Construction of Minimal Catmull-Clark’sSubdivision Surfaces with

Given Boundaries

Qing Pan1) ? Guoliang Xu 2) ??

1)College of Mathematics and Computer Science,Hunan Normal University, Changsha, 410081, China

2)LSEC, Institute of Computational Mathematics, Academy of Mathematicsand System Sciences, Chinese Academy of Sciences, Beijing 100190, China

Abstract. Minimal surface is an important class of surfaces. They arewidely used in the areas such as architecture, art and natural scienceetc.. On the other hand, subdivision technology has always been activein computer aided design since its invention. The flexibility and highquality of the subdivision surface makes them a powerful tool in ge-ometry modeling and surface designing. In this paper, we combine thesetwo ingredients together aiming at constructing minimal subdivision sur-faces. We use the mean curvature flow, a second order geometric partialdifferential equation, to construct minimal Catmull-Clark’s subdivisionsurfaces with specified B-spline boundary curves. The mean curvatureflow is solved by a finite element method where the finite element spaceis spanned by the limit functions of the modified Catmull-Clark’s subdi-vision scheme.

Key words: Minimal Subdivision Surface, Catmull-Clark’s Subdivision,Mean Curvature Flow.MR (2000) Classification: 65D17

1 Introduction

Surfaces whose mean curvature H is zero everywhere are minimal surfaces. Min-imal surfaces are often used as models in architecture because of having severaldesirable properties. Most important of all, minimal surfaces have the least sur-face area, which makes them almost indispensable in large scale and light roofconstructions. Secondly, minimal surfaces are separable. Any sub-patch, no mat-ter how small, sheared from a minimal surface still has the least area of all surfacepatches with the same boundary. Thirdly, minimal surfaces have the balancedsurface tension in equilibrium at each point on the roof, as on a soap film, whichstabilizes the whole construction. Finally, there are no umbilicus points on a

? Supported in part by NSFC grant 10701071 and Program for Excellent Talents inHunan Normal University (No. ET10901). E-mail address: panqing@lsec.cc.ac.cn.

?? Supported in part by NSFC under the grant 60773165, NSFC key project under thegrant 10990013). Corresponding author. E-mail address: xuguo@lsec.cc.ac.cn.

2 Qing Pan & Guoliang Xu

minimal surface; hence no water can stay on the minimal surface roof. Architec-ture inspired from minimal surfaces embodies the unity of economy and beauty.The most representative buildings of that architectural style are the roofs ofthe Munich Olympic stadium, the former Kongreßhalle in Berlin. In art worldwe see plenty of ingenious sculpture works playing the ultimate of minimal sur-faces. Scientists and engineers have anticipated the nanotechnology applicationsof minimal surfaces in the areas of molecular engineering and materials science.

Studies on minimal surfaces was traced back 250 years ago (1744) with Euleras the forerunner, whose research focused on the rotation surface with minimalarea. Since then the research of minimal surfaces has been active for several hun-dred years. In 1760, Lagrange derived the equation minimal surfaces satisfy. Thewell-known Plateau (1855-90) problem is the existence problem of constructinga piece of surface that interpolates the given boundary curve and has minimalarea. This problem, though raised by Lagrange in 1760, was named after Plateau,who created several special cases experimenting with soap films and wire frames.Various special forms of this problem were solved, but it was only in 1930 thatgeneral solutions were found independently by Douglas and Rado. The generalsolution of the equation H = 0 was given by Weierstrass (1855-90).

The construction of minimal surfaces have been a heat topic in the area ofcomputer aided design. According to Consin and Monterde [4], there are certainconditions that the control points of Bézier surfaces must satisfy, and in thebicubical case all minimal surfaces are pieces of the Ennerper surfaces up toan affine transformation. Using the four-sided Bézier surface to approximate theminimal surface, Monterde (see [12]) solved Plateau-Bézier problem by replacingthe area functional with the Dirichlet functional. Triangular Bézier surface basedon a variational approach was constructed by Arnal et al.[1]. Much has beendone (see [8], [10], [11]) on the use of minimal surfaces in geometry modelingand shape design. Discrete minimal surfaces were studied by Polthier in [15].Minimal surfaces as the steady solution of the mean curvature flow (see [16])were also produced, where they can be both continuous and discrete, usuallyBézier surfaces, or B-spline surfaces for the former.

B-splines have been widely accepted as representation tools for curves andsurfaces in the industrial design, however there is a serious limitation for de-signing minimal surfaces with any shaped boundaries using Bézier, B-spline andNURBS because they require the surface patch to be three- or four-sided. In1974, Charkin first brought the concept of discrete subdivision into the area ofcomputer graphics. Doo-Sabin (see [6]) and Catmull-Clark (see [3]) respectivelyproposed the subdivision schemes of biquadratic and bicubic B-spline for quadri-lateral mesh in 1978. The quartic triangular B-splines was developed by Loop(see [9]) in 1987. Henceforth, subdivision surfaces have rapidly gained popularityin computer graphics and computer aided design. Subdivision algorithms haveno limitation on the topology of the control mesh. They can efficiently generatesmooth surfaces from arbitrary initial meshes through a simple refinement algo-rithm, and they are flexible in creating the features of surface without difficulty.

Finite Element Methods for Geometric Modeling and Processing ... 3

It is obvious that these well-known subdivision algorithms suffer from seri-ous problems when applied to a control mesh with a boundary because theyare suitable for the interior control mesh. Boundary subdivision rules are veryimportant: a plenty of surface designing work deals with the input mesh withboundaries, marked edges and vertices, and the specific treatment for the fea-tures of boundaries, such as concave corners, convex corners, sharp creases andsmooth creases etc., is always necessary in order to satisfy the designing require-ment. For many surface modeling problems, such as the construction of bodiesof cars, aircrafts, machine parts and roofs, surfaces are usually piecewise con-structed with fixed boundaries. The following are the related works. Subdivisionrules of Doo-Sabin surfaces for the boundaries were discussed by Doo (see [5])and Nasri (see [14]). Based on the work of Hoppe et al.(see [7]) and Nasri (see[13]), Biermann et al.(see [2]) extended the well-known subdivision schemes ofCatmull-Clark and Loop. They solve some problems of the original ones, suchas lack of smoothness at extraordinary boundary vertices and folds near concavecorners, and improve control of the surface shapes with prescribed normals bothon the boundary and in the interior.

In this paper, we construct minimal subdivision surfaces based on the mod-ified Catmull-Clark’s subdivision algorithms [2] which improves the subdivisionscheme around boundaries, and it is preferable and acceptable to use B-spline torepresent surface boundary. The well-known mean curvature flow with Dirichletboundary condition is our evolution model. We adopt the finite element method,where the finite element space spanned by the limit functions of the modifiedCatmull-Clark’s subdivision scheme, as the discretization tool. All the aboveframeworks contribute to our target, successful construction of desirable mini-mal subdivision surfaces.

The remainder of this paper is organized as follows: Section 2 is a brief reviewof the Catmull-Clark’s subdivision scheme and its modification of the boundaries,as well as the evaluation of standard and nonstandard Catmull-Clark’s subdivi-sion surfaces. In Section 3 we provide the mean curvature flow used to constructthe minimal surfaces, and the details of its discretization and numerical com-putation. Section 4 show several graphic examples and some error comparingresults to illustrate the effects of our method. Section 5 is the conclusion.

2 Evaluation of Catmull-Clark’s Subdivision Surfaces

Our goal is to construct Catmull-Clark’s subdivision surface with specified bound-ary curves and minimal area. The subdivision surface is defined as the limit of aniterative refinement procedure starting from an initial control mesh where a se-quence of increasing refined meshes can be achieved according to the subdivisionscheme. The Catmull-Clark’s subdivision scheme requires all faces of the initialcontrol mesh must be quadrilaterals. The subsequent refined meshes consist ofonly quadrilaterals. The control vertices of the refined meshes are generated fromthe control vertices of the previous step by a portfolio of weight coefficients. Fi-

4 Qing Pan & Guoliang Xu

(a) (b)

(c) (d)

Fig. 1. (a): A regular patch over the shaded quadrilateral with its neighboring 16control vertices. (b): An irregular patch over the shaded quadrilateral with an extraor-dinary vertex labeled ’1’ whose valence is 5. (c): Subdividing this irregular patch oncegenerates 3 shaded sub-patches, and enough control vertices for evaluating them. (d):A unit square is subdivided into unlimited group of quadrilateral sub-domains.

nally, this sequence of meshes converges to a limit surface composed of unlimitednumber of surface patches.

We can refer to [3] for the standard Catmull-Clark’s subdivision scheme, andits modification proposed by Biermann et al. is described in [2]. we need classifythe control mesh into two groups, i.e., standard mesh and nonstandard mesh.Nonstandard mesh includes boundary quadrilaterals and sub-boundary quadri-laterals. Standard mesh consists of only interior quadrilaterals. The quadrilater-als containing boundary vertices are named as boundary quadrilaterals, the onesadjacent to the boundary quadrilaterals are called sub-boundary quadrilaterals,and all others are called interior ones.

2.1 Evaluation of Standard Catmull-Clark’s Subdivision Surface

In this section, we briefly describe the evaluation of the standard Catmull-Clark’ssubdivision surface whose control mesh consists of only interior quadrilaterals.

Each quadrilateral of the control mesh corresponds to one quadrilateral patchof the limit surface. The quadrilateral of the control mesh is regarded as the

Finite Element Methods for Geometric Modeling and Processing ... 5

parameter domain of the surface patch. We choose a unit square

Ω ={(u, v) ∈ R2 : 0 ≤ u ≤ 1, 0 ≤ v ≤ 1}

as the local parametrization for each quadrilateral tα and (u, v) as its barycentriccoordinates. A regular patch whose four control vertices have a valence of 4 canbe represented by 16 basis functions and their corresponding 16 control vertices:

xα(u, v) =16∑

i=1

Bi(u, v)xi, (1)

where the label i refers to the local sorting of the control vertices shown inFig.1(a). The bicubic B-spline basis functions Ni are:

Bi(u, v) = N(i−1)%4(u)N(i−1)/4(v), i = 1, 2, · · · , 16,where ”%” and ”/” stand for the remainder and division respectively. The func-tions Ni(t) are the cubic uniform B-spline basis functions:

N0(t) = (1− 3t + 3t2 − t3)/6,N1(t) = (4− 6t2 + 3t3)/6,N2(t) = (1 + 3t + 3t2 − 3t3)/6,N3(t) = t3/6.

If a quadrilateral is irregular, i.e., at least one of its control vertices has avalence other than 4, the resulting patch is not a bicubic B-spline. Now we assumeextraordinary vertices are isolated, i.e., there is no edge in the control mesh suchthat both of its vertices are extraordinary. This assumption can be fulfilled bysubdividing the mesh once. Under this assumption, any irregular patch has onlyone extraordinary vertex. In order to evaluate the surface at any parametricvalue (u, v) ∈ tα, the mesh needs to be subdivided repeatedly until the parametervalues of interest are interior to a regular patch. Each subdivision of an irregularpatch produces three regular sub-patches and one irregular sub-patch (see Fig.1(b) and (c)). Repeated subdivision of the irregular patch produces three groupsof regular patches. This irregular surface patch can be piecewise parameterizedas shown in Fig.1(d). The sub-domains Ωnj , n ≥ 1, j = 1, 2, 3, which can beevaluated, are given as:

Ωn1 = {(u, v) : u ∈ [2−n, 2−n+1], v ∈ [0, 2−n]},Ωn2 = {(u, v) : u ∈ [2−n, 2−n+1], v ∈ [2−n, 2−n+1]},Ωn3 = {(u, v) : u ∈ [0, 2−n], v ∈ [2−n, 2−k+1]}.

(2)

They can be mapped onto the unit square Ω through the transform

t1,n(u, v) = (2nu− 1, 2nv), (u, v) ∈ Ωn1 ,t2,n(u, v) = (2nu− 1, 2nv − 1), (u, v) ∈ Ωn2 ,t3,n(u, v) = (2nu, 2nv − 1), (u, v) ∈ Ωn3 .

6 Qing Pan & Guoliang Xu

The surface patch xα(u, v) is then defined by its restriction to each quadrilateral

xα(u, v)|Ωnj =16∑

i=1

Ni(tj,n(u, v))xj,ni , j = 1, 2, 3; n = 1, 2, · · · , (3)

where xn,ji are the properly chosen 16 control vertices around the irregular patchat the subdivision level n = floor(min(−log2(u),−log2(v))). Three sets of controlvertices are (see Fig.1(c))

{x1,ni }= [ xn8 ,xn7 ,xn2N+5,xn2N+13,xn1 ,xn6 ,xn2N+4,xn2N+12,xn4 ,xn5 ,xn2N+3,xn2N+11,xn2N+7,x

n2N+6,x

n2N+2,x

n2N+10 ],

{x2,ni }= [ xn1 ,xn6 ,xn2N+4,xn2N+12,xn4 ,xn5 ,xn2N+3,xn2N+11,xn2N+7,xn2N+6,xn2N+2,xn2N+10,x

n2N+16,x

n2N+15,x

n2N+14,x

n2N+9 ],

{x3,ni } = [ xn2 ,xn1 ,xn6 ,xn2N+4,xn3 ,xn4 ,xn5 ,xn2N+3,xn2N+8,xn2N+7,xn2N+6,xn2N+2,xn2N+17,x

n2N+16,x

n2N+15,x

n2N+14 ].

With the subdivision matrix A and the extended subdivision matrix Ā, wecan get these control vertices by

Xn = AXn−1 = · · · = AnX0

andX̄n+1 = ĀXn = ĀAnX0

where Xn = [xn1 , · · · ,xn2N+8]T and X̄n = [xn1 , · · · ,xn2N+17]T .

2.2 Evaluation of Nonstandard Catmull-Clark’s Subdivision Surface

As noted above, for the nonstandard Catmull-Clark’s subdivision surface, whosecontrol mesh includes boundary quadrilaterals and sub-boundary quadrilaterals,we adopt the modified Catmull-Clark’s subdivision rules. Subdividing a sub-boundary quadrilateral once will result in four interior quadrilaterals, so it iseasy to evaluate their corresponding patches using the evaluation method of thestandard Catmull-Clark’s subdivision surface.

The condition of boundary quadrilaterals is a little complicated, howeverwe can repeatedly subdivide it till its sub-patches belong to the class of sub-boundary quadrilaterals. The patches for sub-boundary quadrilaterals can beevaluated using the method stated in the previous paragraph. The boundaryquadrilaterals may need to be further subdivided if the parameter values, wherethe surface patch need to be evaluated, are in this domain. This process arecarried through repeatedly till the parameter values to be evaluated are withina sub-boundary quadrilateral.

In the next section, we will introduce the evolution equation and its finiteelement method based on the modified Catmull-Clark’s subdivision scheme.

Finite Element Methods for Geometric Modeling and Processing ... 7

3 Minimal Surface Construction

Let M0 be a compact immersed orientable surface in R3 and x ∈ M0 be a generalsurface point. We intend to find a family {M(t) : t ≥ 0} of smooth orientablesurfaces in R3 which evolve according to the mean curvature flow

∂x∂t

= 2Hn, M(0) = M0, (4)

where H and n are the mean curvature and the surface normal of M respectively.It is well known that the mean curvature flow is area reducing. The area reducingstops when H = 0. Since

∆sx = 2Hn,

the steady solution of the following mean curvature flow

∂x∂t

= ∆sx, M(0) = M0, (5)

is the minimal surface. We use a finite element method to obtain the numericalsolution of (5), and our finite element basis functions are the limit form of themodified Catmull-Clark’s subdivision scheme.

3.1 Finite Element Method for the Mean Curvature Flow

Let M be the limit surface of the modified Catmull-Clark’s subdivision schemefor the control mesh Md. We multiply a trial function ψ for (5) and apply theGreen’s formula, then we obtain the following weak form equation

Find x(t) ∈ V 3M(t), such that∫

M(t)

[∂x(t)

∂tψ + (∇sx(t))T∇sψ

]ds = 0, ∀ψ ∈ VM(t) ∩ C10 (M(t)),

M(0) = M0, ∂M(t) = Γ, ∀x ∈ Γ,(6)

where VM(t) ⊂ C1(M(t)) is a finite dimensional function space defined by themodified Catmull-Clark’s subdivision scheme for the discrete function values onthe vertices. C1(M(t)) is the function space consisting of C1 smooth functionson M(t), and C10 (M(t)) consists of functions of C

1(M(t)) with compact support.Let φi be a basis function of VM(t) corresponding to the control vertex xi

(i = 1, · · · ,m) of the surface M(t), where we assume {xi}m0i=1 are the interiorvertices, and the remaining {xi}mi=m0+1 are the boundary vertices. Then x(t)can be represented as

x(t) =m0∑

i=1

xi(t)φi +m∑

i=m0+1

xi(t)φi, xi(t) ∈ R3.

Take trial function ψ to be φj(j = 1, · · · ,m0), (6) can be rewritten as

8 Qing Pan & Guoliang Xu

m0∑

i=1

x′i(t)∫

M(t)

φiφjds +m0∑

i=1

xi(t)∫

M(t)

(∇sφi)T∇sφjds

= −m∑

i=m0+1

xi(t)∫

M(t)

(∇sφi)T∇sφjds, j = 1, · · · ,m0,

xj(0) = xj , j = 1, · · · ,m,

(7)

where xj is the j-th control vertex of the initial surface M(0). (7) is a set ofnonlinear ordinary differential equations for the unknowns xi(t), i = 1, · · · ,m0.The system is nonlinear because the domain M(t), over which the integrationsare taken, is also unknown. We use forward Euler scheme to discretize x′i(t)

as xk+1i −xki

τ for a given temporal step-size τ , and use a semi-implicit scheme todiscretize the remaining terms. A linear system is obtained

m0∑

i=1

xk+1i

∫

Mkφiφjds + τ

m0∑

i=1

xk+1i

∫

Mk(∇sφi)T∇sφjds

=m0∑

i=1

xki

∫

Mkφiφjds− τ

m∑

i=m0+1

x0i

∫

Mk(∇sφi)T∇sφjds, j = 1, · · · ,m0,

x0j = xj , j = 1, · · · ,m,

(8)

for the unknowns xk+1i , where Mk is the limit surface of the control vertices xki .

System (8) is iteratively solved for k = 0, 1, · · · , using GREMS method till thetermination condition

maxi‖xk+1i − xki ‖ ≤ ²

(² is a given small value) is satisfied.

3.2 Definition of Basis Functions

As mentioned above, the basis functions of our finite element function spaceVM(t) is the bicubic B-spline. We use φi to represent the basis function asso-ciating with the control vertex xi of the surface M , including its interior ver-tices, corner vertices and boundary vertices. The basis function φi is defined bythe limit of the modified Catmull-Clark’s subdivision scheme where its functionvalue is one at this vertex xi, but zero at any other vertices. The support of φiis compact and it covers the 2-ring neighborhoods of vertex xi.

It needs to evaluate φi and its partial derivatives in forming the linear system(8), whose parameter values are chosen to be the Gaussian quadrature knotswithin a unit square. Therefore we only need a few subdivision steps so as tobring these Gaussian quadrature knots into the interior of a regular quadrilateral.Let ej , j = 1, · · · ,mi be the 2-ring neighborhood elements of xi. If ej is regular,the expression (1) exists for φi on ej . If ej is irregular, local subdivision, asdescribed in §2.1 and §2.2, is needed around ej until the parameter values ofinterest are interior to a regular patch.

Finite Element Methods for Geometric Modeling and Processing ... 9

3.3 Parametrization of Subdivision Surface and Functions on theSurface

In Riemannian geometry, differentiable functions are smooth and C∞. However,our discretized version of the diffusion problem will be in the class C1. As wementioned earlier, the functions are defined by the limit form of the modifiedCatmull-Clark’s subdivision. Such a function is C2 smooth everywhere except atthe extraordinary vertices, where it is C1. The function is locally parameterizedas the image of the unit square defined by

Ω = {(u, v) ∈ R2 : 0 ≤ u ≤ 1, 0 ≤ v ≤ 1}.

That is, (u, v) is the barycentric coordinate of the quadrilateral. Using thisparametrization, our discretized representation of M is

M =k⋃

α=1

Tα, T̊α ∩ T̊β = ∅ for α 6= β,

where T̊α is the interior of the quadrilateral function patch Tα. Each quadrilateralsurface patch is assumed to be parameterized locally as

xα : Ω → Tα; (u, v) 7→ xα(u, v), (9)

where xα(u, v) is defined by (1) and (3). Function itself on the surface and itspartial derivatives, such as tangents and gradients, can be computed directly.The integration of a function on the surface M is calculated as

∫

M

fdx :=∑α

∫

Ω

f(xα(u, v))√

det(gij)du dv, (10)

where gij are the coefficients of the first fundamental form of the surface M . Theintegration on the square Ω is computed adaptively using Gaussian quadratureformulas (see [17]).

4 Experimental Results

In this section, we present several graphical and numerical results to show thatthe proposed method for constructing minimal subdivision surface is effective.

4.1 Graphical Examples

We firstly show three models of minimal surfaces with the analytic forms, Heli-coid, Catenoid and Ennerper. In Fig. 2, we discretize these three analytic surfacesat a rough level and perturb their interior domain as shown in the first column,then we linearly refine them several times as the initial constructions of our equa-tion evolution. The minimal subdivision surfaces as the steady solutions of (6)

10 Qing Pan & Guoliang Xu

(a) (a′) (a′′) (a′′′)

(b) (b′) (b′′) (b′′′)

(c) (c′) (c′′) (c′′′)

Fig. 2. The first row is the Helicoid surface model, the second row is the Catenoidmodel and the third row is Ennerper model. (a), (b) and (c) are their initial roughmeshes. (a′), (b′) and (c′) are the initial constructions for the equation evolution bylinearly refining the meshes in the first column. (a′′), (b′′) and (c′′) are the Catmull-Clark’s surface resulting from refining the rough constructions in the first column by themodified Catmull-Clark’s subdivision scheme. On the base of the initial constructionsin the second column, we show their corresponding minimal subdivision surfaces by ourequation evolution in (a′′′), (b′′′) and (c′′′). The density of meshes in the third columnand in the forth column is the same.

are presented in the forth column. We show their corresponding Catmull-Clark’ssurfaces in the third column which are obtained by refining the rough meshes inthe first column according to the modified Catmull-Clark’s subdivision scheme

Finite Element Methods for Geometric Modeling and Processing ... 11

until they have the same density as the corresponding meshes do in the secondcolumn. It is clear to see that the Catmull-Clark’s surfaces are very differentfrom the final minimal subdivision surfaces.

Fig. 3 shows three examples with fixed boundaries and arbitrary genus. Weconstruct their initial surfaces only from the boundary information at a roughlevel in the first column. We refine the initial meshes several times by linearmethod and show the results in the second column which are the initial con-structions of the equation evolution. The boundary curves can have discontinu-ity on its tangent direction, as shown in Fig. 3 (a′), and some model mesheshave extraordinary vertices clearly presented in Fig. 3 (b′) and (c′) where theface valence of some control vertices is 6. Similarly we also compare the result-ing minimal subdivision surfaces in the forth column with their correspondingCatmull-Clark’s surfaces in the third column where they have the same density,but the difference of them is very clear.

4.2 Refinement and Convergence

In order to further show the proposed method is effective, we compute the max-imum values of |H| from the discrete solutions of our numerical method for thesix models used above. We construct the initial surfaces of these models as theinitial value of the PDE evolution by subdividing the six models at graduallymore and more dense level according to the modified Catmull-Clark’s subdivisionscheme. The maximal asymptotic values of |H| are presented in Table 1. Fromthe numerical results, we can see that the maximal values of |H| monotonouslydecline as the increasing of subdivision times k. Hence, our numerical method isconvergent.

Asymptotic maximal values of |H|Models k k + 1 k + 2 k + 3 k + 4 k + 5

Fig 2(a) 3.783E-2 1.858E-2 1.009E-2 6.092E-3 4.236E-3 3.467E-3

Fig 2(b) 5.748E-2 2.972E-2 1.658E-2 1.077E-2 8.132E-3 6.787E-3

Fig 2(c) 4.638E-2 2.321E-2 1.418E-2 9.558E-3 7.628E-3 6.650E-3

Fig 3(a) 6.223E-2 3.015E-2 1.684E-2 9.713E-3 6.787E-3 5.595E-3

Fig 3(b) 1.073E-1 5.327E-2 2.916E-2 1.631E-2 1.061E-2 7.837E-3

Fig 3(c) 3.234E-1 1.628E-1 8.674E-2 5.607E-2 4.132E-2 3.314E-2

Table 1. k describes the subdivision times, where we subdivide the six models at 6more and more dense levels respectively. The data from the second to the seven roware the maximum approximate errors of the mean curvature |H| computed from thediscrete solutions of the PDE evolution.

5 Conclusions

Extensive research work has been done about minimal surfaces. The fascinatingcharacters of minimal surfaces make them widely used in shape designing and

12 Qing Pan & Guoliang Xu

(a) (a′) (a′′) (a′′′)

(b) (b′) (b′′) (b′′′)

(c) (c′) (c′′) (c′′′)

Fig. 3. (a), (b) and (c) are the roughest surface meshes of three models. (a′), (b′)and (c′) are their corresponding initial constructed surface meshes by linearly refining(a), (b) and (c) respectively. (a′′), (b′′) and (c′′) are their subdivision surfaces throughrefining the meshes in the first column according to the modified Catmull-Clark’s sub-division scheme. (a′′′), (b′′′) and (c′′′) are their corresponding minimal subdivisionsurfaces constructed by use of our method based on the initial constructions in thesecond column.

many other areas. Subdivision algorithm is a simple and efficient tool to describefree surfaces with any topology. In this paper we adopt the modification of theCatmull-Clark’s subdivision scheme which improves the boundary subdivisionrules for quadrilateral mesh. We successfully construct minimal Catmull-Clark’ssubdivision surfaces with given boundary curves using the mean curvature flow,and adopt the numerical method of the finite element based on the modifiedCatmull-Clark’s subdivision scheme. Our framework can uniformly and flexibly

Finite Element Methods for Geometric Modeling and Processing ... 13

treat all kinds of boundary conditions. The asymptotic error data show ournumerical method is also convergent.

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