The Visual Computer manuscript No.(will be inserted by the editor)

Harlen Costa Batagelo · Wu, Shin - Ting

Estimating Curvatures and their Derivatives on Meshes ofArbitrary Topology from Sampling Directions

Abstract Estimation of local differential geometry proper-ties becomes an important processing step in a variety ofapplications, ranging from shape analysis and recognitionto photorealistic image rendering. This paper presents yetanother approach to compute those properties, with com-parable numerical and accuracy performance to the previ-ous works. The key difference of our approach is simplicity,which makes it directly implementable on the GPU. Experi-mental results are provided to underpin our statement.

Keywords Curvatures · Principal Directions · DifferentialGeometry Properties · Computational Aided GeometricDesign

1 Introduction

The differential geometric properties characterize the vicin-ity of each point of a surface, such as the area, the shape, themaximum and minimum of the normal curvature, and theprincipal directions associated to these curvatures. Some ofthese quantities are of interest in image synthesis and anal-ysis applications, such as anisotropic fairing of digital mod-els (13), extraction of visually suggestive contours in orderto enhance shape cues (7), enrichment of visual appearancewith fine-scale details (20) and hatching strokes in pen-and-ink style painting (17). In the context of 3D interactions, theyhave also been shown useful for weighting the gravity func-tion that causes the cursor to snap to ridges or valleys of asurface (21).

The advent of highly capable GPUs has propelled re-searches on estimation of local geometric properties on suchhardware. Today, real-time applications may employ GPUshader programs to efficiently deform geometry immediatelybefore rendering. Hence, the associated per-vertex geomet-ric properties, such as the tangent vectors, normal vectors,

Harlen Costa Batagelo and Wu, Shin-TingSchool of Electrical and Computer Engineering, UnicampTel.: +55-19-35213795Fax: +55-19-35213845E-mail: harlen,ting@dca.fee.unicamp.br

Fig. 1 Lighting a bump-mapped sphere deformed on the GPU. Left:Using differential geometry properties of the original non-deformedsphere (inset). Right: Using updated properties computed on the GPUafter the deformation.

principal curvatures and directions, need to be updated ac-cordingly for using 3D detail mapping techniques and per-forming correct lighting calculations. Figure 1 shows a bump-mapped sphere deformed in the vertex shader by a PerlinNoise displacement function and lit using Phong shading. Inthe left, the sphere is lit without updating the tangent framesafter the deformation (the tangent frames are the same ofthe non-deformed sphere, shown in the inset). In the right,the sphere is accurately lit with tangent frames computed inreal-time on the GPU after the per-vertex deformation. Forrendering 3D details with correct silhouettes, higher orderquantities should be computed on-the-fly as well (20).

Basically, we may distinguish three approximation tech-niques for computing differential geometry quantities of dis-crete surfaces: patch-fitting, averaging, and curve samplingmethods. Averaging and curve sampling techniques are de-fended, respectively, by Rusinkiewicz (18), Agam and Tang(1). They claim that patch-fitting techniques tend to smoothsharp corners and ridges, often requiring connectivity datastructure beyond the usual 1-ring of vertices or faces aroundeach vertex. Inspired by the accurate results achieved byMax for the per-vertex estimation of normal vectors (14),Rusinkiewicz proposes to take a weighted average of the ge-ometric quantities of all adjacent faces of each point of in-terest for calculating its curvatures and higher order deriva-

2 Batagelo H.C. & Wu, S.-T.

tives. His proposal requires that normal vectors at each ver-tex are provided and that the weights are appropriately se-lected. Agam and Tang show that the curve sampling methodmay accurately compute both the normal vectors and the cur-vatures in the same framework. They point out that, as thesemethods have a large degree of freedom, they are capableof modeling the local surface geometry much more flexiblycompared with other techniques. However, instead of apply-ing the formulation of the Weingarten equations, they deter-mine the extrema of the normal curvatures by computing thenormal curvatures of all sampling curves and, then, selectingthe directions that presents minimum and maximum normalcurvatures. The quality of the curvature estimation depends,therefore, on the sampling coverage.

We present in this work a proposal for computing the lo-cal geometry properties of second or higher order derivativeson the basis of a set of neighboring sampling points that lieon the distinct directions with respect to a vertex of interest.It differs from the work of Agam and Tang in that we do notselect the extreme values of the normal curvature among thesamples. Adopting the scheme devised by Rusinkiewicz, weuse the quadratic differential forms and the finite-differenceform of the shape operator to compute the maximum andthe minimum of the normal curvature. Consequently, the re-sults tend to be more stable using only the 1-ring neighbor-hood. We, however, take advantage of the curve samplingapproach to improve efficiency of the algorithm devised byRusinkiewicz. Instead of two processing passes – one overadjacent faces and one over vertices, we present a proce-dure that requires only one pass. From the experimental re-sults, we observed that the gain in efficiency compensatesthe slight degradation in accuracy for irregularly sampledmeshes.

To be self-containing, a brief overview of surface differ-ential geometry is provided in Section 2. Section 3 presentsrelated works, while Section 4 describes our proposal for es-timating curvatures and their derivatives on the curve sam-pling basis. Although surface normals play an important rolein the curvature estimation (8), we will consider in this workthat the tangent frames, composed by normal, tangent andbitangent vectors, are somehow accurately computed. Thisis because that there is a variety of efficient algorithms forcomputing these elements of first derivatives. In Section 5,comparisons of our proposal to previous work in terms of ef-ficiency and robustness are given. Finally, some concludingremarks are drawn in Section 6.

2 Background

We may represent a surface S as a net of parametric curves,such that every point P on the surface is a crossing point oftwo of them. Mathematically, we may express S as a func-tion r(u,v) = (x(u,v), y(u,v), z(u,v)) that maps a point (u,v)in a certain closed interval Ω onto a three-dimensional spaceℜ3. The real variables u, v are called coordinates on S . AtP the vector ru = ∂r

∂u is tangent to the curve r(u,vconstant),

and rv = ∂r∂v is tangent to the curve r(uconstant ,v). If the vec-

tors ru and rv do not vanish and have different directions atevery point, we say that r(u,v) is a regular surface.

Let α(t) = r(u(t),v(t)) be a curve on S that passesthrough P . The vector at P

dr = rudu+ rvdv =[

ru rv][

dudv

]. (1)

is tangent to the curve α(t) and therefore to S .The infinitesimal squared length ds of an element of arc

of α(t) in the vicinity of P can then be expressed in aquadratic or bilinear form, in terms of the differentials U =[du dv

]t, which is known as the first fundamental form

ds = dr ·dr =[[

ru rv][

dudv

]]t · [ ru rv][

dudv

]

=[

du dv][

ru · ru ru · rvrv · ru rv · rv

][dudv

]

=[

du dv][

E FF G

][dudv

]= U t

ISU . (2)

Observe that the inner product is symmetric at every pointF = ru · rv = rv · ru.

According to Eq. 1, all vectors dr at P are linear com-binations of ru and rv, thus lying on the plane spanned bythese two vectors. This plane is the tangent plane at P toS . Its unit normal vector is given by

n =ru × rv

‖ru × rv‖ =ru × rv√EF −G2

. (3)

The variations of the tangent vector of the curve α(t) =r(u(t),v(t)) at P define the curvature vector k of α(t). Thus,the curvature vector is an element of second derivative. Itsprojection over the normalized normal vector n of S at Pis called the normal curvature vector

kn(α(t)) = kn(α(t)) ·n ,

where kn(α(t)) is the normal curvature of α(t) at P anddepends on the shape operator −dα(t)n:

kn(α(t)) = −dα(t) ·dα(t)ndα(t) ·dα(t)

= −dα(t) ·dα(t)nIS

. (4)

The normal curvature depends on the shape operator or thederivative of the normal n along the tangent direction ofα(t) = (u(t),v(t)) at P ,

dα(t)n =∂ n∂u

du+∂ n∂v

dv. (5)

Eq. 5 is called the second fundamental form and mayalso be written in the bilinear form

−dα(t) ·dα(t)n =[[−ru −rv

][dudv

]]t · [nu nv][

dudv

]

=[

du dv][−ru ·nu −ru ·nv

−rv ·nu −rv ·nv

][dudv

]

=[

du dv][

e ff g

][dudv

]= U t

IISU . (6)

Estimating Curvatures and their Derivatives on Meshes of Arbitrary Topology from Sampling Directions 3

As the coefficients of IS and IIS transform appropriatelyunder allowable change of coordinates, they are called themetric and the curvature tensor, respectively.

For the basis ru,rv, the Weingarten equations expressthe derivatives of the normal to a regular surface in terms ofthis basis[

nunv

]=

[f F−eGEG−F2

eF− f EEG−F2

gF− fGEG−F2

f F−gEEG−F2

][rurv

]= W

[rurv

](7)

Although there is an infinity of curves on S passingthrough P and possessing the same tangent at this point,the Theorem of Meusnier tells us that all of these curveshave the same normal curvature.

The direction of the normal curvature vector always re-mains that of the surface normal. Only its length may varyfor different directions. The directions in which the normalcurvature becomes extreme at P are called principal di-rections, e1,e2, and the corresponding normal curvaturesare denominated principal curvatures, κmin and κmax. Theprincipal directions and the principal curvatures are, respec-tively, the eigenvectors and the eigenvalues of W. The prod-uct

K = κminκmax. (8)

of the two principal normal curvatures is called Gaussiancurvature and their arithmetic mean

H =κmin +κmax

2(9)

is called the mean curvature.In some applications, the elements of third derivatives,

namely the differentiations of IIS, are of interest. It is be-cause that they are helpful in quantifying the surface fairness(10). To define a differentiation in such a manner that thederivative of any tensor is again a tensor, the concept of ab-solute differentiation or covariant derivatives is introduced.Gravesen and Ungstrup show that the covariant derivativesof IIS is a trilinear symmetric tensor 2× 2× 2 with coeffi-cients ci jk given by (10)

a = c111 = ruuu ·n+3ruu ·nu

b = c112 = c121 = c211 = ruuv ·n+ ruu ·nv +2ruv ·nu

c = c221 = c122 = c212 = ruvv ·n+ rvv ·nu +2ruv ·nv

d = c222 = rvvv ·n+3rvv ·nv (10)

These derivatives define the tensor of curvature derivativeCS.

Using the principal directions e1,e2 as the local basis,these coefficients may be written as the directional deriva-tives of the principal curvature vectors kmin = κminn and kmax= κmaxn along the principal directions, that is

a = De1 kmin b = De2 kmin c = De1 kmax d = De2 kmax (11)

3 Related Work

Given a mesh of points, an accurate estimation of local dif-ferential geometric properties of the underlying surface foreach sample vertex is, despite of a number of works, still achallenge issue.

Several techniques have been proposed for computingthe elements of a tangent reference frame, which is an or-thonormal basis composed of a normal vector at a surfacepoint and two perpendicular vectors (the tangent and bitan-gent vectors). While most of them consider that the tan-gent and bitangent vectors are aligned with the texture co-ordinates (12), there is a variety of proposals for accuratelyestimating the normal vectors on the basis of the adjacentface normals (9; 5; 19; 14). Using the functionality of tex-ture sampling on the vertex shader in Shader Model (SM)3.0 (16), Calver proposes making available to the GPU ver-tex and adjacency data stored in textures. Based on this ap-proach, the author devises an algorithm for computing faceand vertex normals entirely on the GPU, after the vertex-level deformations (4).

To our knowledge, the geometric properties of secondand third order, namely the curvatures and their derivatives,are still not robustly computable on GPUs due to its com-plexity. In this section, a brief description of three paradigmsof techniques for estimation of geometric properties is pro-vided: patch-fitting methods, averaging methods, and curvesampling methods.

3.1 Patch-fitting Techniques

One of the most traditional methods for estimating the dif-ferential properties from triangle meshes is by approximat-ing a parametric surface to a local region of mesh points,then computing the matrix W from Eq. 7 analytically us-ing the parametric derivatives of this surface (11). Mostly,this surface is represented in the tangent frame associated tothe vertex of interest P , so that we may consider that W isequal to −IIS. For estimating properties of second order, thelowest order suitable is a quadratic surface (19).

Goldfeather and Interrante (8) find, however, that the ro-bustness of the tensor IIS depends on how the local regionaround P is approximated by parabolas along the directionof the edges passing through the adjacent vertices. Approxi-mation errors are greater in meshes sampled irregularly sincein such cases the number of analytical surfaces that may fitin the same neighborhood of points is greater. To obtain bet-ter estimates of the principal directions, they propose higherorder surfaces combined with the information of surface nor-mals available at the adjacent vertices. The idea of exploitingthe information of the normals at each adjacent vertex wasfollowed by other techniques based on the 1-ring neighbor-hood beyond patch-fitting methods (18). It has shown to beespecially important for obtaining robust results in irregu-larly sampled meshes.

4 Batagelo H.C. & Wu, S.-T.

According to Agam and Tang, the main drawback of thisclass of techniques is to be incapable of accurately modelingthe local surface geometry and to introduce a strong elementof smoothing t that may reduce the accuracy of the curvatureestimation (1). Rusinkiewicz points out the ambiguity of thecalculus of curvature at configurations of geometry with ver-tices coincident with two intersecting lines. In such regions,either a planar or hyperbolic surface fits the point set (18).We hypothesize that, if we model the vicinity of a point asa set of curves, we may gain flexibility in describing abruptvariations around a vertex.

3.2 Averaging Methods

This approach is based on the ability of estimating a cur-vature tensor for each edge and then averaging the tensorscomputed for the edges within a mesh region around eachvertex. It considers that, for each edge E of a triangle mesh,there is an associated minimum curvature in the directionalong the edge, and a maximum curvature in the perpendicu-lar direction that crosses the edge on the tangent plane. Suchcondition allow us to define a curvature tensor IIS for anypoint on the edge, and the average of the tensors computedfor the edges inside a given region around any vertex P .Because the point has its own tangent frame, the coordinatetransformation law must be applied on IIS of each edge be-fore averaging.

The techniques of approximation by curvature tensor av-eraging do not suffer from the deficiencies of the patch-fitting approach at hyperbolic points. However, other config-urations may accumulate errors that are not minimized evenwhen increasing the number of edges used in the evaluationof the average. One of these configurations is found at thepoles of a geodesic sphere build up from the subdivision ofa tetrahedron (18). At these pole points, the estimation errordoes not decrease as the mesh refinement increases.

Trying to solve the deficiencies of curvature tensor aver-aging techniques (6; 2), Rusinkiewicz proposes a new tech-nique of approximation (18) based on the idea of Goldfeatherand Interrante (8). He uses the already available vertex nor-mals at the vertices in the 1-ring neighborhood. For com-puting weights, he found that the “Voronoi area” weightingpresented by Meyer et al. (15) produces the best estimatesof curvature for triangles of varying size and shape. Similarto the traditional algorithms for computing vertex normals,the tensor IIS is first computed for the faces and then esti-mated for the vertices through averaging of the tensors of theadjacent faces. Hence, two passes are required. Moreover,the computation of the Voronoi area weighting is somehowcumbersome. Our conjecture is that, by applying the curvesampling approach we may simplify the procedure withoutdegrading the accuracy of results.

3.3 Curve Sampling Techniques

Instead of fitting a patch to points in a local neighborhood ofthe point of interest P , a set of curves leaving P in distinctdirections is sampled for curvature estimation.

For discrete surfaces, the estimative of the normal curva-ture in the direction of an edge between a vertex P to q canbe obtained by the formula of the curvature of the osculatingcircle that spans these vertices (5; 15):

κPq = 2(P −q) ·nP

‖P −q‖2 (12)

where nP is the surface normal at P . By combining equa-tions 4, 7 and 12, we obtain a linear equation system. Thissystem can be solved for IIS through linear least squares (5).When comparing this set of equations with those obtainedwith the quadratic surface approximation, we observe thatthey differ only with respect to the type of curves approxi-mated in the direction of each edge: parabolas, for the ap-proximation by quadratic surfaces; circles, for the approx-imation by normal curvature. Therefore, the estimative bynormal curvature suffers from the same weaknesses of theapproximation by analytical surfaces.

To overcome this restriction, Agam and Tang proposeto fit the local neighborhood of the vertex P with a set ofquadratic curves, then calculate the normal curvature of eachcurve separately (1). The directions along the minimum andmaximum curvatures are chosen as principal directions. Theestimation accuracy relies strongly on the sampling coverageand on the estimation accuracy of the normal curvatures.

To be less sensitive to the sampling frequency, we fol-low Goldfeather and Interrante and propose to overcome thementioned restriction by substituting for Eq. 12 the expres-sion of directional derivative of normal curvature given in(18), which also considers the information of the vertex nor-mals at each adjacent vertex.

4 Our Proposal

We propose a new method for computing second and higherorder elements of discrete differential geometry of arbitrarymeshes. The approach is capable of computing, for each ver-tex, the curvature tensor IIS and therefore the principal cur-vatures and directions, but also the coefficients of the tensorof curvature derivative CS. Our proposal focuses on the fol-lowing requisites:

– Efficiency: The total amount of time for estimating thecurvature tensor, principal curvatures and directions, andtensor of curvature derivative should allow the algorithmto be used in real-time for models composed of thou-sands of vertices on current desktop hardware. In orderto do that, we consider the possibility of using the band-width and stream processing power of today’s GPUs forperforming the estimation. The instruction set and datastructures used by the algorithm should be simple enoughto allow such portability.

Estimating Curvatures and their Derivatives on Meshes of Arbitrary Topology from Sampling Directions 5

– Coverage: The algorithm should avoid restrictions withrespect to the topology of the model. In particular, the al-gorithm should be capable of dealing with surfaces withholes, meshes with irregular sampling and different ver-tex valences. It also should handle the lack of data aboutface orientation, so that the only required connectivitydata should be the 1-ring neighborhood of each vertex,possibly without a winding order.

– Robustness: The algorithm should satisfactorily estimatedifferential geometry properties for the configurationsconsidered problematic for the methods cited before: col-linear points and points of intersection between two lines.In digitized models, the presence of outliers should beminimized in comparison with the cited methods.

None of the techniques presented fulfill all the requi-sites exposed above. Though the techniques based on thepatch-fitting and curve sampling approaches are suitable forstream processor architectures and, therefore, are very effi-cient, they do not produce results robust enough for arbi-trary meshes. On the other hand, the techniques based oncurvature tensor averaging are robust, but cumbersome toimplement on a stream processor due to the need of com-plex data structures or multiple execution passes. For in-stance, the technique by Alliez et al. (2) requires a costlypre-processing for determining which edges (and in whichamount) are inside a region around each vertex. Likewise, inthe algorithm proposed by Rusinkiewicz (18), the averagingbetween the curvature tensors at the faces requires at leasttwo processing passes, which hinders its use on the GPU.

In order to satisfy the requisites above, our proposal triesto combine the robustness of the algorithms based on theaveraging method, and the simplicity and efficiency of themethods based on the curve sampling technique. It standsout as an adaptation of the algorithm by Rusinkiewicz, bothfor computing the curvature tensor as well as the tensor ofcurvature derivative.

4.1 Estimating the curvature tensor

For each vertex P , we estimate IIS in the tangent referenceframe U,V,n, so that we may assume that IIS = −WS.Replacing it in Eq. 7, the derivative of the normal vector nin a given direction x on the tangent plane passing throughP is expressed by

IISx = Dxn (13)

On a smooth surface, the vector x may assume any directionperpendicular to the normal vector n. On a mesh with 1-ring neighborhood, these directions can be approximated byvectors from P to each vertex qi of the 1-ring neighborhood.Similarly, the directional derivatives of the normal vectorscan be approximated by the difference of the normal vectorspreviously estimated at the vertices. We remark that both thedifferences of positions and normals should be expressed in

the reference frame U,V,n. Hence, Eq. 13 may be writtenin the following finite-difference form

IIS

[(qi −P) ·U(qi −P ·V

]=

[(nqi −nP) ·U(nqi −nP) ·V

]

which may be expanded into[(qi −P) ·U (qi −P) ·V 0

0 (qi −P) ·U (qi −P) ·V] e

fg

=

=[

(nqi −nP) ·U(nqi −nP) ·V

], (14)

where qi, i = 1,2, ...,n, are n vertices in the 1-ring neighbor-hood of P . This system is solved for IIS using a linear leastsquares algorithm. As in the implementation by (18), we usethe LDT t decomposition method.

Differently from Rusinkiewicz, our proposal performsthe calculation directly with respect to the vertices, in a sin-gle running loop, because it does not require additional com-putations for transforming the tensor IIS computed on thefaces to the tangent plane at the vertices.

4.2 Estimating the tensor of curvature derivative

To estimate the tensor of curvature derivative, we resort toEq. 11 which gives us a simple way to obtain CS in coordi-nates relative to the principal directions. In order to do so,we first compute IIS and then extract its eigenvalues (princi-pal curvatures) and eigenvectors (principal directions). Theprincipal directions become the new basis vectors e1,e2on the tangent plane at P . Now we can express each di-rection Ei =

−−→Pqi in these new coordinates. We then obtain

the following system of equations which expresses, by finitedifferences, the minimal and maximal curvature variationsalong distinct directions Ei with regard to the principal di-rections:

CS

[Ei · e1Ei · e2

]=

[∆κmini ∆κmaxi

∆κmini ∆κmaxi

], i = 1,2, ...,n, (15)

where ∆κmini and ∆κmaxi are, respectively, the difference be-tween the minimal and maximal normal curvatures at ver-tices qi and P , that is,

∆κmini = κminqi−κminP

∆κmaxi = κmaxqi−κmaxP

In an analogous way that we obtain the finite-difference formfor computing IIS, we may expand Eq. 15 to explicit thecoefficients a, b, c and d of the tensor CS

Ei · e1 Ei · e2 0 00 Ei · e1 Ei · e2 00 Ei · e1 Ei · e2 00 0 Ei · e1 Ei · e2

abcd

=

∆κmini

∆κmini

∆κmaxi

∆κmaxi

, (16)

and solve it through a linear least squares technique, againusing LDLt decomposition. The computation of the eigen-vectors and eigenvalues is done by Jacobi iteration.

6 Batagelo H.C. & Wu, S.-T.

Fig. 2 RMS error of Gaussian curvature estimation on a discretizedtorus as a function of increasing sampling randomness. Bottom: Wire-frame torus with minimum sampling randomness (left) and maximumrandomness (right).

5 Experimental Results and Discussions

We have implemented our algorithm both on the CPU andGPU. On the CPU, the algorithm was implemented in C++using the LAPACK library (3). Methods based solely on the1-ring neighborhood of vertices, such as the approximationby quadratic surface, cubic surface (8), normal curvature andtensor averaging (18), were also implemented with this li-brary in order to perform a fair comparison with our algo-rithm when running the robustness tests.

From our experiments, we observed that the techniqueproposed by Agam and Tang, though delivers good results,is sensitive to the sampling coverage and frequency. Thenonexistence of a robust algorithm for adaptively choosingappropriate sampling coverage makes it impractical in somesituations. Hence, we have decided to limit the comparisonsof our model with the algorithms presented by Rusinkiewicz,Goldfeather and Interrante.

Figure 2 shows a comparison, for different techniques, ofthe RMS error for estimating the Gaussian curvature (Eq. 8)of a discretized torus as the surface sampling becomes moreirregular. Two wireframe renderings of the torus show theminimum and maximum mesh sampling irregularity used.The normals were computed as the normalized sum of thenormal vectors at the faces (9). These results show that thealgorithm by Rusinkiewicz has the smaller proportional in-crease in the estimation error due to sampling irregularity.Our algorithm produces similar results, though slightly lessaccurate. The remaining techniques are more accurate only

Fig. 3 RMS error of Gaussian curvature estimation on a discretizedtorus as a function of increasing vertex displacement along the exactvertex normal. Bottom: Wireframe torus with minimum (left) and max-imum (right) displacement.

when the torus is regularly sampled, as they assume the sur-face is locally polynomial, a condition fully satisfied by thetorus. Nevertheless, they are more sensitive to sampling noiseeven when considering polynomial surfaces.

Figure 3 compares the RMS error for estimating the Gaus-sian curvature of the discretized torus as noise is added tothe vertices. The vertices are displaced along fixed normalvectors computed analytically. Our technique produces themost accurate results in this case. In contrast, the techniquesof approximation by quadratic surfaces and normal curva-ture produce gross estimation errors, as they do not use thenormals to counterbalance the displacement noise.

Figure 4 shows the color-coded visualization of the prin-cipal curvatures for a horse model (48,484 vertices, 96,964triangles) and a bunny model (72,027 vertices, 144,460 tri-angles). The colors are modulated with the model’s diffuseshading. The amount of outliers, which come out as smallisolated colored dots, is relatively low in our technique andvisually equivalent to the results of Rusinkiewicz. The re-maining techniques produce more outliers as a result of theexistence of local configurations considered problematic forthese approaches. The results of the estimation by curve sam-pling of normal curvature are not shown, as they are visuallyidentical to the quadratic approximation.

For the timing tests, we have compared our techniquewith that of Rusinkiewicz (18) due to the similarity of ro-bustness results with those of our algorithm. In that case,both algorithms were implemented using optimized code.The test platform was a AMD Athlon XP 2000+ 1.67GHz

Estimating Curvatures and their Derivatives on Meshes of Arbitrary Topology from Sampling Directions 7

Quadratic approximation

Quadratic approximation

Goldfeather and Interrante (8)

Goldfeather and Interrante (8)

Rusinkiewicz (18)

Rusinkiewicz (18)

Our method

Our method

0

0κmin < 0 κmin > 0

κmax < 0

κmax > 0

Fig. 4 Color-coded principal curvatures for a horse and bunny model according to different estimation approaches.

with 512MB RAM, and a low-end GPU NVIDIA GeForceFX 6200 AGP 8x with 256MB VRAM.

On the GPU, the algorithm has been implemented astwo SM 3.0 pixel shaders in HLSL (16), one for estimat-ing the tensor of curvature and another one for the tensorof curvature derivative. The implementation of the linearleast squares algorithm and Jacobi iteration on the GPU isstraightforward, since the operations involved are conven-tional arithmetic ones that are supported by current graph-ics hardware. Following the paradigm of using the GPU asa general-purpose stream processor, geometry and connec-tivity data are encoded previously in floating point textures.These textures are accessed on the pixel shader that outputsthe computed values to render textures which are then usedby the application.

Figure 5 shows the performance for computing the cur-vature tensor (including the computation of principal cur-vatures and directions) and tensor of curvature derivative incomparison to (18). The processing time for our algorithmon the GPU is shown as well and includes the time for trans-ferring the textures to and from the GPU. From these re-sults we see that our method on the CPU is about two timesmore efficient for the tested model. This is mainly becausethe algorithm performs all the computations in a single pass.This simplicity, which allows us to port the algorithm to theGPU, improves the efficiency to real-time rates even in low-end graphics hardware. In general, we have seen that ouralgorithm has a significant increase in performance with re-spect to the technique of Rusinkiewicz, at a minor cost ofdecreasing the accuracy on irregularly sampled meshes. For

Fig. 5 Performance for estimating the curvature tensor and tensor ofderivative curvature on the horse model, in milliseconds.

digitized models, the decrease of accuracy is hardly notice-able.

Figure 6 shows some screenshots of a demonstration thatwe have implemented to visualize the estimated differentialgeometry elements computed in real-time on the GPU. In(a) we show a knot model with principal curvatures visu-alized as colors using the color code shown in Figure 4.In (b) we show the Gaussian curvature of a teapot modelin grayscale. Darker and lighter tones correspond to nega-tive and positive curvatures, respectively. Middle gray cor-responds to zero. Notice that the surface is locally hyper-bolic when K < 0 (darker), parabolic or planar when K = 0(mid-gray), and elliptical when K > 0 (lighter). In (c) we

8 Batagelo H.C. & Wu, S.-T.

(a) (b) (c)

(d) (e) (f)

0

0

< 0

> 0

|CS|

Fig. 6 Visualization of differential geometry properties estimated by our algorithm on the GPU. (a) principal curvatures using the color codeshown in Figure 4; (b) Gaussian curvature, in grayscale; (c) mean curvature, in grayscale; (d, e) principal directions; (f) magnitude of the tensorof curvature derivative.

show the mean curvature (Eq. 9) of a bunny model, alsoin grayscale. The principal directions of a teapot and knotmodel are shown in (d) and (e), respectively. The red ar-row points to the tangent direction of the minimum curva-ture, while the green arrow points to the tangent directionof the maximum curvature. In (f) we show the color-codedvisualization of the magnitude of the tensor of curvaturederivative (|CS| = c2

1 + c22 + c2

3 + c24, where c1, c2, c3 and c4

are the non-symmetrical components of the tensor). Imageswith higher resolution and source code of the demonstra-tion are available at http://www.dca.fee.unicamp.br/projects/mtk/batagelo.

6 Conclusions

We have presented a curve sampling algorithm for estimat-ing second or higher order local elements of discrete differ-ential geometry for arbitrary meshes. The algorithm is sim-ple to implement, as it requires only the 1-ring adjacencyinformation of each point (not necessarily ordered) and theassociated normal vectors to increase accuracy in compari-son to traditional patch-fitting and curve sampling methods.On a single loop over the vertices, it performs the estimationby sampling the curves in the direction of each edge leavingeach vertex, then solving a system of linear equations thatrelate the directional derivative of the normal curvatures tothe tensor.

The algorithm is efficient in comparison to methods withsimilar accuracy, and is suitable for porting to the stream ar-chitecture of current GPUs. As shown in our timing tests,this allows the real-time computation of curvature tensors(including the extraction of principal directions and princi-pal curvatures) and tensors of derivative of curvature evenfor models composed of thousands of vertices. As a furtherwork, it motivates us to develop a 3D interaction toolkit thatuse the differential geometry properties to correctly performdirect manipulation with geometry deformed on the GPU.

According to the robustness tests, the accuracy of ourmethod is greater than previous methods based on 1-ringneighborhood of vertices when considering irregularly sam-pled meshes and digitized models. In comparison to the tech-niques implemented, it produces the most accurate resultson models with displacement noise. For irregular meshes,it produces slightly inferior results to the tensor averagingmethod proposed by Rusinkiewicz. However, the differencesare hardly distinguishable, and the benefits of efficiency (al-most half the time of (18)) and portability (adaptation tostream architectures) greatly overcome this. Nevertheless,we would like to further compare our results with the onescomputed analytically, especially with the values of deriva-tive of curvature.

Acknowledgements This project was supported by the National Re-search Council (CNPq) under the grant number 141685/2002-6 and theState of Sao Paulo Research Support Foundation (FAPESP) under thegrant numbers 1996/0962-0 and 03/13090-6.

Estimating Curvatures and their Derivatives on Meshes of Arbitrary Topology from Sampling Directions 9

References

1. Agam, G., Tang, X.: A sampling framework for accuratecurvature estimation in discrete surfaces. IEEE Trans-actions on Visualization and Computer Graphics 11(5),573–583 (2005)

2. Alliez, P., Cohen-Steiner, D., Devillers, O., Levy, B.,Desbrun, M.: Anisotropic polygonal remeshing. ACMTransactions on Graphics 22(3), 485–493 (2003)

3. Anderson, E., Bai, Z., Bischof, C., Blackford, S., Dem-mel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Ham-marling, S., McKenney, A., Sorensen, D.: LAPACKUsers’ Guide, 3rd edn. Society for Industrial and Ap-plied Mathematics, Philadelphia, PA (1999)

4. Calver, D.: Accessing and modifying topology on thegpu. In: W. Engel (ed.) ShaderX3: Advanced Renderingwith DirectX and OpenGL. Charles River Media (2004)

5. Chen, X., Schmitt, F.: Intrinsic surface properties fromsurface triangulation. In: ECCV ’92: Proceedings of theSecond European Conference on Computer Vision, pp.739–743. Springer-Verlag, London, UK (1992)

6. Cohen-Steiner, D., Morvan, J.M.: Restricted delaunaytriangulations and normal cycle. In: SCG ’03: Proceed-ings of the Nineteenth Annual Symposium on Computa-tional Geometry, pp. 312–321. ACM Press, New York,NY, USA (2003)

7. DeCarlo, D., Finkelstein, A., Rusinkiewicz, S.: Interac-tive rendering of suggestive contours with temporal co-herence. In: NPAR ’04: Proceedings of the 3rd Inter-national Symposium on Non-photorealistic Animationand Rendering, pp. 15–145. ACM Press, New York, NY,USA (2004)

8. Goldfeather, J., Interrante, V.: A novel cubic-order al-gorithm for approximating principal direction vectors.ACM Transactions on Graphics 23(1), 45–63 (2004)

9. Gouraud, H.: Continuous shading of curved surfaces.IEEE Transactions on Computers 20(6), 623–629(1971)

10. Gravesen, J., Ungstrup, M.: Constructing invariant fair-ness measures for surfaces. Advances in ComputationalMathematics 17, 67–88 (2002)

11. Hamann, B.: Curvature approximation for triangulatedsurfaces. Computing Suppl. 8, 139–153 (1993)

12. Kilgard, M.J.: A practical and robust bump-mappingtechnique for today’s gpus. In: Game Developers Con-ference 2000: Advanced OpenGL Game Development.NVIDIA Corporation (2000)

13. Lange, C., Polthier, K.: Anisotropic smoothing of pointsets. Comput. Aided Geom. Des. 22(7), 680–692 (2005)

14. Max, N.: Weights for computing vertex normals fromfacet normals. Journal of Graphics Tools 4(2), 1–6(1999)

15. Meyer, M., Desbrun, M., Schroder, P., Barr, A.H.: Dis-crete differential-geometry operators for triangulated 2-manifolds. In: H.C. Hege, K. Polthier (eds.) Proceed-ings of Visualization and Mathematics III, pp. 35–57.Springer-Verlag, Heidelberg, Germany (2003)

16. Microsoft: DirectX SDK Programmer’s Reference. Mi-crosoft Corporation, Redmond, WA, USA (2006)

17. Praun, E., Hoppe, H., Webb, M., Finkelstein, A.: Real-time hatching. In: SIGGRAPH ’01: Proceedings of the28th Annual Conference on Computer Graphics and In-teractive Techniques, p. 581. ACM Press, New York,NY, USA (2001)

18. Rusinkiewicz, S.: Estimating curvatures and theirderivatives on triangle meshes. In: 3DPVT ’04: Pro-ceedings of the 3D Data Processing, Visualization,and Transmission, 2nd International Symposium on(3DPVT’04), pp. 486–493. IEEE Computer Society,Washington, DC, USA (2004)

19. Taubin, G.: Estimating the tensor of curvature of asurface from a polyhedral approximation. In: ICCV’95: Proceedings of the Fifth International Conferenceon Computer Vision, p. 902. IEEE Computer Society,Washington, DC, USA (1995)

20. Wang, L., Wang, X., Tong, X., Lin, S., Hu, S., Guo, B.,Shum, H.Y.: View-dependent displacement mapping.ACM Transactions on Graphics 22(3), 334–339 (2003)

21. Yoo, K.H., Ha, J.S.: Geometric snapping for 3d meshes.In: International Conference on Computational Science,pp. 90–97. Krakow, Poland (2004)

Harlen Costa Batagelo is a Ph.D.student in Electrical Engineering atthe School of Electrical and Com-puter Engineering of the State Uni-versity of Campinas, Brazil. He re-ceived his Master’s Degree in 2002from the same institution, and hisBachelor’s in Computer Sciencefrom the University of the West ofSanta Catarina, in 1999. His mainresearch interests are real-time ren-dering, computer games, graphicshardware and interaction.

Wu, Shin - Ting is an associateprofessor of School of Electricaland Computer Engineering (FEEC),State University of Campinas inBrazil. She received Dr.-Ing. in In-formatics from the Technical Uni-versity of Darmstadt, Master inElectrical Engineering from StateUniversity of Campinas, and Bach-elor in Electrical Engineering fromFederal University of Minas Gerais,in 1991, 1985, and 1981, respec-tively. Before she joined the FEEC,she worked as researcher at Re-nato Archer Research Center (Cen-PRA) and did her post-doc fellowin Institute for Computer Graphics

of Fraunhofer Gesellschaft. Her research interests include geometricmodeling, scientific visualization, and 3D interactions.