instead of the surface itself is shownto be a powerful and elegant approach for blend surfacegeneration. The performance of our approach is demon-strated by several examples.

Bohumir BastlUniversity of West Bohemiabastl@kma.zcu.cz

Bert JuettlerJohannes Kepler University of LinzInstitute of Applied Geometrybert.juettler@jku.at

Miroslav LavickaUniversity of West Bohemialavicka@kma.zcu.cz

Tino SchulzJohannes Kepler UniversityLinz, Austriatino.schulz@jku.at

CP0

Inspired Quadrangulation

This paper presents a new approach for the quadrangula-tion of triangular surfaces. We introduce a new example-based quad-meshing paradigm, in order to easily repro-duce the subjective decisions made in the design of refer-ence examples found in a corpus. The algorithm enablesto reproduce the subjective aspects of the example (ex-traordinary vertex layout) while minimizing the induceddistortion; allowing users to leverage reference meshes con-sidered of quality for fast prototyping. Our technique sup-ports localized mesh composition and enables to reproducemeshing styles despite intrinsic reflective symmetry, largevariation from isometry or even topological variation.

Julien TiernyCNRS - Telecom ParisTechtierny@telecom-paristech.fr

Joel DanielsSCI Institute - University of Utahjdaniels@sci.utah.edu

Luis Gustavo NonatoUniversidade de Sao Paulognonato@icmc.usp.br

Valerio PascucciSCI Institute - University of Utahpascucci@sci.utah.edu

Claudio SilvaScientific Computing and Imaging InstituteUniversity of Utahcsilva@sci.utah.edu

CP0

Transfinite Surface Interpolation over Irregular N-Sided Domains

Transfinite surface interpolation is a classic topic of CAGDand many non-quadrilateral schemes are known. Surfacesdefined solely by means of their boundary curves and cross-tangent functions are needed, for examples, in 3D curve-network based design and to fill complex irregular holessuch as with vertex blends. This paper deals with inter-polating so-called tangential ribbons. Former schemes areenhanced and extended in order to minimize shape arti-facts and to provide a more natural patch interior. Theproposed representation is based on irregular convex do-mains that correspond to the lengths and orientations ofthe boundary curves. The mapping of the individual rib-bons within the n-sided domain are calculated by a specialline-sweep method that assures a balanced orientation re-lated to the center of the domain, and avoids parametricshearing. Distance-based blending functions ensure thatmodifying or inserting a small edge will have only a localeffect over the n-sided patch. The construction supports

GD11 Abstracts 31

the generation of one- or two-sided patches, as well. Ex-amples and open research topics conclude the paper.

Tamas VaradyBudapest University of Technology and Economicsvarady@iit.bme.hu

Alyn RockwoodKing Abdullah University of Science and Technologyalyn.rockwood@kaust.edu.sa

Peter SalviBudapest University of Technology and Economicssalvi@iit.bme.hu

CP0

Non-Uniform Recursive Doo-Sabin Surfaces

This paper presents a generalization of Catmull-Clark-variant Doo-Sabin surfaces and non-uniform biquadraticB-spline surfaces called NURDSes (Non-Uniform RecursiveDoo-Sabin Surfaces). One step of NURDS refinement canbe factored into one non-uniform linear subdivision stepplus one dual step. Compared to the prior non-uniformDoo-Sabin surfaces (i.e., quadratic NURSSes), NURDSesare convergent for arbitrary n-sided faces. Closed formlimit point rules, which are important for applications inadaptive rendering and NC machining, are given as well.

Guoping WangPeking University, PR Chinawgp@pku.edu.cn

Zhangjin HuangUniversity of Science & Technology of Chinazhuang@ustc.edu.cn

CP0

Adaptive Isogeometric Analysis Using RationalPht-Splines

Both NURBS-based IGA and T-splines-based IGA showgreat advantageous compared to FEA with respect to theconvergence of the analysis and accuracy of the results, butsuffer from the drawback of a purely local refinements. Inthis paper, we extend PHT-splines to Rational form, andexplore Rational PHT-splines as the basis for analysis. Aresidual-based posteriori error estimator is derived to guidea local h-refinements process adaptively. The numerical re-sults show that RPHT-splines are most promising as basisfor IGA and the given posteriori error estimator and therefinement strategy are efficient and reliable.

Ping Wang, Jinlan XuUniversity of Science and Technology of Chinapingwang@mail.ustc.edu.cn, jinlanxu@mail.ustc.edu.cn

Jiansong Deng, Falai ChenDepartment of MathematicsUniversity of Science and Technology of Chinadengjs@ustc.edu.cn, chenfl@ustc.edu.cn

CP0

Efficiently Computing Geodesic Offsets on TriangleMeshes by Extended Xin-Wang Algorithm

Geodesic offset curves are important for many industrialapplications, such as solid modeling, robot-path planning,

the generation of tool paths for NC machine, etc. Al-though it is well studied in classical differential geometryand computer-aided design, where the underlying surfaceis sufficiently smooth, very few algorithms are available forcomputing geodesic offsets on discrete representation, inwhich the input is typically a polyline curve restricted onthe piecewise linear mesh. In this paper, we propose an ef-ficient and exact algorithm to compute the geodesic offsetson triangle meshes by extending the Xin-Wang algorithmof discrete geodesics. We define new data structure forparallel-source windows, and extend both the “one angleone split’ and the filtering theorem to maintain the win-dows tree. Similar to the original Xin-Wang algorithm,our extended algorithm has an O(n) space complexity andan O(n2 log n) asymptotic time complexity, where n is thenumber of vertices on the input mesh. We tested our al-gorithm on numerous real-world models and showed thatour algorithm is efficient and insensitive to the mesh reso-lution/tessellation.

Shi-Qing Xin, Xiang Ying, Ying HeNanyang Technological UniversitySQXIN@ntu.edu.sg, ying0008@ntu.edu.sg, yhe@ntu.edu.sg

CP0

Computing Shortest Words Via Shortest Loops onHyperbolic Surfaces

Computing the shortest homotopy representations forloops on general surfaces is an NP-hard problem. However,for surfaces allowing hyperbolic metrics and equipped withcanonical homotopic generators, this problem can be re-duced to finding homotopic shortest loops. In this paper,we propose an efficient solution to compute the shortestwords on triangulated surfaces of this special case. Theframework uses hyperbolic metrics and transient embed-ding to compute shortest paths and shortest loops, whichgives rise to a solution of the shortest words problem. Sev-eral techniques are employed to relieve numerical errors.Experimental results are given to demonstrate the perfor-mance in practice.

Xiaotian YinHarvard Universityxyin@math.harvard.edu

Yinghua LiStony Brook Universityyinli@ic.sunysb.edu

Wei HanHarvard Universityweihan@math.harvard.edu

Feng LuoRutgers Universityfluo@math.rutgers.edu

David GuState University of New York at Stony Brookgu@cs.sunysb.edu

Shing-Tung YauDepartment of MathematicsHarvard Universityyau@math.harvard.edu

32 GD11 Abstracts

CP1

Cubic Helical Splines with Frenet-Frame Continu-ity

We present a scheme of interpolating a sequence of pointsin space by a piecewise-cubic-helical spline with continuousFrenet frame. Each cubic helical segment connecting adja-cent data points is constructed such that its end tangentsare symmetric with respect to the displacement vector ofthe data points. The existence condition of such a seg-ment is formulated in terms of the configuration involvingthe displacement vector and the Frenet frame of the seg-ment at the initial point. A Frenet-frame-continuous splineis obtained by matching the Frenet frames of sequentiallyconstructed segments at their juncture points.

Chang Yong HanDepartment of Applied MathematicsKyung Hee Universitycyhan@ymail.com

Song-Hwa KwonThe Catholic University of Koreaskwon@catholic.ac.kr

CP1

Quaternionic Bezier Curves and Surfaces

Quaternionic Bezier (QB) curves and surfaces in 3-spaceare defined by Bezier-like formulas with quaternionicweights. Any rational Bezier curves and surfaces on 2-spheres can be converted into QB-form of twice lower de-gree (e.g., a circular arc is a linear QB-curve). Using im-plicitization formula for general bilinear QB-surfaces it isshown that they are patches on quartic surfaces called Dar-boux cyclides. In particular principal patches of Dupin cy-clides can be parametrized by special bilinear QB-surfaces.

Rimvydas Krasauskas, Severinas ZubeFaculty of Mathematics and InformaticsVilnius Universityrimvydas.krasauskas@maf.vu.lt, severinas.zube@mif.vu.lt

CP1

Linear Methods for Degree Reduction with Geo-metric Continuity

Degree reduction of Bezier curves and surfaces facilitatesdata exchange, compression, transfer, and comparison. Ex-isting methods for degree reduction with geometric conti-nuity constraints at the end points are non-linear. In thistalk, we present linear methods for degree reduction withgeometric continuity constraints at the end points.

Stephen MannUniversity of WaterlooCheriton School of Computer Sciencesmann@uwaterloo.ca

Abedallah RababahDepartment of Mathematics and StatisticsJordan University of Science and Technologyrababah@just.edu.jo¿

CP1

Singularity of Cubic Bezier Curves and Surfaces

Parametric cubic polynomial curves and surfaces are use-ful in applications, being of relatively low dimension, andyet, flexible in their shape. To use these curves and surfacesfully, one must understand the cases of singularity. A para-metric curve is singular where its derivative is zero, and aparametric surface, where its normal vector is zero. Thesesingularities are described in terms of the Bezier form of aparametric polynomial curve and surface.

Edmond NadlerUniversity of Michigannadler@alum.mit.edu

Tae-Wan KimSeoul National University, Seoul, KoreaDepartment of Naval Architecture and Ocean Engineeringtaewan@snu.ac.kr

CP1

Rational Pythagorean-Hodograph Space Curves

A method for constructing rationalPythagorean-hodograph (PH) curves in Euclidean spaceis proposed, based on prescribing a field of rational unittangent vectors. This tangent field, together with its firstderivative, defines the orientation of the curve osculatingplanes. Augmenting this orientation information with a ra-tional support function, that specifies the distance of eachosculating plane from the origin, then completely defines aone-parameter family of osculating planes, whose envelopeis a developable ruled surface. The rational PH space curveis identified as the edge of regression (or cuspidal edge) ofthis developable surface. Such curves have rational para-metric speed, and also rational adapted frames that satisfythe same conditions as polynomial PH curves in order tobe rotation-minimizing with respect to the tangent. Thekey properties of such rational PH space curves are de-rived and illustrated by examples, and simple algorithmsfor their practical construction by geometric Hermite in-terpolation are also proposed.

Zbynek SirCharles University in PragueCzech Republiczbynek.sir@gmail.com

Rida FaroukiUniversity of California, Davisfarouki@ucdavis.edu

CP1

Some Characteristics of Log-Aesthetic PlanarCurves

Log-aesthetic planar curves are spiral curves that are thegeneralization of the Clothoid, Nielsen’s spiral, and the log-arithmic spiral by shape parameter α. We investigate somecharacteristics of log-aesthetic curves, such as their evo-lutes, and the theoretical boundaries of the drawable re-gions of log-aesthetic curves. We first provide a proof thatthe evolutes of log-aesthetic curves are also log-aestheticcurves. We then provide a method for drawing a curve seg-ment with the theoretical boundary when α < 0 or α > 1.

Norimasa Yoshida

GD11 Abstracts 33

Nihon Universitynorimasa@acm.org

Takafumi SaitoTokyo University of Agriculture and Technologytxsaito@cc.tuat.ac.jp

CP2

Advances in Online Reconstruction with Laser-Scanners

The operator of an hand-held laser-scanner for reverse en-gineering needs to scan enough details for CAD reconstruc-tion. We improved our online triangulation method thatsupports the operator in this task. For CAD reconstructionit is not necessary to scan the whole object. Parameters ofregular geometry can be determined from a much smallerset of point data. We are currently working on an onlineCAD reconstruction that segments the point data and fitsregular geometry.

Klaus DenkerTU Kaiserslauternkdenker@htwg-konstanz.de

Georg UmlaufHTWG Konstanzumlauf@htwg-konstanz.de

CP2

Surface Mesh Extrusion on the GPU

This presentation explores the details of surface meshextrusion accelerated using GPU computing throughNVIDIA’s CUDA technology. Topics include choosingGPU-friendly mesh data structures, optimizing GPU ker-nel execution, and detecting mesh self-intersection on theGPU.

Michael S. JefferiesPointwise, Incmsj@pointwise.com

CP2

Designing Multiscale Kernels on Meshes

The recent realization that a multiscale kernel, namely theheat kernel, allows extracting information about shapes atmultiple levels was found useful in applications such as dis-tance measurement, segmentation, shape matching and re-trieval. To increase the scope of applications and to pro-vide a greater assortment of choices, there is a need for newmultiscale kernels. In this talk we introduce a general prin-ciple for constructing multiscale kernels on surface meshes,and present a construction of the multiscale pre-biharmonicand multiscale biharmonic kernels. The resulting kernelshave gradually changing supports, consistent behavior overpartial and complete meshes, and interesting limiting be-haviors (e.g. in the limit of large scales, the multiscalebiharmonic kernel converges to the Green’s function of thebiharmonic equation); in addition, these kernels are basedon intrinsic quantities and so are insensitive to isometricdeformations. We show empirically that our kernels areshape-aware, robust to noise, tessellation, and partial ob-ject, and are fast to compute. Finally, we demonstratethat the new kernels are useful for function interpolation

and shape correspondence.

Raif M. RustamovDepartment of Mathematics and Computer ScienceDrew Universityrrustamov@drew.edu

CP2

Interactive Surface Modeling Using Modal Analy-sis

We propose a framework for deformation-based surfacemodeling that is interactive, robust and intuitive to use.The deformations are described by a non-linear optimiza-tion problem that models static states of elastic shapesunder external forces which implement the user input. In-teractive response is achieved by a combination of modelreduction, a robust energy approximation, and an efficientquasi-Newton solver. Motivated by the observation thata typical modeling session requires only a fraction of thefull shape space of the underlying model, we use secondand third derivatives of a deformation energy to constructa low-dimensional shape space that forms the feasible setfor the optimization. Based on mesh coarsening, we pro-pose an energy approximation scheme with adjustable ap-proximation quality. The quasi-Newton solver guaranteessuperlinear convergence without the need of costly Hessianevaluations during modeling.

Christoph Von Tycowicz, Klaus Hildebrandt, ChristianSchulzFreie Universitat Berlinchristoph.vontycowicz@fu-berlin.de,klaus.hildebrandt@fu-berlin.de,christian.schulz@fu-berlin.de

Konrad PolthierFreie Universitat Berlin, Germanykonrad.polthier@fu-berlin.de

CP2

Noise-Resistant, Feature-Preserving, High-OrderSurface Reconstruction for Surface Meshes

We consider the problem of reconstructing a high-ordersurface from a given surface mesh. We introduce two meth-ods, called Weighted Averaging of Local Fittings (WALF)and Continuous Moving Frames (CMF), both based onweighted least squares polynomial fittings and guaranteeC0 continuity. Our methods are applicable to surfacemeshes composed of triangles and/or quadrilaterals, canachieve third- and even higher order accuracy, and haveintegrated treatments for sharp features.

Duo Wang, Xiangmin JiaoStony Brook Universityduowang@ams.sunysb.edu, jiao@ams.sunysb.edu

CP2

Sharp Feature Preserving Mls Surface Reconstruc-tion Based on Local Neighborhood Modifications

Most point sampled geometries from manufactured and de-signed objects contain sharp features. A standard MLSfitting produces smooth surfaces, but also smoothes sharpfeatures.To preserve those features, we modify the local neighbor-hoods the MLS fitting is applied to. In a first step, sharp

34 GD11 Abstracts

feature points are marked in the point set. Then, the se-lected feature points are used to modify the local neigh-borhoods only in the presence of sharp features. MLS isapplied afterwards.

Christopher WeberTechnische Universitat Kaiserslauternchristopherweber80@googlemail.com

Stefanie HahmannLaboratoire Jean KuntzmannUniversity of Grenoble, FranceStefanie.Hahmann@imag.fr

Hans HagenDepartment of Computer ScienceUniversity of Kaiserslauternhagen@informatik.uni-kl.de

Georges-Pierre BonneauGRAVIR-IMAG, GrenobleFRancegeorges-pierre.bonneau@imag.fr

CP3

Iso-Geometric Flow Simulations Using Catmull-Clark Finite Elements

We present a novel technique to solve the incompressiblesteady-state Navier-Stokes equations for flow simulations.The proposed method is an iso-geometric approach basedon Catmull-Clark subdivision solids to integrate design andthe analysis phase for volumetric models of arbitrary topol-ogy. This allows us to use the same representation for themodeling, the physical simulation, and the visualization,which optimizes the development process and narrows thegap between design and simulation. The underlying ge-ometric models are Catmull-Clark surfaces with optionalcorners and creases. Hence, these models can be simplyrefined to increase the accuracy of the simulation. The cru-cial point in the simulation phase is the ability to performparameter evaluation at any location during the simula-tion at arbitrary parameter values. We propose a methodsimilar to the standard subdivision surface evaluation tech-nique, such that numerical quadrature can be used. Thenon-linear system of equations which arises during the sim-ulation is solved by Newton’s method. Experiments showthat our approach approximately requires the same num-ber of iterations as linear Lagrangian hexahedral finite el-ements for the same number of degrees of freedom. AsCatmull-Clark finite elements are tri-cubic we can achievehigher accuracy with the same number of degrees of free-dom and almost the same computational cost. Increasingly,it is computationally desirable to closely couple geometrydesign, alteration and representation with the actual simu-lation operating on the geometry. Iso-geometric design andmodeling is an evolving area that is aiming to address theissues related to the close coupling and integration of ge-ometry and simulation, and our methods presented providea novel approach in support of this trend.

Daniel BurkhartTU Kaiserslautern, Germanyburkhart@cs.uni-kl.de

Bernd HamannUniversity of CaliforniaDepartment of Computer Sciencehamann@cs.ucdavis.edu

Georg UmlaufHTWG Konstanzumlauf@htwg-konstanz.de

CP3

Subdivision-Based Modeling for IsogeometricAnalysis

The main concept of isogeometric analysis is to directlyuse basis functions of the computer aided design (CAD)geometry for representing solutions of the physical space.This talk presents some ongoing work on isogeometric anal-ysis using subdivision-based representations. The generalmethodology is first discussed. Different modelling optionsare briefly highlighted. Key issues in using subdivision-based modelling for isogeometric analysis are further ad-dressed. Some open problems will also be discussed.

Weiyin MaCity University of Hong KongDept Manufacturing Engineering and EngineeringManagementmewma@cityu.edu.hk

Xiaoyun YuanCity University of Hong Kongxyuan9@student.cityu.edu.hk

CP3

Parallel Particle Tracking Implementation onGPUs for Unstructured Grids

Particle tracking methods are often used to model ground-water contaminant transport since they provide solutionsfree of numerical dispersion. Typical applications involvedomains that are irregularly discretized. Unstructuredmeshes use fewer nodes, but require additional computa-tion in a parallel setting, whereas brute-force techniquesusing highly refined structured meshes are easily paral-lelized. A particle tracking algorithm designed to run inparallel on GPUs is presented. Scenarios with structuredand unstructured meshes are discussed in detail.

Gowri SrinivasanT-5, Theoretical Division, Los Alamos NationalLaboratorygowri@lanl.gov

CP3

Regularity and Approximation Properties for Sin-gular Parameterizations in Isogeometric Analysis

Isogeometric Analysis is a numerical method which usesthe NURBS-based representation of CAD-models. Sin-gularly parameterized surfaces are used to represent non-quadrangular domains without splitting. In the presence ofsingularities the test functions do not necessarily fulfill therequired regularity and approximation properties. Consid-ering two types of singularities, we derive conditions whichguarantee the regularity of the test functions and show ap-proximation properties of the function spaces. We suggesta parameterization strategy depending on the consideredproblem.

Thomas TakacsJohannes Kepler University Linzthomas.takacs@jku.at

GD11 Abstracts 35

Bert JuttlerJohannes Kepler Univ. LinzDepartment of Applied Geometrybert.juettler@jku.at

CP3

Implicitized Distance Fields for ComputationallyEfficient Point Containment Checks in Isogeomet-ric Models

In recent research, we developed a hierarchical compo-sitional philosophy mirroring constructive solid geometry[D. Natekar, X. Zhang, and G. Subbarayan. ”Construc-tive Solid Analysis: A Hierarchical, Geometry based Mesh-less Procedure for Integrated Design and Analysis.” Com-puter Aided Design, vol. 36, pp. 473-486, 2004; M.Rayasam, V. Srinivasan, G. Subbarayan, ”CAD inspiredhierarchical partition of unity constructions for NURBS-based, meshless design, analysis and optimization,” Inter-national Journal for Numerical Methods in Engineering,vol. 72, pp. 14521489, 2007] for isogeometric analysis bycomposing (using function space formalism) functional ap-proximations for the geometry and its associated behav-ioral field. In compositional procedures, point-containmentchecks identify relative contribution of each primitive to theoverall approximation, but are expensive to calculate ateach integration point. Here, we describe an algebraic, im-plicitized procedure to carryout point containment checksinexpensively without surface intersection calculations.

Kritika Upreti, Ganesh SubbarayanPurdue Universitykupreti@purdue.edu, ganeshs@purdue.edu

CP4

Efficient Computation of Electrostatic Interactionsin Bio-Molecules in the Generalized Born ModelVia the R6 Effective Radii

Constructing physically realistic and computationally ef-fective representation of the molecular surface remainschallenging. We illustrate some of the challenges and de-scribe a solution in the context of the generalized Bornmodel, a popular analytical approximation for computingelectrostatic interactions in biomolecules. The proposedsolution is based on the |r|−6 integral over an approxima-tion to the molecular volume. Overall, the solution offersa reasonable balance between efficiency and accuracy, es-pecially if applied to small drug-like molecules.

Boris A. AguilarDepartment of Computer ScienceVirginia Techbaguilar@cs.vt.edu

Alexey V. OnufrievDepartments of Computer Science and PhysicsVirginia Techalexey@cs.vt.edu

CP4

A Semi-Implicit Method of Mean Curvature Flow

We propose a new semi-implicit method for mean-curvature flow using a triangulated surface. For spatialdiscretization, our method computes curvatures using aweighted least-squares approximation. To address the stiff-ness of the differential equation, we propose a semi-implicit

method that enhances the stability and allows for a sub-stantial enlargement of the time step. We present exper-imental comparisons with other methods to demonstratethe accuracy and stability of our proposed method.

Duo Wang, Bryan Clark, Xiangmin JiaoStony Brook Universityduowang@ams.sunysb.edu, blclark@ams.sunysb.edu,jiao@ams.sunysb.edu

CP4

Equilibrium Shapes of Two-Component Vesiclesunder Spontaneous Curvature

The modeling of biological structures, known as vesicles,is based on the minimization of its membrane energy (orshape energy). Most of the current vesicle models assumethat bending energy is the dominating factor in determin-ing vesicle shape. Our study analyzes vesicle shape basedon a different source, known as spontaneous curvature.Through numerical simulation, we show that the presenceof spontaneous curvature leads to many shapes that areboth interesting and, we believe, experimentally relevant.

Geoffrey W. CoxVirginia Military Institutecoxgw10@vmi.edu

CP4

DeWaLoP Robot - Dynamical Independent Sus-pension System

This work describes the geometrical design of the Dynam-ical Independent Suspension System (DISS) for the De-veloping Water Loss Prevention DeWaLoP in-pipe robot,which objective is to redevelop the pipe-joint of fresh watersupply system of Vienna and Bratislava. The DISS enablesthe wheeled-legs from the robot system to expand, creat-ing a rigid structure inside the pipe, so the redevelopmenttools work; or to compress, becoming a mobile platformable to move inside the pipelines.

Luis A. MateosAutomation and Control Institute (ACIN)Vienna University of Technologymateos@acin.tuwien.ac.at

CP4

Configurations of Closed Diblock Fibers

We analyze the conformations of closed diblock fiberswhich consist of two blocks with different bending rigidi-ties and spontaneous curvatures. In each fiber, one blockis a bare polymer while the other is an adsorbed protein-filament complex. The length fraction of each componentand the total fiber length is controlled by tunable chem-ical potentials. We analytically calculate the shape ofthese two-component polymers for all values of the ma-terial parameters and chemical potentials. Our resultsyield: a complete analytical description of all possible two-component polymer conformations in two and three dimen-sions, a phase portrait detailing the parameter spaces inwhich these shapes occur, and the identification of sponta-neous transitions between shapes driven by environmentalchanges.

Sumanth SwaminathanNorthwestern University, Chicagos-swaminathan@northwestern.edu

36 GD11 Abstracts

Francisco SolisArizona State Universityfrancisco.solis@asu.edu

Monica Olvera De La CruzNorthwestern Universitym-olvera@northwestern.edu

CP4

Finite Element Approximation in Shape Optimiza-tion Problems with Neumann Or Mixed BoundaryConditions

For optimal design problems, defined in domains of classC and in arbitrary space dimension, governed by ellip-tic equations with boundary conditions of Neumann ormixed type, we introduce the corresponding discretizedproblems and we prove convergence results. The discretiza-tion method is of fixed domain type, in the sense that it isgiven in the domain that contains all the admissible opensets.

Dan I. TibaInstitute of Mathematics, Romanian AcademyBucharest, RomaniaDan.Tiba@imar.ro

CP5

Using Implicit Equations of Parametric Curves andSurfaces Without Computing Them: PolynomialAlgebra by Values

In many geometric problems involving algebraic curves andsurfaces, such as the point position problem, the equationsto be solved involve polynomials of huge size, which areextremely difficult to work with. We show that, taking asinput a set of values of the considered polynomials, the so-lution of the equations can be reduced to generalized eigen-value problems on matrix pencils, with big, but structuredand sparse numerical matrices.

Mario FioravantiUniversidad de Cantabria, Spainmario.fioravanti@unican.es

Gema M. Diaz-TocaUniversidad de Murcia, Spaingemadiaz@um.es

Laureano Gonzalez-VegaUniversidad de CantabriaDpto. Matematicaslaureano.gonzalez@unican.es

Azar ShakooriUniversidad de Cantabria, Spainazar.shakoori@gmail.com

CP5

Complete Path Planning in the Plane

We present a complete path planning algorithm for a planerobot with three degrees of freedom and a static obsta-cle. The part boundaries consist of linear and circularedges. The algorithm constructs and searches a combinato-rial representation of the robot free space. Its complexity isO((n4 + c3) log n) with n the input size and c3 the numberof configurations with three simultaneous contacts between

robot and obstacle edges. The algorithm is implementedrobustly using our controlled linear perturbation strategy.The program handles passages whose clearance is 10−9.The running time is independent of the clearance.

Victor MilenkovicUniversity of Miamivjm@cs.miami.edu

Elisha SacksPurdue Universityeps@cs.purdue.edu

Steven TracEpicstrac@epic.com

David EinsteinUniversity of Massachusetts Lowelldeinstei@cs.uml.edu

Adam McMahonUniversity of Miami,adam@cs.miami.edu

CP6

Concave and Alpha Hulls of a Set of Freeform Pla-nar Closed Curves

Given a set of points, what is the region occupied by them?Convex hull, concave hull, α-hull, poly hull, r-shape ands-shape have been proposed. In this paper, we extend thequestion to a set of close planar freeform curves (repre-sented as NURBS) without sampling them. We proposeconcave hull by defining it and also present an algorithm.Also the relation between α-hull (extended from point-sets)and concave hulls is explored.

Ramanathan Muthuganapathy, Vishwanath A.V., ArunSrivatsanIndian Institute of Technology Madrasemry01@gmail.com, vishwanathavin@gmail.com, rarunsri-vatsan@gmail.com

CP6

Medial Zones As ’Thick’ Skeletons

We formally define the new concept of a medial zone ofan n-dimensional semi-analytic domain Ω that subsumesthe medial axis MA(Ω) of the same domain as a specialcase, and can be thought of as a ‘thick’ skeleton havingthe same dimension as that of Ω. We will review several oftheir appealing topological and computational propertiesand discuss a paradigm to construct them for 3D semi-analytic domains. Due to the fact that the medial zonesfuse some of the critical geometric and topological prop-erties of both the domain itself and of its medial axis, re-formulating problems in terms of medial zones affords the‘best of both worlds’ in applications such as robotic andautonomous navigation, and design automation.

Horea T. Ilies, Ata EftekharianUniversity of Connecticutilies@engr.uconn.edu, ata.eftekharian@gmail.com

CP6

Estimating Penetration Depth of Convex Polyghe-

GD11 Abstracts 37

dra Using Dynamic Minkowski Sum

In this talk we will present our recent results on estimatingthe penetration depth of two overlapping convex polyhe-dra. In the heart of our approach is a method that canefficiently update the Minkowski sums of the convex poly-hedra without re-computing every Minkowski sum fromscratch. We show that this new penetration depth estima-tion method provides theoretical guarantee to find a globalminimum and is efficient, in particular when the rotationis small between frames.

Yanyan Lu, Stephen Donnelly, Evan Behar,Jyh-Ming LienGeorge Mason Universityylu4@masonlive.gmu.edu, sdonnel4@gmu.edu,behare@gmail.com, jmlien@cs.gmu.edu

CP6

3D Constructive Ma Generation

Medial axis (MA) is a simplified representation of a model.A novel constructive approach for the MA generation of3D models is proposed based on double queue algorithm(DQA). This approach can generate MA of both a singlemodel and a resultant model constructed from modifyingMAs of two operand models via a Boolean operation in-stead of regenerating the MA from scratch.

Yusheng LiuZhejiang Universityysliu@cad.zju.edu.cn

Housheng ZhuState Key Lab. of CAD&CG, Zhejiang Universitycaremoon@126.com

CP7

Smoothness Analysis of Triangular Subdivision Al-gorithms

A C1 analysis technique for infinite classes of subdivisionschemes for arbitrary quadrilateral meshes has been devel-oped by the authors and the class of subdivision schemeswhich includes simplest and midpoint schemes and all theirpossible combinations has been analyzed. In this talk, wediscuss some issues in applying this technique to triangu-lar subdivision algorithms. In particular, a new infiniteclass of triangular schemes is defined, and we analyze thesmoothness of their subdivision surfaces.

Qi ChenApplied Geometry & Computer GraphicsUniversitaet Karlsruhechenqi@ira.uka.de

Hartmut PrautzschUniversitaet Karlsruheprau@ira.uka.de

CP7

Parameterizing Subdivision Surfaces

We present a method for parameterizing subdivision sur-faces in an as-rigid-as-possible fashion. While much workhas concentrated on parameterizing polygon meshes, littlework has focused on subdivision surfaces despite their pop-ularity. We show that polygon parameterization methods

produce suboptimal results when applied to subdivisionsurfaces and describe how these methods may be modi-fied to operate on subdivision surfaces. We also describe amethod for creating extended charts to further reduce thedistortion of the parameterization.

Lei HeTexas A&M Universityleih@cs.tamu.edu

Kai HormannUniversity of Luganokai.hormann@usi.ch

Scott SchaeferTexas A&M Universityschaefer@cs.tamu.edu

CP7

Recent Advances in NURBS-Compatible Subdivi-sion

While subdivision based on uniform B-splines is very wellunderstood now, non-uniform bivariate schemes still posesome challenges. In regular regions, one can rely on uni-variate results and use tensor products. However, in ex-traordinary regions, the situation is much more complex.A real breakthrough came only recently with the thesisof Thomas Cashman and resulted in NURBS-compatiblesubdivision. Although this framework has answered manyquestions, it cannot handle even-degree schemes and mul-tiple knots in extraordinary regions. Solutions to some ofthese shortcomings will be presented in the talk.

Jiri KosinkaUniversity of OsloCentre of Mathematics for Applicationsjiri.kosinka@cl.cam.ac.uk

Neil DodgsonComputer LaboratoryUniversity of Cambridgeneil.dodgson@cl.cam.ac.uk

Malcolm SabinNumerical Geometry Ltdmalcolm@geometry.demon.co.uk

CP7

Error in Multivariate Polynomial Interpolation

Using polynomials to interpolate bivariate data has mixedresults: sometimes the interpolating surfaces are good butother times the interpolating surfaces fluctuate wildly. Inthis talk, we look at the error term of the interpolant, not-ing that any fixed basis will produce poor results for somedata sets, and we explain why schemes such as the Leastusually produce good interpolants.

Stephen MannUniversity of WaterlooCheriton School of Computer Sciencesmann@uwaterloo.ca

Kirk HallerIndustrial Light and Magickirkhaller@gmail.com

38 GD11 Abstracts

CP7

G1 Bzier Surface Interpolation of Boundary Curveswith T-junction

A T-junction is the junction point of two boundary curves,where one boundary curve starts in the middle of an-other boundary curve. We propose the piecewise G1 Bziersurface interpolation method from the boundary curveswith T-junction in which none of the boundary curves arechanged. The key idea of this research is that two mi-cro surfaces that are subdivided by the T-junction can beconsidered one macro surface. This yields a G1 continu-ity problem between two macro patches. With this condi-tion, we add the vertex G1 condition between two micropatches related with the T-junction. A T-junction can oc-cur also at a 3-valent vertex. Two of three boundary curvesthat are collinear at a vertex, also define a T-junction. Wegive a necessary condition to interpolate G1 Bzier surfacesand suggest a subdivision method with three rectangularsub-patches at this type of T-junction. In this research,we solve the vertex G1 continuity constraints and edge G1

continuity constraints independently and we propose con-straints using scalar weight functions for making consistentlinear systems of the vertex and edge G1 systems. We showthe shaded models and the reflection lines, and check theangles of the resulting surfaces to verify the G1 continuity.

Min-jae OhSeoul National UniversityRepublic of Koreamjoh80@snu.ac.kr

Kittichai SuthunyatanakitSeoul National Universitysuthunyatanakit@gmail.com

Tae-Wan KimSeoul National University, Seoul, KoreaDepartment of Naval Architecture and Ocean Engineeringtaewan@snu.ac.kr

CP7

A Herarchical Approach for Generalised B-Splines

One promising alternative to classical tensor-productsplines is given by Hierachical B-splines recently discussedin Juttler et al. (2011). In this approach B-spline ba-

sis functions have been considered as the main ‘buildingblock”. In this talk we shall present a first study about thepossibility to extend the Hierachical approach to General-ized B-splines which have been recently used as an alter-native to NURBS in IgA.

Francesca PelosiUniversity of Rome ”Tor Vergata”Department of Mathematicspelosi@mat.uniroma2.it

Carla ManniDepartment of MathematicsUniversity of Romemanni@mat.uniroma2.it

Maria Lucia SampoliUniversity of SienaDepartment of Mathematics and Computer Sciencesampoli@unisi.it

CP8

Biharmonic Volumetric Mapping Using Fundamen-tal Solutions

We propose a biharmonic volumetric mapping frameworkbased on fundamental solutions. Unlike the traditional har-monic map that only allows boundary positional conditionswith C0 continuity, biharmonic mapping allows users toflexibly control boundary constraints on both positions andnormal derivatives and obtain C1 transition along bound-ary interfaces. This facilitates the parameterization ofhuge or inhomogeneous volumetric models whose mappingcan be more effectively computed by a divide-and-conquerstrategy with the help of model decomposition.

Huanhuan Xu, Xin LiLouisiana State Universityhuanxu4@cct.lsu.edu, xinli@lsu.edu

CP8

Volume Parameterization Using Sequences of Har-monic Maps

Volume parameterization has attracted significant researchefforts recently due to the emergence of isogeometric analy-sis which requires a structural mesh as an input for numer-ical analysis. We present a 2-step method for constructinga regular parameterization of a simply connected domain.We first build an injective mapping from the domain to theparameter domain by using a sequence of harmonic maps.Then we parameterize the original domain by spline ap-proximation of the inverse mapping.

Thien NguyenJKU, Linz, Austriathien tuan.nguyen@jku.at

Bert JuettlerJohannes Kepler University of LinzInstitute of Applied Geometrybert.juettler@jku.at

CP8

Dynamical System Modelling of Cyanobactria’sCircadian Rhythm

We consider about cyanobactria’s Circadian rhythm by useof its characteristic dynamics. We investigate it from thedeterministic and stochastic point of view to make someimportant properties of cyanobactria’s Circadian rhythmbe comprehensible mathematically.

Isamu OhnishiHiroshima Universityisamu o@math.sci.hiroshima-u.ac.jp

CP8

Adaptive Grid Generation Using the Compositionof Tensor Product B-Splines

We describe an adaptive grid generation technique basedon the composition of tensor product B-spline mappings,T∗ = T◦Φ, whereΦmaps the unit square onto itself and Tmaps the square onto the desired physical domain. Tech-nique allows us to reparameterize grid boundary curveswithout disturbing the accuracy of the spline boundaryapproximation. Application to interactive visualizationsin the NIST Digital Library of Mathematical Functions

GD11 Abstracts 39

(http://dlmf.nist.gov) will be shown.

Bonita V. SaundersNational Institute of Standards and Technology (NIST)bonita.saunders@nist.gov

CP8

Contouring Discrete Indicator Functions

We present a method for calculating the boundary of ob-jects from Discrete Indicator Functions. Although March-ing Cubes is effective at calculating contours of functionssampled over discrete grids, it performs poorly when con-touring non-smooth functions and generates surfaces ex-hibiting aliasing and oscillations. We derive a simple vertexplacement method that removes these artifacts, is efficient,easy to implement, and does not require any optimizationor iteration.

Jason Smith, Josiah M. Manson, Scott SchaeferTexas A&M Universityagjdsmith09@gmail.com, josiahmanson@gmail.com, schae-fer@cs.tamu.edu

CP8

Automating Spline Tessellation

This talk presents tight geometric and parametric boundsthat ensure that splines are rendered without artifacts inthe standard graphics pipeline. An implementation on theGPU shows little overhead.

Young YeoUniversity of Floridayiyeo@cise.ufl.edu

Lihan BinAdvanced Micro Devicelihan.bin@amd.com

Jorg PetersUniversity of Floridajorg@cise.ufl.edu

MS1

Challenges for CAD Systems

This presentation will address some of the issues the CADindustry is currently facing, for example the problem ofcapturing design process knowledge in parametric modelsin a usable form.

George AllenSiemensgeorge.allen@siemens.com

MS1

Challenges in Isogeometric Analysis (IGA)

IGA replaces traditional Finite Elements by watertightstructures of 3-variate NURBS. The introduction ofNURBS in FEA allows accurate representation of CAD-shapes in FEA. However, efficient technology for establish-ing watertight IGA-models from CAD-models has to bedeveloped, local refinement of IGA-models has to be im-proved, and direct GPU-based visualization of IGA-modelshas to be developed. IGA potentially simplifies analysis ofas-is models as the path from measurement through CAD

to analysis is shortened.

Tor DokkenSINTEF ICT, Department of Applied MathematicsOslo, Norwaytor.dokken@sintef.no

MS1

Towards a New Surface Modeling Capability

Four years ago in the Forward Looking Session in this con-ference series, several challenge problems were posed byindustrial participants. This talk presents recent progressmade towards the solution to one of those challenge prob-lems.

Thomas A. GrandineApplied MathematicsThe Boeing Companythomas.a.grandine@boeing.com

MS1

Challenges in Topology-based Modeling

There is a growing need for computational topology algo-rithms in many research areas in engineering and life sci-ences. In this talk we present fundamental techniques andtheir applications in fiber modeling, urban planning andcomputational fluid dynamics.

Hans HagenDepartment of Computer ScienceUniversity of Kaiserslauternhagen@informatik.uni-kl.de

MS2

Parametrization in Isogeometric Analysis

We discuss ways to extend a parametrization of the bound-ary of a domain to the interior. This is important in a shapeoptimization context where the boundary is changed in ev-ery optimization cycle. We will in particular study theWinslow functional which turns out to have some desir-able properties. Minimizing the Winslow functional is toocostly during the optimization. Instead a linearized versioncan be used and the full Winslow functional is only usedwhen necessary.

Jens GravesenTechnical University of Denmarkj.gravesen@mat.dtu.dk

Anton Evgrafov, Dang Manh NguyenDepartment of MathematicsTechnical University of Denmarka.evgrafov@mat.dtu.dk, d.m.nguyen@mat.dtu.dk

MS2

Optimization and Modeling of 3D Scanner Data

As lasers and other scanners become more accurate andaffordable, their use in industry is increasing. This is pre-senting new challenges in generating surface models fromdiscrete data. Creating these models involves data fusion,statistical analysis, constrained data fitting of noisy data,and more. We will present some applications of 3D scan-ners around The Boeing Company and then discuss howwe frame and solve the associated data fitting problems as

40 GD11 Abstracts

problems in constrained optimization.

Thomas A. HoganThe Boeing Math Groupthomas.a.hogan@boeing.com

MS2

Isogeometric Shape Optimal Design Based on aNovel Class of Conservative Convex Approxima-tions

We present an isogeometric shape optimisation frameworkwith multilevel design considerations and shape regulari-sation features. A novel class of globally convergent, con-servative convex approximations is proposed to formulateand solve the design problems. In addition, formulation offrequently employed geometric constraints is considered.Validity and efficiency of the framework is demonstratedthrough structural optimisation of thin-walled shells.

Attila P. Nagy, Mostafa Abdalla, Zafer GurdalDelft University of Technologya.p.nagy@tudelft.nl, m.m.abdalla@tudelft.nl,z.gurdal@tudelft.nl

MS2

B-Spline Based Shape and Topology Optimization

In this talk, I present our recent work on the use of B-splinebased finite elements in shape and topology optimization.In shape optimization, we show how NURBS weights canbe used as design variables and how B-spline mesh can begenerated from Coons patches. Shape optimization exam-ples on linear elasticity and band gap in photonic crys-tals will be demonstrated. In topology optimization, wedemonstrate how mesh-dependency and order-dependencyin B-splines can be avoided by applying filtering in theGaussian quadrature points. Topology optimization exam-ples with minimal length control on the optimized struc-tures will be demonstrated.

Xiaoping QianIllinois Institute of Technologyqian@iit.edu

MS3

Computational Challenges in Bio-Nano Technology

Bio-nanotechnology has started to come of age. Startingfrom the early (1959) visionary ideas of physicist ProfessorRichard Feynman, the nascent field has been propelled byteams of scientists spanning multiple disciplines (biologists,chemists, computer scientists, mathematicians, physicists).There are seemingly daily reports of new discoveries of bio-logical methods for de-nuovo nano-medicine, macromolecu-lar drug delivery mechanisms, DNA computers, nanoscaleinformation storage and retrieval, self-assembly graphenesheets, carbon-nanotubes, and molecular motors, etc. Inthis talk, however I shall focus on some of the necessarycomputational and applied mathematics infrastructure andchallenges that enable and accelerate scientific discovery inbio-nano technology.

Chandrajit BajajComputer Science, ICES, CVCThe University of Texas - Austinbajaj@cs.utexas.edu

MS3

New Challenges in Curve and Surface Modeling

For each advance in curve and surface modeling technol-ogy, new refinements and capabilities are needed to respondto ever growing challenges in modeling applications. Thistalk will explore some of the new capabilities currently be-ing sought in response to new demands being placed ongeometric models.

Thomas A. GrandineApplied MathematicsThe Boeing Companythomas.a.grandine@boeing.com

MS3

Computational Impact of Discrete Differential Ge-ometry

Discrete triangle meshes are the raw input data for higherorder spline representations. Nevertheless many novel al-gorithms from discrete differential geometry are designedfor such raw data. In our symposium contribution we willdiscuss the geometry processing pipeline leading from rawdata to high quality geometry representations. Especiallyimportant is the suitability of various geometry represen-tations for numerical simulations.

Konrad PolthierFreie Universitaet BerlinFB Mathematik & Informatikkonrad.polthier@fu-berlin.de

MS3

Challenges in Virtual Tower Operations for AirTraffic Control

The SESAR (Single European Sky Air Traffic ManagementResearch) programme is a large research and developmentproject launched by the European Community. The pro-gramme is the technological and operational part of theSingle European Sky (SES) initiative to meet future ca-pacity and air safety needs. This talk will describe a smallsub-project of this huge effort, namely one dealing with theintroduction of novel concepts for 3D modelling of remote(unmanned) tower operations.

Ewald QuakInstitute of CyberneticsTallinn, EstoniaEwald.Quak@cs.ioc.ee

MS4

Analysis Aware Representations, Parameteriza-tions, and Models

Isogeometric Analysis (IA) has been proposed as a method-ology for bridging the gap between Computer Aided De-sign (CAD) and Finite Element Analysis (FEA). In orderto support design and full 3D IA, new ab initio designmethods must create suitable representations and approxi-mation techniques must include parameterization method-ologies for volumes. This presentation discusses some ofthe challenges in moving from current representations anddatafitting techniques towards this goal and demonstratesinitial results and analyses.

Elaine CohenUniversity of Utah

GD11 Abstracts 41

cohen@cs.utah.edu

MS4

Splines and Isogeometric Analysis (IGA)

The idea behind IGA was to reuse CAD-type NURBS inanalysis. CAD hides B-spline basis functions from the user,while direct access to basis functions is essential in analy-sis. Refinement of B-spline representation was available in1980 (Oslo-algorithm), local refinement of tensor productB-splines was not important before IGA. IGA has conse-quently triggered new uses of and increased research intosplines, and potentially allows interoperability of designand analysis through new requirements to spline technol-ogy.

Tor DokkenSINTEF ICT, Department of Applied MathematicsOslo, Norwaytor.dokken@sintef.no

MS4

Locally Refined B-splines

We will address local refinement of tensor product B-splines specified as a sequence of inserted line segmentsparallel to the knot lines. We obtain a quadrilateral gridwith T-junctions in the parameter domain, and a collec-tion of tensor product B-splines on this mesh here namedan LR-mesh. The approach applies equally well in dimen-sions higher than two. Moreover, in the two dimensionalcase this collection of B-splines spans the full spline spaceon the LR-mesh.

Tom LycheUniversity of OsloDepartment of Informatics and CMAtom@ifi.uio.no

MS4

Translation of CAD Models to Block StructuredIsogeometric Models

A CAD solid is implicitly represented by its bounding sur-faces while a model intended for isogeometric analysis needa trivariate representation. Moreover, CAD models arerepresented by different types of surfaces, not only NURBS,and the surfaces are frequently trimmed, while block struc-tured isogometric models are regular collections of non-trimmed NURBS volumes. It is not trivial to translate anarbitrary CAD model into an isogeometric model. SomeCAD solids, however, employ a certain regularity whichmakes an automatic translation into a model fit for isogeo-metric analysis feasible. This talk will focus on removal oftrimming from CAD surfaces by block structuring and howa block structured face set can be turned into a trivariatemodel.

Vibeke SkyttSINTEF ICT, Department of Applied MathematicsOslo, Norwayvibeke.skytt@sintef.no

MS5

Comparing Solid Representations of Protein Cavi-ties to Reveal Influences on Protein-Ligand Bind-

ing Specificity

This talk presents VASP, a new method that dissectsbiochemically active cavities in protein structures to iso-late structural elements that influence binding specificity.VASP enables these new capabilities by exploiting a novelrepresentation of protein structures based on Computa-tional Solid Geometry and Boolean set operations. Ap-plying VASP to an analysis of the major serine proteases,VASP isolated individual amino acids and regions of func-tional sites that enable the serine proteases to preferentiallybind specific molecules.

Brian ChenDept. of Computer Science and EngineeringLehigh Universitychen@cse.lehigh.edu

MS5

Molecular Surfaces for Poisson-Boltzmann Electro-statics

The Poisson-Boltzmann equation provides a model for theelectrostatic interaction of a protein with an implicit sol-vent. The resulting energy computations depend on the in-terface between the protein and the solvent, the molecularsurface. There are several molecular surface constructionseach designed for a different goal: preservation of ideal-ized geometry, smoothness, ease of computation, etc. Wecompare molecular surfaces based on their impact on thesolvation energy for a practical set of proteins.

Alexander RandUniversity of Texas-AustinInstitute for Computational Engineering and Sciencesarand@ices.utexas.edu

Chandrajt BajajThe University of Texas at AustinDepartment of Computer Sciencebajaj@cs.utexas.edu

MS5

The Geometry of Biomolecular Solvation: Hy-drophobicity and Electrostatics

Biomolecules adopt stable 3D structures and perform theirfunctions in water. Modeling the water-biomolecule inter-actions, namely, solvation, is essential for understandingthe folding, stability and activity of proteins. In implicitsolvent model, the water effects are largely determined bythe geometry of the molecular-water interface. We use theskin surface to model the surface of biomolecules and de-velop new approaches for computing the contribution ofhydrophobicity and electrostatics to form an effective sol-vation potential.

Xinwei ShiUniversity of California, DavisGenome Centerxshi@ucdavis.edu

Patrice KoehlGenome Center and Department of Computer ScienceUniversity of California, Daviskoehl@cs.ucdavis.edu

42 GD11 Abstracts

MS5

Evolution Based Skeletonization of 3D Images forSecondary Structure Elucidation in 3D Cryo EMDensity Maps

Skeletonization has been widely used as one compact wayto represent shape in many fields. This has led to a numberof computational techniques for computing skeletons. Wepropose an evolution based skeletonization method for 3Dimages which utilizes both geometric and volumetric repre-sentations of the object. Important topological informationis preserved during the evolution. We apply this approachto elucidating the secondary structures from 3D cryo EMdensity map of intermediate resolution. Comparison withseveral other elucidating approaches will be presented.

Qin ZhangCVC, Institure for Computational Engineering andSciencesthe University of Texas at Austinzqyork@ices.utexas.edu

Chandrajit BajajComputer Science, ICES, CVCThe University of Texas - Austinbajaj@cs.utexas.edu

MS6

Variational Space Deformation with BarycentricCoordinates

An important problem in Computer Graphics is shape de-formation: given a source shape and user defined con-straints, the goal is to generate a deformed shape whichfulfills the constraints and preserves the local details of theoriginal shape. We show how complex-valued barycentriccoordinates provide a natural solution to this problem intwo-dimensions, and discuss the modifications required for3D deformation, and deformation transfer.

Mirela Ben-ChenComputer Science DepartmentStanford Universitymirela@stanford.edu

MS6

Geometric Criteria for Generalized Barycentric Fi-nite Elements

Generalized barycentric coordinate functions allow fornovel, flexible finite element methods accommodatingpolygonal element geometries. The Sobolev-norm error es-timates associated to such methods, however, require vary-ing levels of geometric criteria on the polygons, dependingon the definition of the coordinate functions. In this talk, Iwill derive these criteria for a variety of coordinate defini-tions and discuss the practical tradeoffs between enforcinggeometric constraints and computing finite element basisfunctions over polygons.

Andrew GilletteUniversity of California, San DiegoDepartment of Mathematicsagillette@math.utexas.edu

MS6

Injective Barycentric Mappings on Convex Do-

mains

Wachspress and mean value coordinates and their exten-sions to smooth convex domains are generalizations of tri-angular barycentric coordinates and have recently beenused to construct mappings between convex polygons ordomains, with applications to curve deformation and im-age warping. We will show that Wachspress mappings be-tween convex domains are always injective but that meanvalue mappings can fail to be so in extreme cases. Possibleextensions of our results to convex polyhedra will also bediscussed.

Jiri Kosinka, Michael FloaterUniversity of OsloCentre of Mathematics for Applicationsjiri.kosinka@cl.cam.ac.uk, michaelf@ifi.uio.no

MS6

Constructing Barycentric Coordinates on Surfaces

This talk introduces a method for defining and efficientlycomputing barycentric coordinates with respect to poly-gons on general surfaces. Our construction is geared to-wards injective polygons (polygons that can be enclosed ina metric ball of an appropriate size) and is based on re-placing the linear precision property of planar coordinatesby a requirement in terms of center of mass, and generaliz-ing this requirement to the surface setting. We show thatthe resulting surface barycentric coordinates can be com-puted using planar barycentric coordinates with respect toa polygon in the tangent plane. We prove theoreticallythat the surface coordinates properly generalize the planarcoordinates and carry some of their useful properties suchas unique reconstruction of a point given its coordinates,uniqueness for triangles, edge linearity, similarity invari-ance, and smoothness; in addition, these coordinates areinsensitive to isometric deformations and can be used toreconstruct isometries. We show empirically that surfacecoordinates are shape-aware with consistent gross behav-ior across different surfaces, are well-behaved for differentpolygon types/locations on variety of surface forms, andthat they are fast to compute. Finally, we demonstrateeffectiveness of surface coordinates for interpolation, decalmapping, and correspondence refinement.

Raif M. RustamovDepartment of Mathematics and Computer ScienceDrew Universityrrustamov@drew.edu

MS7

Variational Hexahedral Subdivision of VolumetricDomains

In this talk I shall briefly survey the approach of varia-tional methods in hexahedral mesh quality improvement.I shall then contrast the choice of a few different high-orderenergy functionals by their Euler-Lagrange equation i.e anon-linear elliptic or biharmonic partial differential equa-tion (PDE), and the discretization formulas of the differen-tial operators. Finally, I shall describe several applicationsof these methods in producing feature preserving, qualityhexahedral parameterizations.

Chandrajt BajajThe University of Texas at AustinDepartment of Computer Sciencebajaj@cs.utexas.edu

GD11 Abstracts 43

MS7

Hex-based Representations of Volumes from Mid-structures

In this talk we discuss techniques to construct hex-basedrepresentations from input tetrahedral meshes. In partic-ular, we present methodologies to decompose a volumetricdomain into simpler pieces that employ mid-structures toestablish a structured hex-based representation. The de-composition is based on a hybrid parameterization whichmixes a polycube-like with a polar-like parameterizationto employ each one’s advantages. It guarantees that ad-jacent components have a matching parameterization onshared boundaries. The presented methodologies are basedon a midstructure which is constructed for the object in ad-vance. We discuss strategies for constructing midstructuresand look at generalizations.

Elaine Cohen, Tobias MartinUniversity of Utahcohen@cs.utah.edu, martin@cs.utah.edu

MS7

Polycube Parameterization for HexahedralRemeshing

We study the volumetric parameterization on polycube do-main and discuss its application on hexahedral remeshing.Uniform hexahedral meshes can be generated on a poly-cube domain, and by mapping this polycube to a givenvolumetric shape, we can transfer the regular hex structurefrom the polycube to the given shape. The main pipelinefor such an approach usually contains two steps: (1) con-structing the polycube domain (which determines the topo-logical structure and singularities of the hex-mesh), and (2)the volumetric mapping (which affects the angle/volumedistortions of the transferred hex-grids and thus the mesh-ing quality). We will introduce and briefly discuss severalalgorithms of hexahedral remeshing based on polycube pa-rameterization.

Xin Shane LiLouisiana State Universityxinli@cct.lsu.edu

MS7

Hexahedral Parameterization

We discuss novel techniques to fill a bounded volumetricshape with a (preferably coarse) cubical voxel structure.Among the optimization goals are alignment of the vox-els with the bounding surface as well as simplicity of thevoxel grid. Mathematical analysis of the possible singular-ities is given. The algorithm is uses a tetrahedral volumemesh plus a user given guiding frame field as input. Thenit constructs an atlas of chart functions, i.e. the parame-terization function of the volume, such that the images ofthe coordinate lines align with the given frame field. For-mally the function is given by a first-order PDE, namelythe gradients of the coordinate functions are the vectorsof the frame. In a first step, the algorithm uses a dis-crete Hodge decomposition to assure local integrability ofthe frame field. A subsequent step assures global integra-bility along generators of the first homology group andalignment a face of the boundary cube with the originalsurface boundary. All steps can be merged into solvinglinear equations. Conceptually the presented CubeCover-algorithm extends the known QuadCover-algorithm from

surface meshes to volumes meshes.

Konrad PolthierFreie Universitaet BerlinFB Mathematik & Informatikkonrad.polthier@fu-berlin.de

MS8

Improved T-splines

In this talk, I will present a variant version of T-splines.The new spline improves T-splines in the following aspectsby consctructing new spline functions. The blending func-tions: (1) are polynomial instead of rational; (2) are lin-early independent; (3) have good properties like B-splines,such as nonnegativity, partition of unity and compact sup-port. Examples are provide the modelling ability of thenew splines.

Falai ChenDepartment of MathematicsUniversity of Science and Technology of Chinachenfl@ustc.edu.cn

MS8

Isogeometric Simulation of Turbine Blades

Isogeometric analysis is a novel approach to numerical sim-ulation that has the potential to bridge the gap betweengeometric design and numerical analysis. It uses the sameexact geometry representation in all stages of the productdevelopment. It is particularly promising to apply IGA tofunctional free-form surfaces, where small changes of thegeometry can lead to major changes in the performance.The talk reports recent results obtained for the isogeomet-ric simulation of turbine blades.

Bert JuettlerJohannes Kepler University of LinzInstitute of Applied Geometrybert.juettler@jku.at

David GrossmannMTU Aero Engines, Germanydavid.grossmann@mtu.de

Helena Schlusnus, Anh-Vu VuongTU Munichn/a, vuong@ma.tum.de

MS8

Efficient Algorithms for Freeform Geometric Mod-els

We present a new approach to the development of efficientgeometric algorithms for freeform curves and surfaces. Pre-processing the given freeform geometric models and repre-senting them in a hierarchical data structure, we show thata variety of geometric algorithms can be greatly acceler-ated. We demonstrate the effectiveness of this approachby developing a real-time dynamic simulation of freeformgeometric models that is based on efficient algorithms forcollision detection and minimum distance computation.

Myung-Soo KimSchool of Computer Science and EngineeringSeoul National University, Koreamskim@snu.ac.kr

44 GD11 Abstracts

Yong-Joon KimSeoul National Universitykyj24182@naver.com

Gershon ElberTechnion, HaifaIsraelgershon@cs.technion.ac.il

MS8

Robust Modeling of Minimal Surfaces

Centroidal Voronoi tessellation (CVT) is an optimalVoronoi diagram such that each generator coincides withthe centroid of its Voronoi cell. The CVT has widely beenused for data visualization and mesh generation but seldomfor shape modeling. In this talk we show how the CVT canbe applied to modeling discrete minimal surfaces with agiven boundary. The variational formulation of the prob-lem and efficient optimization algorithms are presented.The proposed approach generates triangle meshes approxi-mating minimal surfaces with better distributed mesh ver-tices than previous methods. The extension to computingconstant mean curvature surfaces will also be discussed.

Wenping WangUniversity of Hong KongDepartment of Computer Sciencewenping@cs.hku.hk

MS9

Harmonic Analysis Meets Stochastic Geometry

In the field of Stochastic (or Integral) Geometry, the Princi-pal Kinematic Formula describes in closed form the volumein the group of rigid-body motions corresponding to whentwo convex bodies are in collision. But obtaining exactresults for nonconvex bodies has been elusive for the pastcentury. In this talk, lower and upper bounds are derivedin this case using concepts of convolution and Fourier anal-ysis on the group of rigid-body motions. Applications inautomated assembly planning will be discussed.

Greg ChirikjianJohns Hopkins UniversityDept of Mechanical Engineeringgregc@jhu.edu

MS9

Lie Group Morphology with Convolutions

There are at least three applications of Minkowski sumsin geometric modeling: sweeping, configuration-space-obstacles, and computation of workspaces . Group mor-phology generalizes these applications to non-commutativetransformations, such as rotations or affinities, by mak-ing a careful distinction between transformations and theobjects on which they act. We show that binary groupmorphology can be understood in terms of representationtheory and convolutions, which leads to new efficient andpractical algorithms for all types of Minkowski operations.

Saigopal NelaturiUniversity of Wisconsin-Madisonsaigopal@sal-cnc.me.wisc.edu

Vadim Shapiro, Mikola LysenkoUniversity of Wisconsin - Madison

vshapiro@engr.wisc.edu, mikolalysenko@gmail.com

MS9

Uniform Deterministic Sampling of RotationGroups from Successive Orthogonal Images

The ability to construct uniform deterministic samples ofrotation groups is useful in many contexts, such as molec-ular docking and medical imaging, but there are inherentmathematical difficulties that prevent an exact solution.The talk will present successive orthogonal images, an ef-fective means for uniform deterministic sampling of orthog-onal groups. The method is valid in any dimension, andanalytical bounds are provided on the sampling uniformity.Numerical comparisons with other sampling methods aregiven for the special case of SO(3). We make use of non-Riemannian distance metrics that are group-invariant andlocally compatible with the Haar measure. In addition, ourresults provide a semi-unique decomposition of any orthog-onal matrix into the product of planar rotations.

Julie MitchellDepartments of Mathematics/Biochemistry,University of Wisconsin-Madisonmitchell@math.wisc.edu

MS9

Steady Affine Motions: Interpolation, Properties,Applications

The morph between a shape S and a shape E may bedecomposed into a shape deformation and a carrier mo-tion. Most shape deformations either assume or computetheir a correspondence. Minkowski morphs compute cor-respondence based on surface orientation. Closest projec-tion morphs compute correspondence based on distance.The ball morph takes both distance and orientation intoaccount. Other approaches use curvature or saliency tocompute correspondence. The focus on this presentationis on the carrier motion. We propose to use a Steady AffineMorph (SAM), which we compute as At, where A is theaffinity that minimizes the discrepancy between A(S) andE. We present the beautiful properties of SAMs and theircomputation. We invite a discussion on the relation be-tween the particular morph used, the particular correspon-dence, the discrepancy measure, and the goodness measurefor the resulting morph.

Jarek RossignacGeorgia Institute of Technologyjarek@cc.gatech.edu

Alvar VinacuaUPC BarcelonaSpainalvar@lsi.upc.edu

MS10

Delaunay-based Meshes

Delaunay meshes are versatile structures for partition-ing domains with complex geometry into simple elements.They also come with practical and efficient constructionalgorithms that offer mathematical guarantees on the ele-ment quality. We will highlight the results on generatingDelaunay meshes for polyhedral volumes and surfaces. Adomain may change its shape with time. We will outlineour progress on tracking a Delaunay-based mesh for a de-

GD11 Abstracts 45

forming surface specified by dense point samples.

Siu-Wing ChengThe Hong Kong University of Science and Technologyscheng@cse.ust.hk

MS10

Optimal Homology Cycles and Knots

On a simplicial complex, can one efficiently find the small-est homology cycles (as measured by number or size)?In mod 2 homology, the problem is NP-hard in general.However, for integer coefficients, if the complex satisfies acertain topological condition (being relatively torsion-free)then optimal homology cycles can be found in polynomialtime in any dimension. The algorithm is simply linear pro-gramming. Being relatively torsion-free is equivalent to awell-known condition in linear programming that guaran-tees integer solutions. This result is then applied to solvea problem that was open in computational knot theory:for knots embedded in trivial homology manifolds, such asin ordinary three-dimensional space, the minimal spanningarea problem can be solved in polynomial time. The resulton optimal homology cycles is joint work with T. Dey andB. Krishnamoorthy. The knots result is joint work with N.Dunfield.

Anil N. HiraniUniversity of Illinois at Urbana-ChampaignDepartment of Computer Sciencehirani@cs.illinois.edu

MS10

Robust Computation of Delaunay Triangulationsby Degree Driven Algorithm Design

Algorithms are designed for machine models that havethree potentially unbounded quantities: running time,memory cells, and bits per cell. Designers optimize thefirst two, but often don’t consider the third. This can cre-ate. in geometry processing, a large effort gap between atheoretically correct algorithm and a robust implementa-tion. Liotta, Preparata and Tamassia suggested that thethird could also be considered a resource by limiting the al-gebraic degree of predicates. We use this idea in the designand implementation of Delaunay triangulation algorithms.

Jack SnoeyinkUniversity of North Carolina, Chapel HillDept of Computer Sciencesnoeyink@cs.unc.edu

MS10

Spectral Processing Based on Laplace-BeltramiOperator

The Laplace-Beltrami operator of a given manifold (e.g, asurface) is a fundamental object encoding the intrinsic ge-ometry of the underlying manifold. It has many propertiesuseful for practical applications from areas such as graphicsand machine learning. For example, its relation to the heatdiffusion makes it a primary tool for surface smoothing ingraphics. It has recently been used for a broad range ofgraphics and geometric optimization applications, such asmesh editing, compression, matching, segmentation, andetc. In this talk, I will first describe our recent resultson approximating the Laplace operator for general meshor point clouds by a geometric approach. I will then talk

about several applications based on the Laplace-Beltramioperator, including information estimation, shape match-ing, and shape deformation.

Yusu WangThe Ohio State University, Columbus, Ohioyusu@cse.ohio-state.edu

MS11

Space-efficient Maintenance of Nonbonded Lists forFlexible Molecules using Dynamic Octrees

The most time consuming calculations in molecular sim-ulations are associated with nonbonded interactions, suchas van der Waals and electrostatic energy functions. Thetruncation of nonbonded interactions based on distancecutoffs is widely used to reduce the high computationalcosts of such interactions, which is traditionally handledthrough the use of a precalculated and periodically up-dated explicit list of interacting atom pairs, known as the”nonbonded list” or nblist. The size of an nblist grows lin-early with the number of atoms in the system (say, n), andcubically with the distance cutoff (say, d). We propose adynamic octree data structure that can be used for implicitmaintenance of nblists using space linear in n but indepen-dent of d, and can be updated very efficiently as the shapeof the molecule changes. Dynamic octrees are more space-efficient, update-efficient and cache-efficient compared totraditional explicit nblists. Also unlike explicit nblists asingle octree can be used for all distance cutoffs. Thoughoctrees are slightly slower than explicit nblists when thedistance cutoff is small, they outperform nblists for largecutoffs.

Rezaul ChowdhuryUniversity of Texas at Austinshaikat@cs.utexas.edu

MS11

Signatures and Basis Set of Protein Binding Sur-faces: Surface Matching in Sequence Order Inde-pendent Fashion for Function Prediction

Three dimensional structures of proteins and their dynamicchanges are the basis of their biological functions. Detect-ing similarities between local binding surfaces can facili-tate identification of binding sites, prediction of proteinfunctions, as well as aid in our understanding of biochem-ical mechanisms. Local binding sites often take place ina concave surface pocket. Using Delaunay triangulationof the three-dimensional protein structure and the alphacomplex, we compute all surface pockets and other localconcave regions on a protein. We discuss how new insightsinto protein function can be gained by reconstructing theevolutionary history of binding pockets, and how to predictprotein functions from structures at large scale by findingstructurally invariable regions within the binding pocket.To address the challenging task of automatically construct-ing a template of local surface characteristics of a specificbiological function or binding activity, we discuss the modelof signature binding pockets and signature basis sets, aswell as a computational method called Solar (Signature OfLocal Active Regions) for their automatic constructions.We discuss how automatically constructed signatures andsignature basis sets of functional pockets can reveal signifi-cant biological insight. In addition, we assess the feasibilityof deriving signatures of binding pockets through the aid ofthe bootstraps of comparatively modeled structures. Ourresults using 26,590 modeled protein structures show that

46 GD11 Abstracts

when the sequence fragments of the binding surface sharesufficient similarity, informative signature binding pocketscan be derived, with most important atoms in the signa-tures of binding pockets captured.

Joseph DundasUniversity of Illinois at Chicagojdunda1@gmail.com

Jie LiangUniversity of Illinois at ChicagoDepartment of Bioengineeringjliang@uic.edu

MS11

Analytical Electrostatics: Methods and BiologicalApplications

Electrostatic forces are the strongest interactions at molec-ular level, and one of the hardest to compute for complexbiomolecular shapes. I will begin with a brief overviewof the benefits, successes and challenges in developingmethods that provide analytical approximate description ofcharge-charge interactions in biomolecular systems, as op-posed to direct numerical solutions. Construction and bio-physical applications of approximate analytical solutions tothe Poisson equation for boundary conditions relevant tobiomolecules will be discussed. I will conclude by present-ing my view on the current challenges in the field, includingthe need for a new generation of physically realistic math-ematical descriptions of biomolecular shape.

Alexey V. OnufrievDepartments of Computer Science and PhysicsVirginia Techalexey@cs.vt.edu

MS11

Visualizing and Exploring Molecular SimulationData via Topological Methods

Current computation power enables researchers to producea huge amount of folding simulation data. Hence there isa pressing need to be able to analyze such data to enhanceour understanding of molecular dynamics. Such simula-tion data can be considered as a massive set of high di-mensional points, with each point representing a proteinconformation. Each conformation also has an energy as-sociated with it. In this talk, we will describe our recentwork on using a topological method based on the contourtrees to construct a visualization platform for such simula-tion data, as well as how to use this tool for data analysis.

Yusu WangThe Ohio State University, Columbus, Ohioyusu@cse.ohio-state.edu

MS12

On the Numerical Analysis of Some T-splinesBased Isogeometric Methods

This talk will be about some new results on the mathemat-ical foundations of Isogeometric analysis. We will focus onthe use of T-splines as a refinement tool and we will il-lustrate the main steps which are needed to develop thenumerical analysis of T-splines based methods.

Lourenco Beirao Da Veiga

Universita’ degli Studi di Milanolourenco.beirao@unimi.it

Annalisa BuffaI.M.A.T.I. - C.N.R.Pavia, Italyannalisa@imati.cnr.it

Durkbin ChoDipartimento di MatematicaUniversita’ di Milano, Italydurkbin@imati.cnr.it

Giancarlo SangalliDipartimento di MatematicaUnversita’ di Pavia, Italygiancarlo.sangalli@unipv.it

MS12

On the Normalization of Hierarchical B-splines

The hierarchical model allows us to control the localityof the refinement through an adaptive procedure that isexclusively based on basis refinement. It can be suitablymodified in order to define locally supported basis func-tions that form a partition of unity. Reducing the supportof basis functions defined on coarse grids, in accordancewith finer levels in the hierarchy, not only decreases theoverlapping of basis supports, but it also improves stabil-ity properties.

Carlotta GiannelliJohannes Kepler University LinzAustriacarlotta.giannelli@jku.at

Bert JuettlerJohannes Kepler University of LinzInstitute of Applied Geometrybert.juettler@jku.at

Hendrik SpeleersKatholieke Universiteit LeuvenHendrik.Speleers@cs.kuleuven.be

MS12

Analysis-suitable Local Refinement of T-splines

Isogeometric analysis has emerged as an important alterna-tive to traditional engineering design and analysis method-ologies. In isogeometric analysis, the smooth geometricbasis is used as the basis for analysis. T-splines were in-troduced as a superior alternative to NURBS. T-splinescan model complicated designs as a single, watertight ge-ometry. Unlike NURBS, T-splines can be locally refined.These properties make T-splines an ideal technology forisogeometric discretizations. In this talk, we will briefly re-view T-spline technology and then we will discuss analysis-suitable local refinement. We will describe the underlyingalgorithm and demonstrate its behavior in the context ofa demanding phase-field fracture model.

Michael ScottInstitute for Computational Engineering and SciencesU.T. Austinmscott@ices.utexas.edu

Michael J. Borden

GD11 Abstracts 47

ICES, UT Austinmborden@ices.utexas.edu

Thomas HughesInstitute for Computational Engineering and SciencesThe University of Texas at Austinhughes@ices.utexas.edu

Thomas W. SederbergBYUtom@cs.byu.edu

MS12

On the Use of Powell-Sabin B-splines for Local Re-finement in Advection-diffusion-reaction Problems

Powell-Sabin splines are C1 quadratic splines defined ontriangulations. They can be represented in a basis withthe same nice properties of the classical B-spline basis. Wediscuss the use of these splines for the numerical solutionof advection-diffusion-reaction problems. They constitutea natural bridge between classical FEM and the NURBS-based Isogeometric Analysis. They offer the flexibility ofFEM with respect to local refinements and share the in-creased smoothness with NURBS.

Hendrik SpeleersKatholieke Universiteit LeuvenHendrik.Speleers@cs.kuleuven.be

Carla ManniDepartment of MathematicsUniversity of Romemanni@mat.uniroma2.it

Francesca PelosiUniversity of Rome ”Tor Vergata”Department of Mathematicspelosi@mat.uniroma2.it

Maria Lucia SampoliUniversity of SienaDepartment of Mathematics and Computer Sciencesampoli@unisi.it

PP1

Modeling Wing Surfaces During Collision Avoid-ance Behaviors of Flying Locusts

Collision avoidance is critical in evading predation, espe-cially during flight. We presented looming stimuli (simu-lated colliding objects) to locusts (Schistocerca americana)while the subjects were flying under a variety of tether con-figurations in a wind tunnel. Stereoscopic high-speed videowas used to record their behavior as they sought to avoida collision. The collected video data was then used to re-construct the flight trajectory, body direction, and wingbeat patterns and deformations as the subject undertookcollision avoidance responses.

Raymond Chan, Fabrizio GabbianiBaylor College of MedicineDept. of Neurosciencerchan@cns.bcm.edu, gabbiani@bcm.edu

PP1

A Predictive Model for Geographic Statistical Data

Any planar map may be transformed into a graph. If weconsider each country to be represented by a vertex (ornode), if they are adjacent they will be joined by an edge.To consider how trends migrate across boundaries, we ob-tain relevant measures of the statistic we want to consider;namely, the index of prevalence, and the index of incidence.We define a cycle by a given unit of time, usually a year.We then propose various alternate equations whereby, byparametrizing various variables, such as population size,birth rate, death rate, and rate of immigration/emigration,we calculate a new index of prevalence/index of incidence,for the next cycle. For a given data set, each statistic weconsider may propagate by a different equation, and/or adifferent set of parameters; this will be determined empiri-cally. What we are proposing is, technically, to model howa discrete stochastic process propagates geographically, ac-cording to geographical proximity. Very often, statisticsthat depend on geographical proximity are tabulated byvariables that are not; i.e., alphabetically. Such a predic-tive model would be relevant in areas such as public health;and/or crime mapping, for law enforcement purposes. Wepresent an application using a GIS (geographic informationsystem).

Jorge Diaz-CastroUniversity of P.R. School of LawUniversity of P.R. at Rio Piedrastranscripciones@mail.com

PP1

High Accuracy NC Milling Simulation Using Dis-tance Fields of Swept Volumes

We introduce a new 3D shape representation called a Com-posite Adaptively Sampled Distance Field and describe itsapplications to swept volumes. In the case of NC millingsimulation, the distance fields represent both the work-piece and the swept volumes generated by the 3-axis or5-axis motion of the milling tool along its prescribed path.The resulting representation enables reconstruction of themilled surface with micron accuracy while having modestmemory requirements for large and complex shapes. Thesystem provides significant improvements over previous ap-proaches to milling simulation.

Huseyin ErdimMember Research Stafferdim@merl.com

Alan Sullivan, Ronald N. PerryMitsubishi Electric Research Laboratoriessullivan@merl.com, perry@merl.com

Sarah Frisken261 Solutions Inc.sfrisken@gmail.com

PP1

3D-Cad Data for Massive Assembly Models Con-sidering Feature of Production Facility Models

Using 3D-CAD systems for production facilities has a prob-lem to process massive number of parts. We take noticeof the shapes of parts on production facility models: mostof them are united and subtracted projections, revolutions,and simple holes at right angles, and we propose the new

48 GD11 Abstracts

data structure with small data size, specialized for the pro-duction facility models. This enables us to process up to400,000 parts on our 3D-CAD system.

Hidekazu Matsunami, Akio Koda, Narumasa OdaharaiCAD Limitedmatsunami@icad.jp, koda@icad.jp, odahara@icad.jp

PP1

Hybrid Geometric Modeling for OrthopedicSurgery Planning

Surgery planning for internal fractures involves aligningbone fragments (reduction), and deforming fixation platesto fit patient anatomy (adaptation). We propose a hybrid-geometry based planning system with standard plate CADmodels and polygonal bone data as input. A novel featureof the system is the creation of NURBS-based models ofpatient-specific plates semi-automatically that can then beproduced with CNC machines. The system improves plan-ning accuracy, reduces burden on surgeons and ultimatelyimproves patients outcome.

Suraj MusuvathyUniversity of Utahsrm@cs.utah.edu

Sergei AzernikovSiemens Corporate Researchsergaz@gmail.com

Tong FangSiemens Corporate Research, Inc.tong.fang@siemens.com

PP1

Shape Optimization of Sub-Wavelength AntennasUsing Isogeometric Analysis

We aim to design an antenna that concentrates the mag-netic energy from an incoming electromagnetic wave in avery small region of space. This relates to the problem ofwireless energy transfer. We use shape optimization andisogeometric analysis to attack the problem. Even thoughwe only consider the problem in 2D the results are verypromising. Compared to earlier attempts using topologyoptimization we increase the energy in the designated areawith factor of one million.

Dang Manh Nguyen, Anton EvgrafovDepartment of MathematicsTechnical University of Denmarkd.m.nguyen@mat.dtu.dk, a.evgrafov@mat.dtu.dk

Jens GravesenTechnical University of Denmarkj.gravesen@mat.dtu.dk

PP1

Regularization in Isogeometric Shape Optimizationfor Fluids

The goal in shape optimization for fluids is to find anoptimal boundary of the flow domain that minimizes aprescribed objective, while satisfying suitable constraints.Inclusion of an artificial objective term is often neededto avoid inappropriate boundary parametrizations andthereby regularize the optimization problem. This work

uses isogeometric analysis as framework for the numericalmethod and discusses various regularizations by comparingtheir effects in different examples.

Peter N. NielsenTechnical University of Denmarkp.n.nielsen@mat.dtu.dk

Thomas A. GrandineApplied MathematicsThe Boeing Companythomas.a.grandine@boeing.com

Jens GravesenTechnical University of Denmarkj.gravesen@mat.dtu.dk

Thomas A. HoganThe Boeing Math Groupthomas.a.hogan@boeing.com

PP1

Applied Computational Homology Toward Struc-tural Study to Protein Science

In protein science, one of important problems is that struc-tural property of each protein should be determined. Weadopt homology group (Rips homology, or ) to calculatethe number of ”holes” and ”cavities” in each protein. Thisis very important information about the structure of theprotein.

Isamu Ohnishi, Fumihide NonoHiroshima Universityisamu o@math.sci.hiroshima-u.ac.jp,m104418@hiroshima-u.ac.jp

Hiroaki HiraokaKyushu Universityhiraoka@math.kyushu-u.ac.jp

Shunsuke IzumiHiroshima Universitysizumi@sci.hiroshima-u.ac.jp

PP1

Multiresolution Methods in Isogeometric Analysis

In this research we implement subdivision schemes andwavelets for isogeometric finite element analysis. Thesubdivision schemes are routinely employed in computergraphics and animation and their applications in finite el-ement modeling has recently started. The application ofwavelets in numerical solution of PDEs has been so farlimited to simple geometries. We outline application ofwavelet based multiresolutional methods in modeling arbi-trary topology via use of subdivision schemes.

Sarosh M. QuraishiDepartment of Mechanical Engg., Institute of TechnologyBanaras Hindu Universitysarosh.quraishi@gmail.com

PP1

High-Order Numerical Integration over DiscreteSurfaces

We investigate the problem of numerical integration over a

GD11 Abstracts 49

discrete surface to high-order accuracy. Presently, integra-tion over discrete surfaces (such as a surface triangulation)is in general limited to only second-order accuracy. Wepresent a novel method that can achieve high order accu-racy, by combining a stabilized least squares approxima-tion for high-order surface and functional reconstructions,a blending procedure, and high order numerical quadraturerules.

Navamita Ray, Duo Wang, Xiangmin JiaoStony Brook Universitynray@ams.sunysb.edu, duowang@ams.sunysb.edu,xjiao@sunysb.edu

James GlimmSUNY at Stony BrookBrookhaven National Laboratoryglimm@ams.sunysb.edu

PP1

Simple Determination of Geometric Characteris-tics of Bezier Conics

We show how to compute in a straightforward mannerthe geometric characteristics of a conic segment in ratio-nal Bezier form, by employing complex arithmetic. For acentral conic, a simple quadratic equation defines the focilocation, and its solution furnishes not only an explicit for-mula for the foci, but also for the centre, axis directionand linear eccentricity. Other characteristics, such as axislengths or eccentricity, result from traditional geometry.

Javier Sanchez-ReyesUniversidad de Castilla-La Mancha, SpainIMACI, ETS Ingenieros IndustrialesJavier.SanchezReyes@uclm.es