A General Sixth Order Geometric Partial Differential

Equation and Its Application in Surface Modeling ?

Dan Liu, Guoliang Xu∗,LSEC, Institute of Computational Mathematics, Academy of Mathematics and System Sciences,

Chinese Academy of Sciences, Beijing, 100080, China

Abstract

In computer aided geometric design and computer graphics, high quality fair surfaces with G2 smoothnessare sometimes required. In this paper we derive a general sixth order geometric partial differentialequation from minimizing a curvature integral functional. The obtained equation is used to solve severalsurface modeling problems such as free-form surface design, surface blending and N-side hole filling,with G2 boundary constraints. We solve the equation numerically using a generalized divided differencemethod, where a quadratic fitting scheme is adopted to discretize several used geometric differentialoperators consistently. The experiments show that the proposed method is efficient and yields highquality surfaces.

Keywords: Surface Modeling; Euler-Lagrange Equation; Geometric PDE.

1 Introduction

The PDE based approach, which is used to handle surface modeling problems as an alternativeto spline-based method, begins with Bloor et al’s work in 1980s [2, 3]. They use the solutions ofbiharmonic equations to generate surfaces, where the boundary constraints are used to control thesurface shape. Surfaces produced by this method are smooth and meet the boundary constraintswith a certain degree of continuity.

One of the disadvantages using these equations is that the biharmonic equation is not geometricintrinsic and therefore its solution depends greatly on the used parametrization. In [7], meancurvature flow ∂tp = ∆Mp is used to substitute the diffusion equation, where M is a surface, pis a surface point on M and the Laplace-Beltrami operator ∆M is a generalization of Laplacianfrom flat space to manifold. Mean curvature flow has been widely used in surface processing anddenoising. Clarenz et al. [6] introduce an anisotropic geometric diffusion to enhance featureswhile smoothing. Ohtake et al. [17] combine an inner fairness mechanism in their fairing processto increase the mesh regularity. Bajaj and Xu [1] smooth both surfaces and functions on surfaces,

?Project supported in part by NSFC grant 10371130 and National Key Basic Research Project of China(2004CB318000).

∗Corresponding author.Email addresses: liudan@lsec.cc.ac.cn, xuguo@lsec.cc.ac.cn (Dan Liu, Guoliang Xu).

1

in a C2 smooth function space defined by the limit of triangular subdivision surfaces. Escher etal. [9] introduce the average mean curvature flow, which is volume preserving and area shrinking.Similar to the diffusion equation using the Laplacian, different kinds of flows on surface have beendeveloped (see [24] for references).

All above mentioned equations have been shown to be efficient for surface processing. However,these second order flows cannot be used to construct surfaces satisfying the boundary constraintswith G1 continuity. Fourth order flows have been recently used in the surface blending and free-form surface design. Surface diffusion flow is presented by R. Schneider et al. [20] to create fairsurfaces with G1 boundary conditions. The algorithm in this work creates meshes with subdivisionconnectivity composed of regular patches. The algorithm presented in their later work [19] allowsto completely separate outer and inner fairness. The outer fairness is based on the same equationwith G1 boundary constraints and the inner fairness leads to regular distribution of mesh verticeswithin the surface. A method for surface restoration based on a elastic gradient descent flowhas been presented by Yoshizawa et al. [30]. This scheme uses the classical formulation of theWillmore flow, which is an another well-known fourth order flow. In [5], the Willmore flow isdiscretized in space consistently using a finite element method and in time applying a semi-implicitEuler discretization. The resulting scheme is easy to implement and allow for large time step.Furthermore, the constructed surface satisfies the smooth boundary conditions. In [28], severalgeometric flows, including the second, fourth and sixth order flows have been used for surfaceblending, N-side hole filling and free-form surface design. The proposed approach is general,solves several surface modeling problems in the same manner and gives very desirable results fora range of complicated free-form surface models. Furthermore, the sixth order flow is used for G2

continuity in each of the surface modeling problems.

Surface modeling problem can also be solved by an energy-based variational approach [10, 13,22, 23]. The variational approach searches a surface with minimal energy of a certain type, suchas thin plate energy, membrane energy [12], total curvature [13, 23], or sum of distances [15].Among them, the area functional and the total curvature functional (see [11])

E1(M) :=

∫

MdA, E2(M) :=

∫

M(k2

1 + k22)dA

are the most frequently used energies, where k1 and k2 are the principal curvatures. The min-imizing surfaces of E1(M) and E2(M) are known as minimal surface and Willmore surface, re-spectively. The energy

E3(M) :=

∫

M

[(dk1

de1

)2

+(dk2

de2

)2]

dA (1)

proposed by Moreton et al. in [16] is also used in surface design which punishes the variation ofthe principal curvatures, where e1 and e2 are principal directions corresponding to the principalcurvatures k1 and k2, respectively. Functional

E4(M) :=

∫

M‖∇(Hn)‖2dA,

is proposed by Greiner in [10] for simplifying energy (1) and is used to fair spline surfaces byan optimization approach. The optimization approach starts from a given surface and searchesiteratively a next surface that has less energy. Using local interpolation or fitting, or replacing dif-ferential operators with divided difference operators, the optimization problems are discretized toarrive at finite dimensional linear or nonlinear systems. Approximate solutions are then obtainedby solving the constructed systems. In [29], a general framework for surface modeling using a

2

fourth order curvature driven partial differential equation is presented. The equation is obtainedby minimizing the second order functional

E5(M) :=

∫

M[f(H, K) + h(x,n)]dA.

The functional is general, which contains several well known geometric partial differential equa-tions as its spacial cases, such as mean curvature flow and Willmore flow etc.

To solve geometric PDE using divided-different-like method, discrete approximations of dif-ferential operators, such as mean curvature, Gaussian curvature and Laplace-Beltrami operatoretc., are required. Many discrete schemes have been proposed for these differential operators fromdifferent points of view (see [8, 14, 26]). Asymptotic error analysis for some of these schemes havebeen conducted under various conditions (see [4, 14, 25, 26]). Except for the schemes based on theinterpolation or fitting, non of these schemes converges without any restriction on the regularityof the considered meshes. In this paper, all the used differential operators are approximated basedon a parametric quadratic fitting.

Main Contributions. In this work we construct a general sixth order geometric partialdifferential equation for creating high quality and high order continuity surfaces. The PDE isobtained by minimizing a given general form third order functional

∑m`=1

∫M ‖∇f `‖2dA, where

the function f `(H, K) depends on the two variables: mean curvature and the Gaussian curvatureof the surface M. The derived PDE is solved numerically using a generalized divided differencediscretization under a general framework.

The discretization of the PDE yields a sparse linear system which is solved using an iterativeapproach. We apply the proposed algorithm to solve several problems of surface modeling. Theexperiments show that this sixth order geometric PDE can solve fair surface modeling problemssatisfying G2 boundary constraints.

The rest of the paper is organized as follows: Section 2 derives the Euler-Lagrange equationfrom a curvature integral functional. Section 3 presents the geometric PDE and its discretizaiton.Comparative examples are given in section 4. Section 5 concludes the paper.

2 An Euler-Lagrange Equation

In this section, we derive a sixth order Euler-Lagrange equation from a generic curvature integralfor surfaces. Let M be a regular smooth parametric surface in R3 represented as x(u1, u2),(u1, u2) ∈ Ω ⊂ R2. Let ti = xui , tij = xuiuj , i, j = 1, 2. Then the unit normal vector of x(u1, u2)

is given by n = t1×t2

‖t1×t2‖ . Let gij = tTi tj, bij = nT tij be the coefficients of the first and second

fundamental form, and set g = det (gαβ), (gαβ) = (gαβ)−1, b = det (bαβ), (hαβ) = (bαβ)−1. Thenwe have the mean curvature H and the Gaussian curvature K

H :=1

2bαβgαβ, K :=

b

g,

where the Einstein summation convention is used. In what follows, we assume all the involveddifferential operators, such as ∇, ∆ and 3, are defined on surface M. Consider a function fwhich depends on two variables H and K, and the corresponding functional E (M) defined by

E (M) :=

∫

M‖∇f‖2dA =

∫

Ω

‖∇f‖2√gdu1du2, (2)

3

where ∇f = gαβ ∂f∂uα xuβ is the gradient operator acting on f .

Theorem 2.1. Let E (M) be defined as (2). Then the Euler-Lagrange equation of F (M) is

2H‖∇f‖2 − 2〈∇f, 3f〉 − 4HKfK∆f − 2(2H2 −K)fH∆f − 22(fK∆f)−∆(fH∆f) = 0. (3)

Proof. Consider an extremal M of the functional (2) and a family of normal variations x(w, ε)of M defined by

x(w, ε) = x(w) + εϕ(w)n(w), w ∈ Ω, |ε| ¿ 1,

where ϕ ∈ C∞c (Ω) = ϕ ∈ C∞(Ω,R); suppϕ ⊂ Ω. Then we obtain

0 =d

dεE (M(·, ε))

∣∣∣ε=0

=: δE (M, ϕ) =

∫

Ω

[δ(‖∇f‖2) + ‖∇f‖2 δ√

g√g

]√

gdu1du2. (4)

It is easy to derive that

δ(gαβ)=4gαβHϕ− 2hαβKϕ, δ(g) = −4gHϕ, (5)

δ(H)=(2H2 −K)ϕ +1

2∆ϕ, δ(K) = 2HKϕ + 2ϕ, δ(xuα) = ϕnuα + ϕuαn. (6)

where

2f =1√g

∑ij

∂

∂ui(√

gKhij ∂f

∂uj), ∆f =

1√g

∑ij

∂

∂ui

( √ggij ∂f

∂uj

). (7)

We call the second order differential operator 2 Giaquinta-Hildebrandt operator because it isfirstly introduced by Giaquinta and Hildebrandt. ∆ is Laplace-Beltrami operator. Then we have

δ∇f =δ(gαβDαfxuβ)

=δ(gαβ)Dαfxuβ + gαβDα(δf)xuβ + gαβDαfδ(xuβ)

=2ϕH∇f − ϕ3f + gαβDαfϕuβn + gαβDα(fKδK + fHδH)xuβ .

where 3f := Khij ∂f∂ui xuj . It is easy to see that 3f is a tangent vector of the surface M. Hence

3 is called the second tangential gradient operator (see [31]).

Substituting the corresponding terms into (4), we have

δE (M, ϕ)=

∫

Ω

[2H‖∇f‖2 − 2〈∇f, 3f〉]ϕ√gdu1du2 + 2

∫

Ω

gαβDβfDα(fKδK + fHδH)√

gdu1du2

=

∫

Ω

[2H‖∇f‖2 − 2〈∇f, 3f〉]ϕ√gdu1du2 − 2

∫

Ω

(fKδK + fHδH)∆f√

gdu1du2

=

∫

Ω

[2H‖∇f‖2 − 2〈∇f, 3f〉 − 4HKfK∆f − 2(2H2 −K)fH∆f

−22(fK∆f)−∆(fH∆f)]ϕ√

gdu1du2, (8)

Since δE (M, ϕ) = 0 holds for any ϕ ∈ C∞c (Ω), we obtain, from (7), the Euler-Lagrange equation

(3) of functional E (M).

Obviously, equation (3) is of sixth order if f is not a constant.

Example 2.1. Take f = H, then the Euler-Lagrange equation (3) is simplified as

2H‖∇H‖2 − 2〈∇H, 3H〉 − 2(2H2 −K)∆H −∆2H = 0.

4

Example 2.2. Take f = K, then the Euler-Lagrange equation can be written as

2H‖∇K‖2 − 2〈∇K, 3K〉 − 4HK∆K − 22(∆K) = 0.

Now we consider a more general case. We define the functional F (M) :=∫M

∑m`=1 ‖∇f `‖2dA.

By the same derivation, the Euler-Lagrange equation of this functional is obtainedm∑

`=1

[2H‖∇f `‖2−2〈∇f `, 3f `〉−4HKf `

K∆f `−2(2H2−K)f `H∆f `−22(f `

K∆f `)−∆(f `H∆f `)

]= 0.

(9)

3 Geometric PDE and Its Discretization

In this section, we describe the curvature driven geometric PDE. The approach adopted for solvingthe PDE is based on a generalized divided difference method.

3.1 Geometric Partial Differential Equation

Let M be a triangulation of surface M and xiNi=1 be the vertex set of M . For the vertex

xi, N(i) = 1, 2, · · · , n denote the set of the vertex indices of one-ring neighbors of xi. Let[xixikxik+1

] be a neighbor quadrilateral of xi in M . We assume that i1, · · · , in are arranged suchthat the triangles [xixikxik−1

] and [xixikxik+1] are in M , and xik−1

,xik+1opposite to the edge

[xixk]. Let M0 be a given initial surface in R3 with boundary Γ, the curvature driven geometricflow consists of finding a family M(t) : t ≥ 0 of smooth orientable surfaces in R3 which evolveaccording to the following equation

∂x

∂t= n(x)Vn(H, K), M(0) = M0, (10)

where Vn(H, K) denotes the normal velocity of the surfaceM(t). The sixth order flow constructedfrom the Euler-Lagrange equation (3) is

∂x

∂t= n(x)[2H‖∇f‖2−2〈∇f, 3f〉−(4HKfK +2(2H2−K)fH)∆f−22(fK∆f)−∆(fH∆f)],

M(0) = M0, ∂M(t) = Γ(11)

The sixth order flow constructed from the Euler-Lagrange equation (9) is

∂x

∂t= n(x)

m∑

`=1

[2H‖∇f `‖2−2〈∇f `, 3f `〉−(4HKf `

K +2(2H2−K)f `H)∆f `

−22(f `K∆f `)−∆(f `

H∆f `)],

M(0) = M0, ∂M(t) = Γ

(12)

3.2 Discretizaion of Differential Geometric Operators

To solve the geometric PDEs (11) and (12), discrete approximations of the mentioned differentialgeometric operators are needed. We discretize these operators into the following forms

H(xi)(or K(xi)) =∑

j∈N(i)

ωijxj, ωTij ∈ R3, ∆f(xi)(or 2f(xi)) =

∑

j∈N(i)

$ijxj, $ij ∈ R. (13)

5

There are several discrete schemes in the required form of Laplace-Beltrami operator and meancurvature. However, the existing discrete schemes of Gaussian curvature are not in this form andmay not be consistent with the discretization of mean curvature and Laplace-Beltrami operator.Here we use a biquadratic fitting (details can be found in [27]) of the surface data and thefunction data to approximate these differential operators. Then we get the approximation ofabove differential operators as follows:

∆f(xi) ≈∑

j∈N(i)

ω∆ij f(xj), 2f(xi) ≈

∑

j∈N(i)

ω2ijf(xj), (14)

∇f(xi) ≈∑

j∈N(i)

ω∇ij f(xj), 3f(xi) ≈∑

j∈N(i)

ω3ijf(xj). (15)

The definitions of ω∆ij , ω2

ij, ω∇ij and ω3ij are also given in [27]. Using the relations ∆x = 2H = 2Hn

and 2x = 2Kn, we have the following approximations

H(xi) ≈ 1

2

∑

j∈N(i)

ω∆ijxj, H(xi) =

1

2

∑

j∈N(i)

ω∆ijn(xi)

Txj, K(xi) ≈ 1

2

∑

j∈N(i)

ω2ijn(xi)

Txj. (16)

Remark 3.1. The reasons why we approximate the used differential operators based on the pa-rameter fitting have been explained in [27]. In a word, the scheme we adopted leads to convergent,consistent and required form approximations.

3.3 Numerical Solution of Geometric PDE

In this section, we present a semi-implicit discretized scheme for solving the proposed geometricPDEs. For simplicity, we just discretize the equation (11). The discretization of (12) is similar.In the following, we write the function f(H, K) as

f(H, K) = α(H, K)H + β(H, K)K + φ(H, K),

where α, β and φ are certain continuous functions.

Suppose we have the discrete solutions x(k)i of the geometric PDE at the time t = tk for

interior vertices. For each interior vertex xi, the operators n(xi), H(xi), K(xi), ω∆ij , ω2

ij, ω∇ij andω3

ij are computed using (14)-(16). Then we get f, α, β and φ at (H(xi), K(xi)) denoted as fi, αi, βi

and φi, respectively. To simplify the notation, n(xi), H(xi) and K(xi) are denoted by ni, Hi andKi, respectively. We compute the discrete solutions xk+1

i for the next time step tk+1 = tk + τ ,where τ is the time step-size, using semi-implicit Euler scheme.

In order to form a linear system, we discretize each term of the equation (11) as∑

j∈N(i) ωijxj.

Using (14)-(16), we discretize the first term of the right-handed side of (11) as

‖∇fi‖2∑

j∈N(i)

ω∆ijx

k+1j .

Since

2〈∇fi, 3fi〉 =∑

j∈N(i)

〈ω∇ij , 3fi〉(αjHj + βjKj + φj) +∑

j∈N(i)

〈∇fi, ω3ij〉(αjHj + βjKj + φj),

the second term of the right-handed side of (11) can be discretized as

6

∑

j∈N(i)

∑

l∈N(j)

(〈ω∇ij , 3fi〉+ 〈∇fi, ω

3ij〉

)(1

2αj(nin

Tj )ω∆

jl + βj(niυTjl)

)pk+1

j

+∑

j∈N(i)

(〈ω∇ij , 3fi〉+ 〈∇fi, ω

3ij〉

)φjni.

Similarly, the rest terms of the right-handed side of (11) are discreted as follows

n (4HKfK + 2(2H2 −K)fH)∆f

≈ (4HiKi(fK)i + 2(2H2i −Ki)(fH)i)

∑

j∈N(i)

∑

l∈N(j)

(1

2αj(nin

Tj )ω∆

ij ω∆jl + βj(nin

Tj )ω∆

ij ω2jl

)pk+1

j

+(4HiKi(fK)i + 2(2H2i −Ki)(fH)i)

∑

j∈N(i)

φjω∆ijni,

2n2(fK∆f)≈ 2∑

j∈N(i)

∑

l∈N(j)

∑

m∈N(l)

(fK)jω2ij

(1

2αlω

∆jlω

∆lm(nin

Tl ) + βlω

∆jlω

2lm(nin

Tl )

)pk+1

k

+ 2∑

j∈N(i)

∑

l∈N(j)

(fK)jω2ijφlω

∆jlni,

n∆(fH∆f)≈∑

j∈N(i)

∑

l∈N(j)

∑

m∈N(l)

(fH)jω∆ij

(1

2αlω

∆jlω

∆lm(nin

Tl ) + βlω

∆jlω

2lm(nin

Tl )

)pk+1

k

+∑

j∈N(i)

∑

l∈N(j)

(fK)jω∆ij φlω

∆jlni.

Therefore, the geometric PDE (11) can be discretized as the following form

xk+1i + τ

∑ωijx

k+1j = xk

i + τbi, bi ∈ R3 (17)

Obviously, the coefficient matrix of this linear system for the unknowns xk+1i , i = 1, · · · , n, is

sparse. We solve it using Saad’s iterative method [18], named GMRES, with the incomplete LUdecomposition. The boundary treatment method adopted in this work is the same as that in [28].The experiments show that this iterative method works well.

4 Examples

In this section, we present several examples to show the applications of the derived geometricPDEs in surface processing and modeling.

4.1 Smoothing

First we show that geometric PDEs (11) and (12) can be used to smooth noising surfaces efficiently.Several functions are used to generate the smoothing surfaces from noising inputs. Fig. 1 andFig. 2 show some results of the smoothing, where (a) shows an input, (b)–(f) are evolution resultsat time t = 0.02, using the sixth order flow with different f . We can see from these examplesthat there is no significant difference between the surfaces generated using different functions f ,since the evolution time is short.

7

To illustrate the sixth order flow is more efficient than the lower order flows, we give a smoothingresult in figure (g) of Fig. 1 generated by using mean curvature flow (a second order flow) atthe same time t = 0.02. It is easy to see that at this time moment, the resulted surface is notsmooth enough. To achieve a similar smoothing effect to the sixth order flow, we need to furthersmooth the surface. Figure (h) shows a desirable result at t = 0.06. But a shrinkage side-effect isobserved.

(a) (b) (c) (d) (e) (f) (g) (h)

Fig. 1: (a) shows an input noising surface. (b)-(f) are the surfaces after smoothing using differentfunctions. (b) shows the result of f = |H|− 1

2 H after 10 iterations with τ = 0.002. (c) shows the resultof f = H after 2 iterations with τ = 0.01. (d) is the result of f = |H|H after 100 iterations withτ = 0.0002. (e) shows the result of the functions f1 = H and f2 = K after 4 iterations with τ = 0.005.(f) shows the result of the functions f1 = k1 and f2 = k2 where k1 and k2 are the principal curvaturesafter 4 iterations with τ = 0.005. (g) and (h) show the results of mean curvature flow after 2 and 6iterations, respectively, with τ = 0.01.

(a) (b) (c) (d) (e) (f)

Fig. 2: (a) shows an input noising surface. (b)-(f) are the surfaces after smoothing using differentfunctions. (b) shows the result of f = |H|− 1

2 H after 20 iterations with τ = 0.002. (c) shows the result off = H after 4 iterations with τ = 0.01. (d) is the result of f = |H| 15 H after 4 iterations with τ = 0.01.(e) is the result of f = |H|H after 400 iterations with τ = 0.0001. (f) shows the result of the functionsf1 = H and f2 = K after 4 iterations with τ = 0.01.

4.2 Blending

Given two surface meshes with boundaries, we construct a fair surface to blend these meshes withG2 continuity constraints at the boundaries. For this blending problem, a long-time evolution isconducted to get the steady solutions. Fig. 3 shows the blending surfaces between a sphericalcap and a circular disk (figure (a)) using different f(H, K), where an initial G0 minimal surface

construction is given as shown in figure (b). The functions are selected as 4|H|H + 2K, |H|− 12 H,

|H| 12 H and H for figures (c), (d), (e) and (f), respectively. Fig. 4 shows the blending surface

8

between two spherical caps (figure (a)). Figure (b) is the initial minimal surface construction.

Functions |H|− 12 H, H, |H| 12 H, and H2 are used to generate figures (c), (d), (e) and (f), respec-

tively. Figures (g) of Fig. 3 and Fig. 4 show the blending surface generated by equation (12)where the functions are selected as f 1 = H and f 2 = K.

(a) (b) (c) (d)

(e) (f) (g)

Fig. 3: (a) is the surface to be blending with an initial minimal surface construction (b). (c)-(g) are theblending surfaces using different functions f . (c) shows the result of f = 4|H|H + 2K. (d) shows theresult of f = |H|− 1

2 H. (e) is the result of f = |H| 12 H. (f) is the result of f = H. (g) is the result offunctions f1 = H and f2 = K.

(a) (b) (c) (d) (e) (f) (g)

Fig. 4: (a) is the surface to be blending with an initial minimal surface construction (b). (c), (d), (e)and (f) are the blending surfaces with different functions f = |H|− 1

2 H, f = H, f = |H| 12 H and f = H2

respectively. (g) is the result of functions f1 = H and f2 = K.

4.3 N-sided hole filling

We construct a fair surface with G2 continuity on the boundaries to fill the holes in a given surfacemesh. In Fig. 5, a head mesh is given with holes, which is filled with a minimal surface (the

brass areas of figure (a)). Different functions |H|− 12 H, H and |H| 15 H are used to generate the fair

surface, as shown in figure (b), (c) and (d), respectively. Figure (e) and (f) show the filled surfacegenerated by equation (12) where the functions are selected as f 1 = H, f 2 = K and f 1 = k1,f 2 = k2, respectively.

9

(a) (b) (c) (d) (e) (f)

Fig. 5: (a) is the surface to be filled with a minimal surface as an initial construction, (b)-(f) are thefilled surfaces. (b) shows the result of f = |H|− 1

2 H. (c) shows the result of f = H. (d) is the resultof f = |H| 15 H. (e) shows the result of the functions f1 = H and f2 = K. (f) shows the result of thefunctions f1 = k1 and f2 = k2 where k1 and k2 are the principal curvatures.

4.4 Comparing with the lower order flow

Now we illustrate the difference between the sixth order flow and some lower order flows. InFig. 6, (a) shows the input mesh. (b) shows the mean curvature plot of (a). (c) and (d) showthe mean curvature plots of the evolution results of Willmore flow and the sixth order flow withf = H, respectively. From this example we can see that the surface generated by the sixth orderflow is more smoothing than that generated by the lower order flows. Fig. 7 shows the differentresults of recovering the cylinder, where mean curvature flow (figure (a)), Willmore flow (figure(b)) and the sixth order flow with f = H (figure (c)) are used. Figure (d) and (e) show the meancurvature plots of (a) and (b), respectively. Fig. 8 shows the mean curvature plots of the ringsinterpolating two circles (the red line) generated by mean curvature flow (figure (a)), Willmoreflow (figure (b)) and sixth order flow with f = H (figure (c)), respectively. We can see from Fig.7 and Fig. 8 that the mean curvature flow makes the cylinder and the ring shrinking, while theWillmore flow makes them inflating. The sixth order flow recover the cylinder and the ring.

(a) (b) (c) (d)

Fig. 6: (a) shows the input mesh. (b) shows the mean curvature plot of (a). (c) and (d) show the meancurvature plots of the evolution results of Willmore flow and the sixth order flow with f = H from (b),respectively.

10

(a) (b) (c) (d) (e)

Fig. 7: (a) the cylinder generated by mean curvature flow. (b) the cylinder generated by Willmore flow.(c) the cylinder recovered by the sixth order flow with f = H. (d) and (e) are the mean curvature plotsof (a) and (b), respectively.

(a) (b) (c)

Fig. 8: (a), (b) and (c) show the mean curvature plots of the evolution results of mean curvature flow,Willmore flow and sixth order flow with f = H , respectively.

5 Conclusion

By minimizing a given functional∫M

∑ml=1 ‖∇f `‖2dA, we have derived a general sixth order ge-

ometric partial differential equation. We use finite difference method to discretize the geometricPDE, therefore a sparse linear system is obtained. The implementation and the illustrative ex-amples show that this sixth order geometric partial differential equation can be used to efficientlysolve problems of surface processing and modeling, and yields high order continuity surfaces.

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