A face-based curvature estimation on triangle meshes is presented in
this paper. A flexible disk is laid on the mesh around a given
triangle. Such a bent disk is used as a geodesic neighborhood of the
face for approximating normal and principal curvatures. The radius
of the disk is free input data in the algorithm. Its influence on
the curvature values and the stability of estimated principal
directions are investigated in the examples.U članku je prikazana procjena zakrivljenosti na trokutnim mrežama, bazirana na stranicama. Gipki disk položen je na
mrežu oko danog trokuta. Takav prilagodljiv disk koristi se
kao geodetska okolina stranice za aproksimaciju normalnih i
glavnih zakrivljenosti. Polumjer diska je nezavisni ulazni
podatak u algoritmu. U primjerima se istražuje njegov utjecaj na vrijednosti zakrivljenosti i na stabilnost procijenjenih glavnih
smjerova

Márta Szilvási-Nagy

English

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KoG•10–2006 Márta Szilvási-Nagy: About Curvatures on Triangle MeshesOriginal scientific paperAccepted 1. 12. 2006.MÁRTA SZILVÁSI-NAGYAbout Curvatures on Triangle MeshesAbout Curvatures on Triangle MeshesABSTRACTA face-based curvature estimation on triangle meshes ispresented in this paper. A flexible disk is laid on the mesharound a given triangle. Such a bent disk is used as ageodesic neighborhood of the face for approximating nor-mal and principal curvatures. The radius of the disk is freeinput data in the algorithm. Its influence on the curvaturevalues and the stability of estimated principal directionsare investigated in the examples.Key words: triangle mesh, curvatureMSC 2000: 68U05, 68U07, 65D20O zakrivljenostima na trokutnim mrežamaSAŽETAKU članku je prikazana procjena zakrivljenosti na trokut-nim mrežama, bazirana na stranicama. Gipki disk položenje na mrežu oko danog trokuta. Takav prilagodljiv diskkoristi se kao geodetska okolina stranice za aproksimacijunormalnih i glavnih zakrivljenosti. Polumjer diska je neza-visni ulazni podatak u algoritmu. U primjerima se istražujenjegov utjecaj na vrijednosti zakrivljenosti i na stabilnostprocijenjenih glavnih smjerova.Ključne riječi: trokutna mreža, zakrivljenost1 IntroductionTriangle meshes are the most frequently used surface rep-resentations in many surface-oriented applications. Sur-face curvature properties have been successfully employedfor solution of different practical problems, as smooth-ing or simplifying meshes in modeling and manufacturing,also for surface classification and 3D object recognitionin computer vision research, etc. Discrete counterparts ofcontinuous definitions of differential operators, curvaturevalues, geodesic curves and Dirichlet energy, etc. havebeen given and derived for arbitrary triangle meshes in [3],[10], [13] and [14].Almost all methods for surface derivative and curvature es-timations on meshes have been vertex connectivity based.In these approaches a specified neighborhood of verticesformed by adjacent vertices, edges and faces is used to ap-proximate the surface normal, surface derivatives and cur-vature values at a vertex. The algorithms use either analyticmethods based on surface fitting, or they work with dis-crete differential operators. The crucial first step in thesealgorithms is the computation of a vector at each vertexthat approximates the true normal vector at this point of thesurface represented by the mesh. This problem is equiv-alent to the computation of the best tangent plane to themesh at a given vertex. Most methods compute a weightedaverage of facet normals in a one-ring neighborhood of thevertex.N =∑ω jNjmi, j = 1. . .mi ,wheremi is the number of edges emanating from the ver-texvi , ω j = k ·Area(trianglej), k > 0 andArea(trianglej)is either surface area or Voronoi-surface area or a mixedsurface area of the triangle{vi,v j ,v j+1} (Fig. 1).v ijvv j+1N jN?Figure 1: One-ring neighborhood of a vertex13KoG•10–2006 Márta Szilvási-Nagy: About Curvatures on Triangle MeshesA number of proposals have been published for choosingthe weights or the neighborhood for determining the bestnormal vector ([15], [11], [9]).An other problem is to estimate normal curvatures, whichis equivalent to the definition of osculating circles produc-ing a second-order approximation to those curvature values([12], [19]).A normal curvature estimation in a one-ring neighborhoodcan be given simply by defining the osculating circle ina normal plane through the verticesvi , v j and the normalvectorN (Fig. 2).vijvNTFigure 2: Osculating circle in a normal planeThe curvature of this circle isκn(vi) ≈2 < N,(v j −vi) >|v j −vi|2,where<,> denotes the dot product.Instead of circle in a normal plane, interpolating quadraticpolynomial curve is also used e.g. in [7]. The one-ring neighborhood of a specified vertex is replaced by aVoronoi or mixed surface area [10], it is extended in [7]and geodesic neighborhoods are also used ([12], [16]) inthe computations. The selection of neighborhood size canaffect results significantly: small neighborhoods providebetter estimates for clean data, while increasing the neigh-borhood size smoothes the estimates, leading to less sen-sitivity to noise. Obviously, small errors in these approxi-mations lead rapidly to unreliable, noisy curvature values.Comparisons of five frequently used methods are given in[5].Analytic methods are also applied for curvature estima-tions by fitting a surface to the mesh in the neighborhoodof the point of interest and evaluating its curvatures ([5],[6], [19]). Principal curvatures and principal directionscan be determined on the base of Euler theorem ([11], [2],[4]). Mean and Gauss curvature values can be computedfrom the curvature tensor, i.e. from the Weingarten matrixor from its symmetric extension by eigen-decomposition([12], [18]). The Gauss-Bonnet theorem gives a directmethod for the computation of Gauss curvature. It has twodifferent discrete forms which provide good approxima-tions for special triangulations of surfaces [20].In this paper we define normal curvatures on each face ofthe triangle mesh in order to estimate the principal direc-tions and to characterize elliptical, umbilical, flat and hy-perbolic regions. The proposed new method is presentedin Chapter 2. In the examples (Chapter 3) we show theproposed method oǹ‘ synthetić’ and real triangulated sur-faces.2 Curvatures defined on faces2.1 Geodesic circle of a triangleInstead of computing surface properties at vertices in ver-tex neighborhoods we define curvature values ordered tofaces. The center of the defined region is the barycentriccenter of the given triangle. We intersect the mesh withnormal planes passing through the face normal of the tri-angle, then we measure a given geodesic radius along thepolygonal lines of intersection in both directions from thecenter point. In this way we get a number of curved diame-ters of the geosedic circle bent on the mesh around the face.We call this geodesic neighborhood “splat” after Kobbelt[8] and the set of the polygonal diameters “spider” afterSimari [16] (Fig. 3). Then we compute the chord lengthsof the constructed diameters in order to estimate normalcurvature values.NFigure3: Geodesic circle of a face14KoG•10–2006 Márta Szilvási-Nagy: About Curvatures on Triangle Meshes2.2 Osculating circle and normal curvatureWe define in each normal plane an osculating circle to theface determined by the endpoints of the bent diameter andthe face normal (Fig. 4).rrd/2gnFigure 4: Osculating circle in a normal sectionDenoterg the given geodesic radius,d the chord lengthbetween the endpoints of the curved diameter, 2α the un-known central angle andrn the required radius of the oscu-lating circle. From the equationsrnα = rg and rn sinα =d2we get forαd2rg=sinαα.We apply the approximationsinα ≈ α−α36, 0 < α << 1and getα ≈√(1−d2rg)6,consequently,rn ≈rgαif α 6= 0 and κn ≈αrgis the radius of the osculating circle and the normal curva-ture, respectively.2.3 Principal directions and principal curvaturesRepeating this computation for a set of normal sections inthe geodesic neighborhood of the given triangle we obtainnormal curvature valuesκn,i, i = 1, . . .k. If the mesh isa dense triangulation of a regular surface then the normalplanes belonging to the minimal and maximal normal cur-vatures are orthogonal to each other. They determine theprincipal directions.We select the maximal curvature and define it as first prin-cipal curvatureκ1 and the corresponding directionT1 inthe plane of the current triangle as first principal direc-tion. This direction is fairly stable, even if we computewith smaller geodesic circles. The second principal direc-tion T2 is orthogonal to it. In the case of properly definedgeodesic circle and nearly regular triangulation it is the di-rection belonging to the maximal chord length (Fig. 5).NT1T2dmaxdminFigure 5: Principal directions3 ExamplesIn order to compute a geodesic disk around a triangleface in a mesh we have to construct a flexible polyhe-dral data structure on the mesh which differentiates innerand boundary edges, moreover “feature edges” along sharpridges. Then we have to implement an algorithm for com-puting the lines of intersection of the mesh and the definednormal planes [17]. The normal sections on the mesh arepolygonal lines, and the arc length on the approximatedsurface is measured along these polygonal lines.The triangulated cylindrical meshes in Fig. 6 and 7 aregenerated from the analytical description of a cylindricalsurface. The vertices of such a “synthetic” mesh are ly-ing exactly on the surface approximated by the mesh. Thegeodesic radius in Fig 6 is 3.5 times the average size ofthe triangles. The number of the computed diameters is24. In Fig 7 the same cylindrical surface is shown with dif-ferent triangulation. The geodesic radius is 0.6 times theaverage size of the triangles in the mesh. The largest cur-vature computed by the proposed method are 0.0402 and0.0405, respectively, instead of 0.0400. Hence the relativerrors in the curvature estimation are 0.5% and 1.2%, re-spectively. The corresponding principal directions are inboth cases the most perpendicular ones to the axis of thecylinder among the 24 tangent directions. These directionsare shown in the figures with longer segments.15KoG•10–2006 Márta Szilvási-Nagy: About Curvatures on Triangle MeshesFigure 6: Splat on a cylindrical surfaceFigure 7: The same cylindrical surfaceThe second example shows that our curvature estimationworks also in the case, where one-ring neighborhoods can-not be applied.Figure 8: Splat on the sphereIn Fig. 8 a real triangle mesh approximating a sphere isshown. In this mesh there is a noise in the vertex data,and in the size of the triangles. The chosen geodesic ra-dius is three times the average triangle size. The differencein the computed normal curvatures is about 3%, and thisrelative error does not change when the geodesic radius isvarying between 2.5 and 7 times the average triangle size.This accuracy in the computation is comparable to the pub-lished results in the literature analysing curvature estima-tion methods [5].In Fig. 9 and 10 a synthetic mesh of one eighth of a torus isshown. In Fig 9 the geodesic radius is three times the aver-age triangle size, and the results in the principal directionsare very good, considering the relative coarse approxima-tion with 24 directions in the disk. In Fig 10 the geodesicdisk is apparently too big to get reliable approximations(5 times the average triangle size) for the specified triangleface. In this picture only boundary and silhouette edges areshown besides the 24 diameters of the geodesic disk.Figure 9: Splat on the torusFigure 10: The same facette with a bigger splat16KoG•10–2006 Márta Szilvási-Nagy: About Curvatures on Triangle MeshesAccording to our experience the best measurement of theradius of a geodesic disk is between 2.5 and 3.5 times theaverage triangle size in a dense mesh. Of course, the meshshould provide a proper surface approximation. The exam-ples have shown that our face-based curvature estimationworks better than vertex-oriented methods using a one-ringneighborhood, especially in the case when the mesh con-tains long narrow triangles.Some more investigations have to be done in the future,e.g., in the analysis of relative errors and in further appli-cations.4 ConclusionWe have introduced curvature values ordered to faces intriangle meshes by laying a flexible circular disk with user-specified radius onto each face of the mesh. From thechords of such a bent disk and from the face normal wehave defined normal curvatures of the current face. Theexamples have shown that the obtained principal curva-ture values and the corresponding principal directions arequite reliable if the radius of the disk achieves an optimalsize. Our method provides a good classification of elliptic,parabolic, flat and hyperbolic regions of the mesh.We have implemented the algorithm for constructinggeodesic disks on triangle meshes and for estimating nor-mal curvatures and principal directions in Java.References[1] DO CARMO, M.P., Differential Geometry of Curvesand Surfaces, Prentice-Hall, Englewood Cliffs, NJ.,1976.[2] CHEN, X., SCHMITT, F., Intrinsic surface propertiesfrom surface triangulation, Proceedings of the Euro-pean Conference on Computer Vision, 1992, pp.739-743.[3] DESBRUN, M., MEYER, M., SCHRÖDER, P., BARR,A.H., Implicit fairing of irregular meshes using diffu-sion and curvature flow, Computer Graphics Proceed-ings (SIGGRAPH’99), 1999, pp.317-324.[4] DONG, C.S, WANG, G.Z.,Curvatures estimation ontriangular mesh, Journal of Zhejiang University SCI-ENCE 6ASuppl.I (2006) 128-136.[5] FLYNN , P.J., JAIN , A.K., On reliable curvature es-timation,IEEE Conf. Comput. Vision Patt. Recogn.1989, pp. 110-116.[6] GOLDFEATHER, J., INTERRANTE, V., A novel cubic-order algorithm for approximating principal directionvectors, ACM Transactions on Graphics 23/1 (2004),45-63.[7] HAMEIRI , E., SHIMSHONI, I., Estmating the princi-pal curvatures and the Darboux frame from real 3Drange data, Computers & Graphics 28 (2004), 801-814.[8] K OBBELT, L., BOTSCH, M., A survey of point-based techniques in computer graphics, Computers &Graphics 28 (2004), 801-814.[9] M AX , N., Weights for computing vertex normals fromfacet normals, Journal of Graphics Tools 4 (2000), 1-6.[10] MEYER, M., DESBRUN, M., SCHRÖDER, P.,BARR, A.H., Discrete differential-geometry operatorsfor triangulated 2-manifolds, Visualization and Math-ematics III. (H.C. Hege and K. Polthier, Ed.) SpringerVerlag 2003, pp. 35-57.[11] MORETON, H.P., ŚEQUIN, C.H., Functional opti-mization for fair surface design, SIGGRAPH’92 Con-ference Proceedings 1992, pp. 167-176.[12] PAGE, D.L., SUN, Y., KOSHAN, A.F., PAIK , J.,ABIDI , M.A., Normal vector voting: Crease detec-tion and curvature estimation on large, noisy meshes,Graphical models 64 (2002), 199-229.[13] PINKALL , U., POLTHIER, K., Computing discreteminimal surfaces and their conjugates, ExperimentalMath. 2 (1993), 15-36.[14] POLTHIER, K., SCHMIES, M., Straightest geodesicson polyhedral surfaces, Mathematical visualization(K. Polthier and H.C. Hege Ed.) pp.391-409, SpringerVerlag 1998.[15] RUSINKIEWICZ, S.,Estimating curvatures and theirderivatives on triangle meshes, Proc. 3DPVT’04 2004,486-493.[16] SIMARI , P., SINGH, K., PEDERSEN, H., SPI-DER, A rpbust curvature estimator for noisy, irregularmeshes, Tecnical report CSRG-531, Dynamic Graph-ics Project, Dept. of Comp.Sci., Univ. Toronto, 2005.[17] SZILV ÁSI-NAGY, M., Removing errors from trian-gle meshes by slicing, Third Hungarian Conference onComputer Graphics and Geometry, 17-18. Nov. 2005Budapest, Hungary, 125-127.17KoG•10–2006 Márta Szilvási-Nagy: About Curvatures on Triangle Meshes[18] TAUBIN , G., Estimating the tensor of curvature ofa surface from a polyhedral approximation, ICCV’95Proceedings of the Fifth International Conference onComputer Vision, IEEE Computer Society, Washing-ton DC, USA, 902.[19] TODD, P.H., MCLEOD, R.J.Y.,Numerical estima-tion of the curvature of surfaces, computer-aided de-sign 18 (1986) 33-37.[20] XU, G., Convergence analysis of a discretizationscheme for Gaussian curvature over triangular sur-faces,Computer Aided Geometric Design 23 (2006)193-207.Márta Szilvási-Nagye-mail: szilvasi@math.bme.huDepartment of GeometryBudapest University of Technology and EconomicsH-1521 Budapest, HungarySupported by a joint project between the TU Berlin and the BUTE, by the Hungarian National Foundation OTKA No.T047276 and by the Hungarian-Croatian Intergovernmental S& T Cooperation Programme18

O zakrivljenostima na trokutnim mrežama

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