J Sci ComputDOI 10.1007/s10915-014-9872-7

Design of Loop’s Subdivision Surfaces by Fourth-OrderGeometric PDEs with G1 Boundary Conditions

Guoliang Xu · Qing Pan

Received: 25 August 2013 / Revised: 20 March 2014 / Accepted: 24 May 2014© Springer Science+Business Media New York 2014

Abstract In this paper, we present a method for constructing Loop’s subdivision surfacepatches with given G1 boundary conditions and a given topology of control polygon, usingseveral fourth-order geometric partial differential equations. These equations are solved by amixed finite element method in a function space defined by the extended Loop’s subdivisionscheme. The method is flexible to the shape of the boundaries, and there is no limitation onthe number of boundary curves and on the topology of the control polygon. Several propertiesfor the basis functions of the finite element space are developed.

Keywords Subdivision surface · Geometric partial differential equations · G1 continuity

1 Introduction

A surface satisfying a geometric partial differential equation (PDE) is referred to as a geomet-ric PDE surface in this paper. Geometric PDE surfaces, such as minimal surfaces (see [19]),constant mean-curvature surfaces (see [11,24]), Willmore surfaces (see [3,14,15,28]) andminimal mean-curvature variation surfaces (see [33]), are important and preferred in the shapedesigning and modeling because they share certain optimal properties. For instance, the min-imal surfaces have minimal area, the Willmore surfaces have minimal total squared meancurvature and minimal mean-curvature variation surfaces have minimal total mean-curvaturevariation. Here the terminology total means the integration over the surfaces. Various typegeometric PDE surfaces have been constructed in the literatures (see [31]). Most of them arediscrete surfaces (triangular or quadrilateral control polygons), a few of them are continuous

G. XuLSEC, Institute of Computational Mathematics, Academy of Mathematics and System Sciences,Chinese Academy of Sciences, Beijing 100190, China

Q. Pan (B)Key Laboratory of High Performance Computing and Stochastic Information Processing,College of Mathematics and Computer Science, Hunan Normal University, Changsha, Chinae-mail: panqing@lsec.cc.ac.cn

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Fig. 1 a, b and c The boundary curves with co-normals, the initial control mesh with perturbation and thegeometric PDE subdivision surface respectively

surfaces. Usually, the representation of the continuous surfaces are Bezier (see [12]), rationalBezier, B-spline (see [17,18]) and NURB surfaces.

Obviously, the boundaries of the Bezier surface, B-spline and NURB surface patches con-sist of three or four curves (three- or four-seded). This is a serious limitation for designinggeometric PDE surfaces with arbitrary shaped boundaries. In this paper, our intention is toconstruct geometric PDE subdivision surfaces with piecewise B-spline curve boundaries andtangent conditions. There is no limitation on the number of spline pieces. B-spline represen-tation for curves and surfaces have been widely accepted in the CAD and industrial design.Using B-spline to represent surface boundary is acceptable and preferable. To represent asurface patch with any topology, subdivision surfaces are the best candidates, since there isno limitation on the topology of the control polygon. However, subdivision surfaces, such asLoop’s subdivision surfaces and Catmull–Clark subdivision surfaces are traditionally closed,which cannot be used directly for serving our purpose.

For many free-form surface modeling problems, such as the construction of the bodies ofcars and aircrafts, machine parts and roofs, surfaces are usually constructed in a piecewisemanner with fixed boundaries for each of the pieces. In such a case, Loop’s subdivisionscheme cannot be applied near the boundary of the control polygon. Therefore, an extensionof the Loop’s subdivision scheme to control polygon with boundaries is required. On thisaspect, an excellent work has been done by Biermann et al. [2] and that is just sufficient forachieving the goal of constructing piecewise smooth surface.

The G1 condition, consisting of boundary curves and co-normals, plays an important rolefor designing the desirable shape of the constructed surface. It not only affects the shape ofeach surface patch, but also determines how the surface patches join together. For instance,to continuously join two surface patches, we need to make the two patches share a commonboundary curve. To smoothly join two surface patches, we only need to use a commonboundary curve and a common co-normal. To continuously join two surface patches but witha sharp edge, we only need to use a common boundary curve and different co-normals. Theangle between the co-normals determines the dihedral angle of the sharp edge. Therefore,the G1 boundary condition provides us with a powerful mechanism to control the shape ofthe object to be designed.

In this paper we construct geometric PDE subdivision surface (see Fig. 1) patches withgiven G1 boundary conditions and a given topology of the control polygon using severalfourth-order geometric PDEs. These equations are solved by a mixed finite element methodin a function space defined by the extended Loop’s subdivision scheme. By the term topology

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of the control polygon, we mean the connection mode among the vertices of the controlpolygon.

Fourth-order geometric flows have been used to solve the problems of discrete surfaceblending, N-sided hole filling and the free-form surface fitting (see [5,25,26,32]). In [25,26],the surface diffusion flow has been used for fairing/smoothing meshes while satisfying theG1 boundary conditions. The finite element method is used by Clarenz et al. [5] to solve theWillmore flow equation, based on a new variational formulation of this flow, for the discretesurface restoration.Problem Description.

Input: Given an initial open control polygon (a piece of triangular mesh) of a surfacepatch (see Fig. 1b) with fixed boundary control points (see Fig. 1a), and some of theboundary control points are to be interpolated. The boundary curve is defined as piecewisecubic B-spline with the boundary control points as the B-spline control points and equalspaced knots for each piece. The interpolated boundary control points are served as theend-points of the B-spline curves. On each of the boundary curves, we are also given atangential vector (co-normal) curve (named as the tangential curve), which is representedin the same form as the boundary curve (see Fig. 1a).Output: We want to construct a geometric PDE subdivision surface that interpolates theboundary curves and tangents, at the same time its control polygon has the same topologyas the initial one (see Fig. 1c).

Contribution: The contribution of this paper includes: proposing a method for constructingsubdivision surfaces with G1 boundary conditions. The method is flexible to the shape ofthe boundaries. Several schemes, such as extended Loop’s subdivision, fast evaluation ofthe basis functions and finite element method for the initial-boundary problem of severalfourth-order geometric flows with G1 boundary condition, are combined together to form anefficient and mathematically sound approach. Several properties for the basis functions ofthe finite element space are developed.

The rest of the paper is organized as follows. In Sect. 2, we introduce some used notationsand three fourth-order geometric PDEs. Numerical methods for solving these equations aredescribed in Sect. 3. Illustrative examples to show the effectiveness of our methods arepresented in Sect. 4. Section 5 concludes the paper.

2 Geometric PDEs and Their Weak-Form Formulations

To construct smooth geometric PDE subdivision surface patches with G1 boundary condi-tions, we use three fourth-order equations, namely surface diffusion flow (SDF), Willmoreflow (WF) and quasi-surface diffusion flow (QSDF). To describe these equations precisely,we need to introduce a few notations. The details of them can be found in [7,31].

2.1 Notations and Preliminaries

Let S := {x(u1, u2) ∈ R3 : (u1, u2) ∈ D ⊂ R

2} be a parametric surface. For simplicity, weassume it is sufficiently smooth and orientable. Let gαβ = 〈xuα , xuβ 〉 and bαβ = 〈n, xuαuβ 〉be the coefficients of the first and the second fundamental forms of S with

xuα = ∂x∂uα

, xuαuβ = ∂2x∂uα∂uβ

, α, β = 1, 2,

n = (xu × xv) /‖xu × xv‖, (u, v) := (u1, u2),

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where 〈·, ·〉, ‖·‖ and ·×· stand for the usual inner product, Euclidean norm and cross productin R

3, respectively. The tangent space TxS is defined by span{xu, xv}.For surface S, the mean curvature and the mean curvature normal are denoted by H and

H, respectively. Gaussian curvature is denoted by K . We use ∇s, �, divs and �s to denotethe tangential gradient operator, the third tangential operator, the divergence operator and theLaplace–Beltrami operator (LBO), respectively. All these operators are defined on the surfaceS. For deriving the weak forms of the used equations, we introduce the Green’s formulas inthe following.

Theorem 2.1 (Green’s formula for LBO) LetS be an orientable surface, and� is a subregionof S with a piecewise smooth boundary ∂�. Let nc ∈ TxS (x ∈ ∂�) be the outward unitnormal (also named as co-normal) along the boundary ∂�. Then for a given C1 smoothvector field v on S, we have∫

�

[〈v,∇s f 〉 + f div(v)] dA =∫

∂�

f 〈v,nc〉ds. (2.1)

Let g ∈ C2(S). Taking v = ∇s g in (2.1), we have∫

�

[〈∇s g,∇s f 〉 + f �s g] dA =∫

∂�

f 〈∇s g,nc〉ds.

2.2 Used Geometric PDEs

For completeness, we describe briefly the used equations and their behaviors. More detailson these equations can be found in [31].

Surface Diffusion Flow

∂x∂t

= −2�s Hn. (2.2)

This flow was introduced by Mullins in 1957 (see [20]), to describe the interface motion lawof the growing crystal. If S is a closed surface where we use A stand for its area and V standfor its enclosed volume, then by Green’s formula we obtain (see [6,23] for the change ratesof the surface area and the enclosed volume of the evolved surface in the general case):

d

dtA(t) = 2

∫

S(t)�s H HdA = −2

∫

S(t)‖∇s H‖2dA ≤ 0,

d

dtV (t) = −2

3

∫divs(∇s H)dA = 2

3

∫(∇s H)T∇s(1)dA = 0.

Hence surface diffusion flow is volume preserving and area shrinking. The area shrinkagestops when H is a constant. It is easy to see that surfaces with the constant mean curvatureare the steady solution of (2.2).

Willmore Flow

∂x∂t

= −2[�s H + 2H(H2 − K )

]n. (2.3)

Willmore flow was derived from minimizing the total squared mean-curvature∫S H2dA.

Notice that a factor 2 is added to the original Willmore flow for comparability with the other

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equations used in this paper. Since the constant factor can be absorbed by the parameter t inthis equation, there is no influence on the behavior of the equation by adding this factor. Thisflow is also used in [3,14,15,28]. There is no volume/area preserving/shrinking property forthis flow. However, if the initial surface is a sphere, Willmore flow keeps the spherical shapeunchanged. Moreover, surfaces with zero mean curvature are the steady solution of (2.3). Atorus with R/r = √

2 is a steady solution of (2.3), where the torus is defined by rotating acircle with radius r along another circle with radius R.

Quasi-surface Diffusion Flow

∂x∂t

= −�2s x. (2.4)

This flow was introduced in [32], and used in discrete surface design. It is well-known thattangent motion of a surface does not alter the surface shape (see [10]). Hence if we removethe tangential movement of (2.4), we obtain the following geometric flow

∂x∂t

= −2[�s H − 2H(2H2 − K )

]n. (2.5)

To derive (2.5) from (2.4), we need to use the following facts:

�s(Hn) = �s Hn + 2∇snT ∇s H + H�sn,

�sn = 2(K − 2H2)n − 2∇s H.

Then

−�2s x = −2�s(Hn)

= −2[�s Hn + 2∇snT ∇s H + H�sn

]

= −2[�s Hn + 2∇snT ∇s H + 2H(K − 2H2)n − 2H∇s H

].

Removing the tangential terms 2∇snT ∇s H and 2H∇s H , we obtain the right-hand side of(2.5).

If S is a closed surface, it is easy to derive that

d

dtA(t) = −2

∫

S(t)

[‖∇s H‖2 + 2H2(2H2 − K )]

dA ≤ 0.

Hence, quasi-surface diffusion flow is area diminishing. Since ‖∇s H‖2+2H2(2H2−K ) = 0if and only if H = 0, and the shrinkage stops when H ≡ 0. Hence, the solution surfaces of(2.4) approach to the minimal surface.

Note that these three flows share the same fourth-order term −2�s Hn, and only thesecond order terms are different. However, their behaviors are quite different. This is thereason we choose to use them for achieving different effect in the shape design.

2.3 Mixed-Form Variational Formulations

Now we present the variational form for these equations. Let y = H , φ ∈ H10 (S) be a test

function. Then by Green’s formula we have

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∫

Snφ�s ydA = −

∫

S[∇s(nφ)]T ∇s y dA +

∫

∂Snφ(∇s y)Tncds

= −∫

S

[φ∇sn∇s y + n(∇sφ)

T∇s y]

dA

(a)=∫

S

[φ � x∇s y − n(∇sφ)

T∇s y]

dA

=∫

S

[φ � y − n(∇sφ)

T∇s y]

dA, (2.6)

where the validity of the equality (a) follows from the following equation (see [31], page 17)

∇sn + �x = 0.

Let y = ny = H(x). Then it is easy to derive that

∇s y = (∇sy)n, �y = (�y)n.

Substituting these into (2.6), we have∫

Snφ�s ydA =

∫

S

[φ � y − n(∇sφ)

T∇sy]

ndA.

On the other hand, it follows from Green’s formula that∫

SHψ dA = 1

2

∫

S�sxψ dA

= −1

2

∫

S(∇sx)T ∇sψ dA + 1

2

∫

∂S(∇sx)Tncψ ds

= −1

2

∫

S(∇sx)T∇sψ dA + 1

2

∫

∂Sncψ ds,

where the equality (∇sx)Tnc = nc is used. Then the mixed variational form of (2.2) is: Find(x, y) ∈ H2(S)3 × H1(S)3 such that⎧⎪⎪⎪⎨

⎪⎪⎪⎩

∫S∂x∂t φ dA + 2

∫S

[φ � y − n(∇sφ)

T∇sy]

n dA = 0, ∀φ ∈ H10 (S),∫

Syψ dA + 1

2

∫S(∇sx)T∇sψ dA − 1

2

∫∂S

ncψ ds = 0, ∀ψ ∈ H1(S),

S(0) = S0, ∂S(t) = , nc(x) = n()c (x), ∀x ∈ ,(2.7)

where n()c is the given co-normal on the boundary curve . The mixed variational form of(2.3) and (2.5) are similar. The difference occurs only in the first equation. For Willmore flow(2.3), the first equation of the mixed variational form is as follows.∫

S

∂x∂tφ dA + 2

∫

S

[φ � y − n(∇sφ)

T∇sy]

n dA

+ 4∫

Sn(H2 − K )φnTy dA= 0, ∀φ ∈ H1

0 (S). (2.8)

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For the quasi-surface diffusion flow (2.5), the first equation of the mixed variational form is∫

S

∂x∂tφ dA + 2

∫

S

[φ � y − n(∇sφ)

T∇sy]

n dA

− 4∫

Sn(2H2 − K )φnTy dA= 0, ∀φ ∈ H1

0 (S). (2.9)

In the next section, systems (2.7)–(2.9) are discretized in a finite element space definedby the extended Loop’s subdivision scheme.

3 Subdivision Surfaces and Finite Element Space

We discretize the geometric PDEs in a function space defined by the limit of the extendedLoop’s subdivision. Subdivision schemes generate smooth surfaces via a limit procedureof an iterative refinement process starting from an initial control mesh of the limit surface.Several schemes of subdivision for generating smooth surfaces have been proposed. Subdi-vision schemes where the vertex positions of the coarse mesh are fixed, and only the newlyadded vertex positions need to be computed are named as interpolatory (see e.g., [13] forquadrilateral meshes [9,34], for triangular meshes), while others are approximatory (see e.g.,[4,8] for quadrilateral meshes [16], for triangular meshes [21], for general polyhedra). Theapproximatory subdivision schemes update both old and new vertex positions at each refine-ment step. Generally speaking, approximatory schemes produce better quality surfaces thanthose produced by interpolatory schemes. Hence, in this work, we shall use an approximatingscheme for triangular meshes.

3.1 Loop’s Subdivision Scheme

We consider only the construction of Loop’s subdivision surfaces for triangular control poly-gons. The idea for constructing Catmull–Clark surfaces for quadrilateral control polygons issimilar. In this subsection, we only review the Loop’s subdivision scheme. In the refinementstep of the Loop’s subdivision, each triangle is subdivided into 4 sub-triangles. Then allthe vertex positions of the refined mesh is calculated as the weighted average of the vertexpositions of the coarse mesh. Let x(k)0 be a vertex at level k with neighbor vertices x(k)i for

i = 1, . . . , n, where n is the valence of vertex x(k)0 . Then the newly generated vertices x(k+1)i

on the edges of the previous mesh are computed as

x(k+1)i = 3x(k)0 + 3x(k)i + x(k)i−1 + x(k)i+1

8, i = 1, . . . , n, (3.1)

where index i is to be understood modulo n. The old vertices get new positions according to

x(k+1)0 = (1 − nω)x(k)0 + ω

(x(k)1 + x(k)2 + · · · + x(k)n

), (3.2)

where ω = 1n

[58 − ( 3

8 + 14 cos 2π

n

)2]

(Loop’s weight) or ω = 38n for n > 3, ω = 3

16 for

n = 3 (Warren’s weight). Note that all newly generated vertices have a valence of 6, whilethe vertices inherited from the initial mesh may have a valence other than 6. The former caseis refereed to as ordinary and the latter case is refereed as extraordinary. The limit surfaceof Loop’s subdivision is C2 everywhere except at the extraordinary points where it is C1

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Fig. 2 A control mesh for extended Loop’s subdivision rules. Vertices are classified into 5 categories. Edgesare classified into 6 categories and sub-boundary edges include 4 categories

(a) (b) (c)

Fig. 3 Vertex subdivision rules of extended Loop’s scheme. a Corner vertex. b Crease vertex. c Dart vertexand interior vertex. n is the valence of the vertex and ω is the Loop’s weight

(see [27]). Fast method exists for evaluating the limit surfaces of the Loop’s scheme at anyparameter value (see [29]).

3.2 Extended Loop’s Subdivision Scheme

The extended subdivision scheme proposed by Biermann et al. [2] is applied to a triangularcontrol mesh that may have boundaries (see Fig. 2). Vertices of the control mesh are taggedas corner vertices (red points), crease vertices (blue points), dart vertices (green points)and interior vertices (black points). Edges are tagged as boundary edges (red lines), sub-boundary edges (blue lines) and interior edges (black lines). One is referred to [2] for theirexact definitions. The subdivision rules for the vertices and edges are detailed in Figs. 3and 4, respectively. In [2], the interior vertices are further classified into interior vertices withvalence 3 and with valence greater than 3 where Warren’s weight is used. Here we adoptLoop’s weight (see Fig. 3). It should be point out that the subdivision rules for the boundariesare the same as the ones of the cubic B-spline functions. Hence, the control vertices on theboundaries are in fact the control vertices of cubic B-spline curves.

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(a) (b) (c)

(d) (e) (f)

Fig. 4 Edge subdivision rules of extended Loop’s scheme. a Boundary edge. b–e Sub-boundary edges,γk = 1/2 − (cos θk )/4; b one endpoint of the edge is a corner vertex and the edge is in a concave sector.θk = (2π − α)/k, k ≥ 2, k is the number of triangles in the concave sector between the boundary edgesand α is the acute angle between the boundary edges. Here k = 5; c one endpoint is a corner vertex and theedge is in a convex sector. θk = α/k, k ≥ 1, k is the number of triangles in the convex sector between theboundary edges and α is the acute angle between the boundary edges. Here k = 3; d one endpoint of the edgeis a crease vertex. θk = π/k, k ≥ 1, k is the number of triangles in the sector, which includes the edge and isbetween the boundary edges. Here k = 4; e one endpoint of the edge is a dart vertex. θk = 2π/k, k ≥ 1, k isthe number of triangles around the dart vertex. Here k = 7; f interior edge

3.3 Basis Functions and Their Properties

Now let us define the basis functions of the finite element function space, denoted as VS(t).For each control point xi , including the corner control point and boundary control points, ofa control polygon Sd , we associate it with a basis function φi , where φi is defined as the limitof the extended Loop’s subdivision scheme applying to the zero control values everywhereexcept at xi where it is one.

The control polygon Sd , as a piecewise linear surface, is served as the definition domainof the basis function φi . The mapping from Sd to φi is defined by a dual subdivision process.More precisely, when the extended Loop’s subdivision scheme is applied to the controlfunction values recursively, the linear subdivision scheme (each triangle is partitioned intofour equal-sized sub-triangles) is applied to the control polygon correspondingly. The limitof the former is φi and that of later is Sd itself.

The basis functions share some properties with the well known B-spline basis. Theseproperties are important in our finite element method. Now let us describe them as follows.

1. Positivity. The weights of the extended subdivision rules are positive. Hence the basisfunction φi is nonnegative everywhere and positive around xi .

2. Locality. It is known that the limit value at a control point is a linear combination of theone-ring neighbor values. Hence, the limit value is zero at a control point if the control

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values on the one-ring neighbor control points are zeros. Therefore, the support of thebasis function is within the two-ring neighborhood.

3. Partition of Unity. Since all the subdivision rules have the properties that the weights aresummed to one. Therefore, if we choose all the control values as one. The control valuesafter one subdivision step are still one. This implies that

∑mi=0 φi = 1. This property is

called partition of unity.4. Interpolatory Properties at the Boundary. The extended subdivision rules on the boundary

do not involve the interior control points. Hence the basis functions for the interiorcontrol points are zero at the boundary. This means that the given boundary curves areinterpolated.

5. Tangential Property. Let xi be a control point, with non of its one-ring neighbor con-trol points are boundary control points. Then ∇sφi vanishes on the boundary. This factcan be observed by considering the eigen-decomposition of the control points. Letp(k) ∈ R

(n+1)×3 be a vector consisting of one-ring neighbor control points of x(k)i atthe subdivision level k, S ∈ R

(n+1)×(n+1) the local subdivision matrix that convert p(k)

to pk+1, i.e.,

p(k+1) = Sp(k) = Skp(1), k = 1, 2, . . .

Here n stands for the valence of x(k)i . Suppose p(1) is decomposed into

p(1) = e0aT0 + e1aT

1 + e2aT2 + · · · + enaT

n , a j ∈ R3,

where e0, e1, . . . , en are the eigenvectors of S. Here we assume that these eigenvectorsare arranged in the order of non-increasing eigenvalues λ j . Then

p(k+1) = λk0e0aT

0 + λk1e1aT

1 + λk2e2aT

2 + · · · + λnnenaT

n ,

where λ0 = 1, λ1 = λ2 < 1. It is well-known that the limit position at the center is a0.The tangent direction at this point are a1 and a2, and a j is given by aT

j = eTj p(1). e j are the

left eigenvectors of S with normalized condition eTj e j = 1. The analysis above is valid

for control function values. The fact that ∇sφi vanishes on the boundary implies that thetangent vector of the subdivision surface on the boundary determined by the boundaryand sub-boundary control points. This is similar to Bézier and B-spline surfaces.

6. Linear Independency. As a set of basis functions, {φi }mi=0 must be linearly independent.

This fact can be derived from the result on the solvability of the following interpolationproblem:

For the given function values { fi }m0 , find the control function values {gi }m

0 such that

m∑j=0

g jφ j (vi ) = fi , i = 0, . . . ,m, (3.3)

where vi , named as vertex, is the limit position of the subdivision surface corresponding tothe control point xi .

Theorem 3.1 The interpolation problem (3.3) always has a unique solution.

Proof Equation (3.3) can be subdivided into three sets. The first set consists of the equationscorresponding to the corner control points. The second set consists of the equations corre-sponding to the boundary control points, except for the corner control points. The third set

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consists of the remaining equations. From property 4 above and interpolatory property ofB-spline functions, the equations in the first set are in the form

giφi (vi ) = fi ,

with φi (vi ) = 1. Hence, gi (gi = fi ) is uniquely determined.

Suppose that the boundary of the surface patch consists of L B-spline curves. The equationsin the second set can be further subdivided into L sub-sets with each sub-set correspondingto the vertices on one curve. Then each set of equations is the classical cubic B-splineinterpolation problems. Hence the solution of each set of the equations exists uniquely.

For the equations in the third set, the coefficient gi corresponding to the corner andboundary control points have been determined through solving the first two sets of equations.Hence, the related terms in these equations can be moved to the right-handed sides. Thenthese equations are the same as the interpolation problem of the Loop’s subdivision scheme.From the result in [30], this interpolation problem has a unique solution. Hence, the theoremis proved.

Theorem 3.1 implies that {φi }mi=0 is linearly independent. In fact, if {φi }m

i=0 is not linearlyindependent, there exists a set of constants {gi }m

i=0 with at least one nonzero element suchthat

∑mj=0 g jφ j (v) = 0 for any v on the subdivision surface. Taking v = vi , i = 0, . . . ,m,

we have∑m

j=0 g jφ j (vi ) = 0. Since the coefficient set with zero elements is obviously thesolution of this homogeneous system, the system has at least two solutions. It contradictsthe uniqueness of the solution of the interpolation problem. Therefore, we have proved thefollowing result.

Theorem 3.2 The functions φi , i = 0, . . . ,m, defined by the extended Loop’s subdivisionscheme, are linearly independent.

3.4 Spatial Discretizations

Suppose φi is a basis function of VS(t) corresponding to control point xi , i = 0, . . . ,m.Assume that x0, . . . , xm0 are the interior control points, and xm0+1, . . . , xm are the boundarycontrol points. Then x(t) ∈ S(t) can be represented as

x(t) =m0∑j=0

x j (t)φ j +m∑

j=m0+1

x j (t)φ j , x j (t) ∈ R3, (3.4)

and therefore,

∇sx(t) =m0∑j=0

∇sφ j [x j (t)]T +m∑

j=m0+1

∇sφ j [x j (t)]T , x j (t) ∈ R3. (3.5)

The mean curvature vector of the surface is represented approximately as

y(t) =m∑

j=0

y j (t)φ j , y j ∈ R3, ∇sy(t) =m∑

j=0

∇sφ j [y j (t)]T ∈ R3×3. (3.6)

Since the boundary control points are fixed and the interior control points are to be deter-mined, the coefficients x j in the first term of (3.4) are unknowns, while the coefficients x j

in the second term are the given control points on the boundary. Furthermore, since the cur-vature on the surface boundary involves the unknown interior control points, hence all thecoefficients in (3.6) are treated as unknowns.

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Now let us discretize Eqs. (2.7)–(2.9) in the finite space VS(t). Since these equations aresimilar in form, we treat them together. Let S be the limit surface of the extended Loop’ssubdivision scheme for the control polygon Sd . Substituting (3.4)–(3.6) into (2.7)–(2.9), andtaking the test functions φ as φi (i = 0, . . . ,m0),ψ as φi (i = 0, . . . ,m), and finally noticing

that∂x j (t)∂t = 0 if j > m0, we obtain the following matrix representations of (2.7)–(2.9):

⎧⎨⎩

M (1)m0

∂Xm0 (t)∂t + L(1)m Ym(t) = 0,

M (2)m Ym(t)+ L(2)m Xm(t) = B,

(3.7)

where

X j (t) =[xT

0 (t), . . . , xTj (t)

]T ∈ R3( j+1),

Ym(t) = [yT

0 (t), . . . , yTm(t)

]T ∈ R3(m+1),

are matrices consisting of the control points for the surface and the mean curvature normals,respectively, and

B = [bT

0 , . . . ,bTm

]T ∈ R3(m+1),

M (1)m0

= (mi j I3

)m0,m0i j=0 , M (2)

m = (mi j I3

)m,mi j=0 ,

L(1)m =(

l(1)i j

)m0,m

i j=0, L(2)K =

(l(2)i j I3

)m,K

i j=0.

are the coefficient matrices. The elements of these matrices are defined as follows:

mi j =∫

Sφiφ j dA,

l(1)i j =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

l(s)i j for SDF,

l(s)i j + 4∫S

[n(H2 − K )φiφ j

]nTdA for WF,

l(s)i j − 4∫S

[n(2H2 − K )φiφ j

]nTdA for QSDF,

l(2)i j = 1

2

∫

S

[(∇sφi )

T∇sφ j]

dA,

bi = 1

2

∫

ncφi ds, (3.8)

with

l(s)i j = 2∫

S

[φi � φ j − n(∇sφi )

T∇sφ j]

nT dA.

Moving the terms relating to the known control points xm0+1, . . . , xm in the second equationof (3.7) to the equations’ right-hand side, we can rewrite (3.7) as

⎧⎨⎩

M (1)m0

∂Xm0 (t)∂t + L(1)m Ym(t) = 0,

M (2)m Ym(t)+ L(2)m0 Xm0(t) = B(2).

(3.9)

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Note that, matrices M (1)m0 and M (2)

m are symmetric and positive definite. The integrations forcomputing the matrix elements are computed by a 12-point Gaussian quadrature rule. Thatis, each domain triangle is subdivided into four sub-triangles and a three-point Gaussianquadrature rule is employed on each of the sub-triangles. The 3-point Gaussian quadraturerule has error bound O(h3), where h is the maximal edge length. The knots in the barycentriccoordinate form and weights of the Gaussian quadrature formulas can be found in [1]. Forevaluating the basis functions defined by the Loop’s subdivision, a fast method is given byStam [29]. We modify this method so that the basis functions defined by the extended Loop’ssubdivision can be evaluated.

In the boundary integrals (3.8), nc is the co-normal of the surface, it is infeasible tocompute these co-normals from the previous approximation, since they do not satisfy thegiven boundary condition. The right way is to replace nc with n()c . That is

bi = 1

2

∫

n()c φi ds.

3.5 Temporal Direction Discretization

Suppose we have obtained approximate solutions X (k)m0 = Xm0(tk) and Y (k)m = Ym(tk) at

t = tk . We want to obtain approximate solutions X (k+1)m0 and Y (k+1)

m at t = tk+1 = tk + τ (k)

using a forward Euler scheme. Specifically, we use the following approximation

Xm0(tk+1)− Xm0(tk)

τ (k)≈ ∂Xm0

∂t.

The matrices M (1), M (2), L(1) and L(2) in (3.9) are computed using the surface data at t = tk .This yields a linear system with X (k+1)

m0 and Y (k+1)m as unknowns:⎡

⎣ M (1)m0 τ (k)L(1)m

L(2)m0 M (2)m

⎤⎦

⎡⎣ X (k+1)

m0

Y (k+1)m

⎤⎦ =

[τ (k)B(1) + M (1)

m0 X (k)m0

B(2)

].

Though the matrices M (1) and M (2) are symmetric and positive definite, the total matrixis neither symmetric nor positive definite. However the coefficient matrix of this systemis highly sparse, hence a stable iterative method for its solution is desirable. We use Saad’siterative method, namely GMRES (see [22]), to solve our sparse linear system. The numericaltests show that this iterative method works well.

The iteration in the temporal direction stops if the termination condition

max1≤i≤m0

‖xi (tk+1)− xi (tk)‖ < ετ(k)

is satisfied, where ε is a given threshold value, we take it as 10−5.

Remark 3.1 In each iteration step, the matrices M (1), M (2), L(1) and L(2) need to be recom-puted since they depend on the evolved surface S(t). However, the basis function φi and theirpartial derivatives (up to the second orders) do not change in the evolution. Hence, functionvalues and partial derivatives of each φi can be pre-computed at the knots of the Gaussianquadrature. Furthermore, the boundary terms bi are fixed during the iterations, since bothn()c and φi are fixed.

At the end of this section, we point out the differences between our geometric PDE methodand other subdivision methods. We compare our method with the one presented in [2] by

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Fig. 5 a The boundary curves. b The initial control polygon. c Boundary curves with different co-normals.e The boundary curves with the same co-normals. d, f The control polygons of the PDE subdivision surfacescorresponding to the boundary curves (c) and (e), respectively

Biermann et al, since both methods can handle boundaries and inner sharp features. Thedifferences are mainly on three aspects: (1) The method in [2] needs to repeatedly subdividethe initial control mesh, so the control mesh will be finer and finer. Our method just moves thevertex position in the mesh, and it does not change the number of vertices. (2) The resultingsurfaces of our method are the solutions of geometric PDES, which share certain optimalproperties. (3) The method in [2] is simpler and more efficient than our method, because inour method, linear systems need to be solved in each iteration.

4 Illustrative Examples

To illustrate our surface construction method is effective. We give several graphical examplesin this section. All the examples are constructed as follows:

1. Given G1 boundary conditions: i.e., specify B-spline boundary curves with co-normalson these curves.

2. Given an initial control triangular mesh that interpolates the boundary B-spline controlpoints.

3. Evolve the triangular control polygon using the geometric flows with fixed G1 boundarycondition.

For easy to illustrate, several existing triangle mesh models are used as the initial control mesh.Boundary control polygon are extracted from these models. The co-normals are computedfrom the initial control mesh. To show the power of our approach, the interior control verticesare sometimes shifted.

4.1 Surface Patch Join

In Figs. 1 and 5, we join several surface patches together to form closed surfaces. For eachpatch, boundary curves and co-normals are provided. At the common boundaries, the co-normals may not be the same. Hence, surfaces with sharp feature can be constructed. Thecontrol mesh of the PDE subdivision surface as shown in Fig. 1d is generated using surfacediffusion flow with 20 iterations and a temporal step-size 0.0001. The control meshes of thePDE subdivision surfaces as shown in Fig. 5d, f are generated using Willmore flow with 20iterations and a temporal step-size 0.01.

4.2 The Effect of Different Geometric PDEs

The aim of Fig. 6 is to illustrate the different evolution effects of the used three forth-order geometric flows. We construct the initial surfaces at a rough level and perturb their

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Fig. 6 a and a′ The boundary curves. b and b′ The prescribed co-normals on the boundary curves. c and c′Are initial constructed surfaces. From the initial surface (c), we achieve three resulting surfaces (d), (e) and (f)after 50, 50 and 25 evolution steps with the time step-size 0.0005, using SDF,WF and QSDF respectively. Fromthe initial surface (c′), we achieve three resulting surfaces (d′), (e′) and (f′) after 100, 100 and 12 evolutionsteps with the time step-size 0.001 by use of SDF,WF and QSDF respectively

Table 1 Time cost for six models of Fig. 6

Examples # unknowns τ (k) Matrix T One GMRES # GMRES Total time

Figure 6d 18,408 0.0005 2.49 50 0.0217 300 450.45

Figure 6e 18,408 0.0005 2.51 50 0.0220 300 455.5

Figure 6f 18,408 0.0005 2.52 25 0.0222 300 229.5

Figure 6d′ 2,688 0.002 0.38 100 0.0027 300 119.0

Figure 6e′ 2,688 0.002 0.40 100 0.0028 300 124.0

Figure 6f′ 2,688 0.002 0.41 12 0.0030 300 15.732

First column: examples. Second column: the number of unknowns. Third column: the temporal step-size (inseconds). Fourth column: the time (in seconds) for computing the coefficient matrix. Fifth column: the numberof evolution steps. Sixth column: the time cost (in seconds) of one step GMRES iteration. Seventh column:The number of GMRES iteration times in every evolution step. Last column: The total time (in seconds) usedfor achieving the corresponding resulting surfaces

interior domain, then linearly refine them several times as the initial constructions of ourevolution equation which are shown in (c) and (c′). WF makes evolved surfaces fat. QSDFmakes the evolved surfaces thin since the solution of QSDF approaches minimal surfaces.While the volume preserving property of SDF makes the evolved surfaces neither too fatnor too thin. Hence, for long-term evolution, the behaviors of the three flows are quitedifferent.

In Table 1 we summarize the computation time needed in the examples in Fig. 6. Thealgorithm was implemented using C++ in Linux system running on a Dell PC with a 2.4 GHzQ6600 Intel CPU. The second column in Table 1 lists the number of unknowns. Thesenumbers are counted as 3n0 + 3n1. Here n0 is the number of interior vertices, n1 is thenumber of the unknown mean curvature normals. The third column is the temporal step-sizes used (in seconds). The fourth column in the table is the time (in seconds) for formingthe coefficient matrix (one time step). The fifth column is the number of evolution steps.The linear systems are solved by GMRES iterative method. The sixth column is the time(in seconds) of one GMRES iteration in every evolution step. The seventh column is thenumber of one GMRES iteration. The last column is the total time (in seconds) used for the

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Fig. 7 a The boundary curves. a′ The prescribed co-normals on the boundary curves. b and c and d Threedifferent initial constructed surfaces. a′′ The final resulting surface from these three initial surfaces after 200evolution steps using SDF with temporal step-size t = 0.01. b′ and b′′ The resulting surfaces after 1 and2 evolution steps from the initial surface (b) respectively. c′ and c′′ The resulting surfaces after 20 and 50evolution steps from the initial surface (c) respectively. d′ and d′′ The resulting surfaces after 10 and 50evolution steps from the initial surface (d) respectively

corresponding resulting surfaces. The threshold value of controlling the iteration-stopping istaken to be 10−5, about the single word-length accuracy of the used computer.

4.3 Stability

Our aim of presenting Figs. 7 and 8 is to illustrate that our method is stable. Given thesame boundary condition with the different initial values, we can achieve the same finalresulting surfaces. In Fig. 7, surface diffusion flow is used to evolve three different surfaces.The final results are the same. We show one of them in Fig. 7(a′′). In Fig. 8, quasi-surfacediffusion flow is used to evolve three different initial surfaces. Again, the final results arethe same. One of them is shown in Fig. 8(a′′). A few transition results between the initialsurfaces and the final surfaces are also presented in Fig. 7 and 8. We can see that the num-ber of the required evolution steps is larger if the initial surface is farther from the steadysolution.

It is easy to see that the initial surfaces are quite different. But the steady solutions are thesame. However, these illustrative results do not imply that the steady solutions of the usedgeometric flows are unique. If the initial surface has a different structure, the final surfacescan be very different. Figure 9 shows such an example. In this figure, the boundary conditionis a circle with co-normals (see Fig. 9a, b). An initial surface shown in (c) is a simple andbowl-shaped surface. Another initial surface shown in (e) is an ashtray-shaped surface wherethe boundary curve is inside the ashtray and could not be seen from outside. Willmore flowis used to evolve these two different initial surfaces. The final results, as shown in Fig. 9d, f,are completely different.

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Fig. 8 a The boundary curves. a′ The prescribed co-normals on the boundary curves. b, c and d Threedifferent initial constructed surfaces. a′′ The final resulting surface from these three initial surfaces after 200evolution steps using QSDF. b′ and b′′ The resulting surfaces after 1 and 2 evolution steps from the initialsurface (b) respectively. c′ and c′′ The resulting surfaces after 20 and 50 evolution steps from the initial surface(c) respectively. d′ and d′′ The resulting surfaces after 20 and 50 evolution steps from the initial surface (d)respectively. The time step-size is chosen as t = 0.04

Fig. 9 a The boundary curve (a circle). b The prescribed co-normals on the boundary curve. c and e Twodifferent initial surfaces constructed. d and f The resulting surfaces after 160 and 80 evolution steps from theinitial surfaces (c) and (e) respectively, using WF with temporal step-size t = 0.02. The view direction for (e)and (f) is perpendicular to the circle

5 Conclusions

Extended mesh subdivision technology can be utilized as a simple and efficient method toconstruct piecewise smooth surfaces with any topology structure and specified boundaryconditions. Geometric PDEs are powerful tools for constructing high quality surfaces. In

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this paper, we combine these two ingredients together. We construct geometric PDE Loop’ssubdivision surfaces, with given G1 boundaries condition, using three fourth-order geometricflows. A numerical solution method of the finite element based on the extended Loop’ssubdivision scheme is adopted, and the geometric PDE subdivision surfaces are thereforeefficiently constructed.

Acknowledgments Guoliang Xu was supported in part by NSFC Funds for Creative Research Groups ofChina (Grant No. 11021101, 11321061). Qing Pan was supported by a National Natural Science Foundationof China (Grants No. 11171103) and Scientific Research Fund of Hunan Provincial Education Department(No. 12K029).

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