-
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: [email protected].
?? Supported in part by NSFC under the grant 60773165, NSFC key
project under thegrant 10990013). Corresponding author. E-mail
address: [email protected].
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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.
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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-
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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
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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
.
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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.
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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
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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.
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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)
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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
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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
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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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