Variational Laplacians for semidiscrete surfaces

We study a Laplace operator on semidiscrete surfaces that is defined by variation of the Dirichlet energy functional. We show existence and its relation to the mean curvature normal, which is itself defined via variation of area. We establish several core properties like linear precision (closely related to the mean curvature of flat surfaces), and pointwise convergence. It is interesting to observe how a certain freedom in choosing area measures yields different kinds of Laplacians: it turns out that using as a measure a simple numerical integration rule yields a Laplacian previously studied as the pointwise limit of geometrically meaningful Laplacians on polygonal meshes.

1 Introduction and preliminaries 1

.1 Introduction
The Laplace-Beltrami operator = − div • grad on smooth surfaces and Riemannian manifolds is an extremely well investigated differential operator, which plays an essential role in many fields including applications. A main strength lies in Riemannian geometry, but it is also relevant to the elementary differential geometry of surfaces in three-dimensional space, e.g. via the equation id = −2H n that relates the Laplacian to the mean curvature and unit normal vector field. Its intrinsic nature makes it very useful for computational applications in geometry processing, see e.g. [16], and it has therefore been extensively discretized. Discrete Laplace operators defined on triangulations share characteristics with graph Laplacians, but ideally maintain as many of the core properties of the original Laplace-Beltrami operator as possible. For contributions to this topic see e.g. [1-3, 11, 12]. Another important aspect of discretizations is a suitable convergence behavior, see e.g. [18,20,21].
A powerful tool to derive Laplace operators on more general surfaces arises from the calculus of variations. The Laplacian of Riemannian geometry can be seen as gradient of the Dirichlet energy, which leads to the famous "cotangent formula" Laplacian on triangle meshes, see e.g. [8,12]. The variational approach is also particularly suited to study the mean curvature normal H = H n, which has an interpretation as the gradient vector field of the area functional.
In this paper we follow the variational approach. Our aim is to define meaningful Laplacians on semidiscrete parametric surfaces, which are represented by a point depending on one continuous and one discrete variable. The reader is reminded that semidiscrete objects occur in the classical theory of transformations of surfaces. For a systematic and unified treatment of continuous and semidiscrete surfaces as limits of a discrete master theory we refer to the textbook [4]. The lowest-dimensional case, i.e., 2-dimensional surfaces, has been investigated from various viewpoints. The semidiscrete incarnation of conjugate surfaces is studied by [13] where piecewise-developable surfaces (including circular and conical semidiscrete surfaces) are considered from the computational viewpoint. Curvatures, in analogy to polyhedral surfaces, are the topic of [10]. Asymptotic surfaces and especially K-surfaces are investigated by [17]. The present paper however, is not concerned with any special class of semidiscrete surfaces.

Outline and results
In Section 2 we define a Laplace operator on semidiscrete surfaces by a variational principle, namely as gradient of an appropriate Dirichlet energy functional. We show that this gradient exists and provide a closed-form expression for the semidiscrete Laplacian in Theorem 1. It turns out that there is quite some freedom in the choice of the particular L 2 space which is basic to the concepts of both gradient and Dirichlet energy. Section 3 investigates the gradient of the area functional to gain a semidiscrete mean curvature normal, and establishes the relation id = −2H for the semidiscrete case (Theorem 2), which in turn implies that linear functions on flat surfaces are in the kernel of the Laplacian (i.e., the linear precision property). Section 4 discusses further properties like locality, symmetry, positive semidefiniteness, and lack of a maximum principle. The last section deals with pointwise convergence of the semidiscrete Laplacian towards the Laplace-Beltrami operator on smooth surfaces (Theorem 3).

Variational properties of the Laplacian
The Laplace-Beltrami operator M on a Riemannian manifold M can be defined via the Dirichlet energy functional It is then given as the gradient of the Dirichlet energy, which means that for smooth test functions u, and all smooth 1-parameter variations u ξ of u, with the property that is compactly supported, we have [9, pp. 89-94]). This relation is basic to the generalization of the Laplace-Beltrami operator to discrete surfaces and will also be used in the present paper. Recall that for a surface M embedded in R 3 , the Laplace operator has a remarkable connection to the mean curvature normal. Applying the Laplacian component-wise to the identity mapping id M , we get [7, p. 22]), where the mean curvature normal H = H n is a unit normal vector n on M scaled by the corresponding mean curvature H . Observe that H is independent of the particular choice of n, as the sign of H depends on the direction of n. This vector field likewise has a variational definition, namely for every smooth 1-parameter variation p ξ : M → R 3 with p 0 = id M (see [7, p. 7]).

Semidiscrete surfaces
The semidiscrete surfaces which constitute our object of study are mappings of the form and where V is a vector space equipped with a positive definite scalar product ·, · V . Throughout this paper we assume that x is at least twice continuously differentiable in the second argument, and we denote the corresponding set of mappings by C 2 sd (U, V ). Accordingly, the set of semidiscrete functions that are merely continuous in the second argument is denoted by C sd (U, V ). With the help of the canonical hat function ϕ(s) := max{1 − |s|, 0}, we extend x to a mapping, again called x, where the domain U is constructed as a disjoint union of strips U k ⊂ R 2 , each strip being defined as In the non-degenerate case, this procedure converts a sequence of curves into a piecewise-ruled surface, connecting corresponding points x(k, t) and x(k+1, t) by straight line segments. For each pair of successive curves x(k, ·) and x(k + 1, ·) there is a ruled surface strip, which is treated separately from the others as far as the domain of definition is concerned. This procedure does not alter the values x(s, t) where s happens to equal an integer k ∈ Z; x(s, t) has the same value regardless of the question if s is considered as an element of [k − 1, k] or as an element of [k, k + 1]. We call the procedure of converting a semidiscrete surface x(k, t) to a piecewise-ruled surface x(s, t) an "extension", even if U is not a subset of U . In order to make the upcoming formulas shorter and thus better readable, we set up the following notation. For the derivatives of x(k, t) with respect to the variable t, we write x , x , and so forth. Finite differences in the discrete direction are denoted by δx(s, t) Note that in contrast to x itself, the discrete derivative δx does have different values for s = k ∈ Z, depending on whether s is thought to be contained in [k − 1, k] or in [k, k + 1]. We resolve this ambiguity by always making it clear which of the two corresponding surface strips we are considering. We call a semidiscrete surface regular, if all its surface strips are regular in the usual sense, i.e., if is linearly independent throughout. Moreover, we call (k, t) ∈ U an inner point, if for some ε > 0. Otherwise it is called a boundary point. The set of inner points of U will be denoted by U inn . Note that we do not make any assumptions on the embeddedness of the surfaces we study. Later, when considering a real-valued function "u" on a semidiscrete surface x, we regard it as defined in U rather than in x(U ). Such a function u therefore formally is a semidiscrete surface in its own right and we use the same notation as for the surface x. We call u smooth, if it is at least twice continuously differentiable in the second argument, i.e., if u ∈ C 2 sd (U, R).

Remark 1
It is easy to see that a semidiscrete surface x(s, t) is regular for all s ∈ [k, k + 1] if x (k, t), x (k + 1, t) and δx(k, t) are linearly independent (in which case the ruled surface strip corresponding to s ∈ [k, k +1] is a regular skew ruled surface).

Variational definition of a semidiscrete Laplace operator
This section aims at a meaningful definition of a Laplace operator on semidiscrete surfaces. Mimicking the smooth case, we define a semidiscrete Laplacian as gradient of an appropriate Dirichlet energy functional. For this purpose we first discuss area measures.

Integration and Laplacian on semidiscrete surfaces
Consider a semidiscrete surface x with open domain U ⊂ Z × R, which has been extended to a piecewise-ruled surface defined in the domain U , as described above (cf. Eq. 1). A reasonable definition of its area obviously is given by the sum of the areas of individual ruled surface strips, which in terms of the matrix I of the first fundamental form is expressed as (see [6, p. 98]). Note that, in order to resolve the ambiguity in the definition of δx, the double integral over U has to be interpreted as the sum of double integrals over the individual strips U k stated in Eq. 2. It makes sense to generalize this definition by replacing Lebesgue measure dsdt by other measures. We start with a Borel measure μ 0 supported on the unit interval [0, 1], whose zeroth and first moments have the following values: That is, we require integration of polynomials up to degree 1 to coincide with integration w.r.t. Lebesgue measure. A stronger property is symmetry of the measure, meaning that Together with m 0 = 1 symmetry implies m 1 = 1/2. This symmetry property is not required except in Theorem 3, where it is explicitly mentioned.
We will see that these assumptions are crucial for some important properties of the Laplacian, and also for convergence. We actually construct an entire family of semidiscrete Laplace operators, depending on the type of integration we employ. Note that in particular the measure μ 0 might be a numerical integration rule, like the midpoint rule or the trapezoidal rule. As it turns out, a particular choice of measure leads to the semidiscrete Laplacian introduced in [5] as a pointwise limit of the discrete construction of Alexa and Wardetzky [1]. We discuss this connection in Section 2.2. Now, by translation, μ 0 acts as a measure on each interval [k, k + 1], and we denote the sum of measures on the disjoint union of intervals [k, k + 1] by μ. With the Lebesgue measure λ on the reals, we consider the product measure μ ⊗ λ on the disjoint union of strips [k, k + 1] × R. It is precisely this measure which we use for integration in the domain U : We use its first fundamental form I and the measure μ ⊗ λ on U to define the surface integral of a function u : U → R: The surface area is given by area μ (x) := x 1 dA.
Again, by the integral over U we mean the sum of integrals over the individual strips U k given by Eq. 2. Note that for nonnegative functions u, the integral always exists and x udA ∈ [0, ∞], whereas for general u, Example 1 This definition in particular applies to a semidiscrete function u : U → R, which has been extended to a piecewise-linear function u : U → R by linear interpolation: If u vanishes at the boundary of U , we can write its surface integral as where π 2 : R × R → R : (s, t) → t, and the semidiscrete function a is defined by Here, the integral over An index shift yields the formula given above. Definition 2 Given a semidiscrete surface x with domain U , we define L 2 inner products for semidiscrete real-valued (resp. V -valued) functions u, v with the same domain by letting The integrals in the previous formulas mean that the semidiscrete functions u, v are multiplied to create a semidiscrete product function (u · v)(k, t) (resp. u, v V (k, t)), which for the purpose of integration undergoes linear interpolation. The inner products are, for instance, well defined for semidiscrete functions that are continuous in the second argument and have finite L 2 norm.
For the Dirichlet energy of a semidiscrete function we use the following definition: As to the gradients of real-valued functions u(s, t), v(s, t) on a parametric surface x(s, t), recall that ∇u, ∇v = ∂u/∂s ∂u/∂t is the matrix of the first fundamental form. This leads to the following explicit expression for the Dirichlet energy: Observe that E μ (u) essentially is the ordinary Dirichlet energy of the piecewisesmooth function u(s, t) over the piecewise-smooth surface x(s, t).
It is tempting to employ L 2 notation for the definition of the Dirichlet energy. We will not do that, since the integrand is not generated by extending a semidiscrete function, and therefore does not fit Definition 2.
Next, we generalize the notion of the gradient of an energy functional to the semidiscrete case. For that we consider "admissible" variations of semidiscrete functions: which depends smoothly on ξ and t, coincides with x(k, t) for ξ = 0, and such that x ξ (k, t) does not depend on ξ outside a compact subset K of U inn . Moreover, we call a subset B ⊂ U associated with the variation x ξ , if it is open and bounded, with For the derivative of x ξ with respect to ξ we use the notationẋ This definition in particular applies to admissible variations u ξ (k, t) of smooth semidiscrete functions u : U → R. In what follows we discuss the variation of energy, and the variation of surface area, even if these quantities are not finite. As admissible variations only take effect on compact parts of the domain U , the corresponding change in energy or surface area can be defined in a meaningful way.
Definition 5 Let x : U → V be a regular semidiscrete surface and let E be a functional on C 2 sd (U, R), with the property that there exists an operator ∇E : , such that for every u ∈ C 2 sd (U, R) and all admissible 1-parameter variations u ξ of u with associated subset B ⊂ U , we have Then ∇E is called the gradient of E. In particular, we define the semidiscrete Laplace operator sd on x as the gradient of the Dirichlet energy functional E μ , i.e., sd := ∇E μ .
Note that this definition is independent of the particular choice of the bounded open subset B ⊂ U associated with a variation u ξ of u.  Proof Let u ξ be an admissible variation of u with derivativeu and let B ⊂ U be associated with u ξ . We compute the derivative of the Dirichlet energy by using the Leibniz rule (which applies because all occurring functions are smooth in the variables ξ and t, andu has compact support): Next we apply integration by parts w.r.t. t to the second summand: Observe that the boundary terms vanish, since the support ofu is contained in U inn . Finally, an index shift yields with b, c, and sd u as stated above (cf. also Example 1).

Example: semidiscrete Laplacians arising as limits of discrete ones
As demonstrated in [5], the discrete Laplace operator L constructed by Alexa and Wardetzky [1] for functions defined on the vertices of a polygonal mesh gives rise to a Laplace operator on semidiscrete surfaces via pointwise limits. We may discretize a regular semidiscrete surface x : U → R 3 and a smooth function u : U → R near a point of interest (k, t) ∈ U inn by letting x ε ij := x(k + i, t + εj ), u ε (x ε ij ) := u(k + i, t + εj ). This defines the vertices x ε ij of a quad mesh with regular combinatorics, and function values on these vertices. The discrete Lapace operator on that mesh is denoted by L ε , and we let Existence and properties of this limit were investigated in [5], in particular independence of the limit from the still remaining degrees of freedom in the construction of L. There is a remarkable connection between our semidiscrete Laplacian sd and the semidiscrete Laplacian lim which arises by pointwise limits. In fact, if the measure μ 0 used to construct sd is taken as the midpoint rule for numerical integration (i.e., ), then they are equal:

ϕ(s − k) δx(s, t) × x (s, t) dμ(s)
where we adopt the notation from [5, Corollary 1]. In particular, By inserting these functions into Eq. 6 and comparing the resulting expression with the formula stated in [5, Corollary 1], we see that sd u = lim u.

Semidiscrete mean curvature normals
Before we analyze further properties of the semidiscrete Laplace operator, we discuss its connection to the mean curvature normal. Recall from the introductory section the relations between the Laplacian and the mean curvature normal, which hold for smooth surfaces embedded in R 3 : On the one hand, M id M = −2H, on the other hand the mean curvature normal itself has the variational definition −2H = ∇area(M). Here we consider the semidiscrete version of these objects and the relations between them. Our notation is not entirely the same as in Section 1.2, because we deal with parametric surfaces.

Variational properties of mean curvature
Definition 6 Let F be a functional on C 2 sd (U, V ) and let x : U → V be a semidiscrete surface with the property that there exists a function ∇F (x) ∈ C sd (U inn , V ), such that for all admissible 1-parameter variations x ξ of x with associated subset Then ∇F (x) is called the gradient of F at x. In particular, the semidiscrete mean curvature normal H sd of a regular semidiscrete surface x is defined as Proof Let x ξ (k, t) be an admissible variation of x. Each semidiscrete surface x ξ (k, t) is extended to a piecewise-ruled surface x ξ (s, t), having first fundamental form I ξ (s, t) (cf. Eq. 3). By definition, x ξ (k, t) is independent of ξ outside a compact subset K of U inn . Thus, by a standard argument, the piecewise-ruled surfaces x ξ (s, t) are regular for all ξ in some interval (−h, h), because det I ξ , i.e., the area spanned by the partial derivatives of x ξ , is positive in a compact set {0} × K ⊂ R × U , thus positive in a neighbourhood of this set, and consequently positive in a product set (−h, h) × K . Thus, we may compute For ξ = 0, this expression is simplified by computing the individual derivatives where the functions b(s, t) and c(s, t) are the same as in Eq. 7, and the previous formula is the same as the expression for the derivative of the Dirichlet energy in the proof of Thm. 1, only with x instead of u, and scalar products of V -valued functions instead of products of real-valued ones. It follows that the gradient of area μ evaluated at x indeed equals sd x.

Mean curvature of extrinsically flat surfaces
We show that the mean curvature normal of a regular semidiscrete surface x vanishes, if that surface is embedded in a 2-dimensional plane. Here, embeddedness means injectivity of the extended surface x(s, t). Besides constituting a sanity check on our definitions, this fact is of importance later when we show the "linear precision" property of the semidiscrete Laplacian.

Lemma 1
If the regular semidiscrete surface x : U → V is embedded in a 2dimensional plane, then its mean curvature normal H sd vanishes.
Proof The general idea of the proof is to show that H sd L 2 (x,V ) = 0 by constructing a variation whose derivative equals H sd . This can be done in the following way. Choose a smooth function v : U → R with compact support contained in U inn . Then is a well-defined 1-parameter variation of x with velocityẋ = v 2 H sd . Now, let denote the plane containing the surface x and assume without loss of generality that 0 ∈ , so is a linear subspace and therefore δx, x ∈ . It follows from Thm. 1 and Thm. 2 that H sd (k, t) ∈ , and consequently, x ξ (k, t) ∈ . Since dim = 2, we can express the above-mentioned area in terms of an appropriate determinant form |·, ·|: det I ξ (s, t) = δx ξ (s, t), x ξ (s, t) = (1 − s + k) δx ξ (k, t), x ξ (k, t) + 1, t) , for s ∈ [k, k + 1], t fixed.
By Eq. 4, integrating det I ξ over [k, k + 1] w.r.t. dμ(s) is the same as integrating w.r.t. Lebesgue measure. Thus, area μ (x ξ | B ) equals the unsigned Euclidean area. Since the variation x ξ leaves the boundary of the surface unchanged, area μ (x ξ | B ) does not depend on ξ , and we get We conclude that vH sd vanishes for all v, i.e., H sd = const. = 0.

Properties of the semidiscrete Laplacian
The classical Laplace operator enjoys several properties like linear precision, symmetry, positive semidefiniteness, and an associated maximum principle for harmonic functions. It is natural to ask if they carry over to the purely discrete or semidiscrete cases (for triangle meshes, these core properties turn out to be incompatible for Laplacians whose definition involves the 1-ring neighbourhood of individual vertices; see [19]). We start by investigating the kernel of the Laplacian. Surely it contains the constant functions. As to linear functions, we have the following result: Lemma 2 For a regular semidiscrete surface x and its corresponding Laplacian sd and mean curvature normal field H sd , the following statements are equivalent: We show that our semidiscrete Laplacian is symmetric and positive semidefinite in the L 2 sense, in a way analogous to the well known Laplace-Beltrami operator (see, e.g. [15]). This follows directly from the variational definition of the Laplacian.

Lemma 3
The semidiscrete Laplace operator sd is symmetric and positive semidefinite. More precisely, for semidiscrete functions u and v, with compact support contained in U inn , we have Proof We use the quadratic form E μ corresponding to the Dirichlet energy (see Definition 3) and compute where we have used the relations given in Definition 5. This implies symmetry and, for u = v, semidefiniteness.
Unfortunately, the maximum principle is not valid for the semidiscrete Laplacian, even for functions on very simple surfaces. This is in contrast to the smooth case, where the maximum principle holds in general; and it is also in contrast to the cotan-Laplacian on triangle meshes (likewise found as gradient of the Dirichlet energy), where a maximum principle holds e.g. if all angles are acute. A counterexample is as follows.
Example 2 Here we construct a semidiscrete harmonic function u with a maximum at the inner point (0, 0) of the semidiscrete surface x(k, t) := (k, t). For this purpose we first derive a more explicit expression for the Laplacian sd u of a semidiscrete function u on x. We extend x to x(s, t) = (s, t) and u to the piecewise-linear function u(s, t) = (k,t)∈Z×R ϕ(s − k)u(k, t). Then I = diag(1, 1), so by the assumptions 4, we get where m 2 = [0,1] s 2 dμ 0 is the second moment of the measure μ 0 . Hence, in this situation, the Laplacian of u is given by The harmonicity condition sd u = 0 thus becomes a system of linear ODEs for the functions t → u(k, t), where k runs through the integers. Observe that the assumptions 4 imply 1 4 ≤ m 2 ≤ 1 2 . The maximum principle obviously holds if m 2 = 1 2 , which applies e.g. to the trapezoidal rule. Otherwise, for m 2 < 1 2 , we can construct a harmonic function u on x with a maximum at (0, 0) as follows. Choosing u(0, t) := −t 2 and assuming symmetry u(±1, t) := φ(t), we find φ(t) easily as φ(t) = 1 − t 2 + γ 1 cos(( 1 2 − m 2 ) −1 /2 t) + γ 2 sin(( 1 2 − m 2 ) −1 /2 t). An appropriate choice of constants, e.g. γ 1 = −2, γ 2 = 0, yields a function u(k, t), which undoubtedly has a local maximum in u(0, 0) = 0. We have thus created a locally defined counterexample to the maximum principle. It can be turned into a globally defined example by constructing u (±2, t), u(±3, t), . . . such that overall sd u = 0: one has to iteratively solve linear ODEs.

Pointwise convergence / consistency
In this section we show that the semidiscrete Laplace operator converges pointwise to its smooth counterpart, as the semidiscrete surface converges to a smooth one. In the Finite Elements literature this kind of convergence is called consistency, while convergence would be reserved for the situation where the solutions of equations involving the semidiscrete Laplacian converge to solutions of equations which involve the continuous Laplacian.
More precisely, the situation in the following theorem is as follows. We fix a point p on a regular surface M, which is assumed to have a local parametrization f . Without loss of generality, p = f (0, 0). Next, we consider the semidiscrete surface which obviously contains the point p = x ε (0, 0) and is inscribed in the surface M. Then we analyze the semidiscrete Laplace operator associated with x ε and its action on functions u ε , and let ε → 0.

Theorem 3 Consider a smooth regular surface M with parametrization f and a real-valued function u(s, t) which represents a function defined on the surface M.
Let p = f (0, 0). Semidiscretize these objects by defining a semidiscrete surface x ε (k, t) := f (εk, t) and a semidiscrete function u ε (k, t) := u(εk, t). Then the corresponding semidiscrete Laplace operator ε sd converges to the Laplace-Beltrami operator M defined on M: (1), as ε → 0. In case the measure μ 0 is symmetric in the sense of Eq. 5, convergence is improved: Theorem 2 immediately implies a convergence statement concerning mean curvature:

Corollary 2
In the situation of Theorem 3, the semidiscrete mean curvature normal H ε sd on x ε converges pointwise to its smooth counterpart (with the rate of convergence depending on the smoothness of the parametrization f).

Proof of Theorem 3
We first set up some notation. For differentiation with respect to s and t we use the notation ∂ 1 and ∂ 2 , respectively. The coefficients of the first fundamental form are denoted by g ij := ∂ i f, ∂ j f . Their determinant is denoted by det I = g 11 g 22 − g 2 12 . We also use the symbols ρ ij k := ∂ i f, ∂ jk f . • Step 1: Overview of the proof.
Integration with respect to dμ(s) and substituting the definitions of α j , β j eventually yields In the same manner as before we get the expansions