Uniform convergence of discrete curvaturesfrom nets of curvature lines

Uniform convergence of discrete curvatures
from nets of curvature lines

Ulrich Bauer Institute for Numerical and Applied Mathematics, University of Göttingen, Lotzestr. 16–18, 37083 Göttingen, Germany. http://ddg.math.uni-goettingen.de/    Konrad Polthier Department of Mathematics and Computer Science, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany. http://geom.mi.fu-berlin.de/    Max Wardetzky 11footnotemark: 1
December 11, 2009
Abstract

We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.

1 Introduction

The field of discrete differential geometry has brought to light intriguing discrete counterparts of classical differential geometric concepts as well as efficient geometric algorithms (see, e.g., Bobenko et al., 2008; Grinspun and Desbrun, 2006). One aspect of this theory is convergence: classical smooth notions should arise in the limit of refinement. Recently, several convergence results have been obtained for curvatures and differential operators defined on polyhedral surfaces. Roughly, one may distinguish three approaches: (i) polynomial surface approximation (see, e.g., Meek and Walton, 2000; Cazals and Pouget, 2005), (ii) geometric measure theory (see, e.g., Fu, 1993; Cohen-Steiner and Morvan, 2006), and (iii) finite element analysis (see, e.g., Dziuk, 1988; Hildebrandt et al., 2006). Among these, (i) provides pointwise convergent curvatures for many, but not all, discrete meshes. In contrast, (ii) and (iii) consider generalizations of integrated, or total, curvatures and yield convergence in the sense of measures or appropriate Sobolev norms, respectively.

Given the convergence of curvatures studied by approaches (ii) and (iii) in an integrated sense, it is natural to ask whether these curvatures can be shown to also converge in a pointwise manner. An affirmative answer can be obtained in some special cases, such as polyhedral surfaces with vertices on the unit -sphere (Xu, 2006). In general, however, the answer to this question is negative: it was observed in Xu et al. (2005) that for general irregular polyhedral surfaces, there exist no -local definitions of discrete curvatures that are pointwise convergent. Here, by -locality we mean that the definition of curvatures associated with a vertex of a polyhedral surface only depends on the -star of , i.e., those vertices that are connected to by a path of at most edges. The concept of -locality is motivated by the smooth setting, where the definition of curvatures and differential operators only depends on local properties of the underlying Riemannian manifold.

Figure 1: A discrete net of curvature lines on an ellipsoid.

Uniform convergence from nets of curvature lines

We provide an affirmative answer to the above question of pointwise convergence of curvatures for a special class of discrete meshes: discrete nets of curvature lines on a given smooth surface that is immersed into Euclidean space (see Figure 1). To obtain approximations of principal curvatures on , we follow a three-step approach. We consider (a) local polyhedral approximations to nets of curvature lines, to which we apply (b) well-known -local integrated notions of discrete curvatures, such as those based on normal cycles (see, e.g., Cohen-Steiner and Morvan, 2006) or those based on the so-called cotangent formula (see, e.g., Pinkall and Polthier, 1993), followed by (c) dividing the resulting integrated curvatures by appropriate area terms. The resulting pointwise curvature approximations are at first only defined on the vertex set of the underlying net. However, we may regard these curvature approximations as functions by extending them from in a piecewise constant manner to the intrinsic Voronoi regions of the set on . Assuming this extension, we show:

Theorem 1.

Let be a smooth compact oriented surface without boundary111Surfaces with nonempty boundary can be treated with minor technical modifications. immersed into . Consider a discrete net of curvature lines on such that at each vertex the sampling condition (10) is satisfied. Let be an upper bound for the edge lengths of the net such that additionally the intrinsic -balls around vertices cover all of . Then

where depends only on properties of and the shape regularity (9) of the net of curvature lines.

A few remarks seem pertinent before proceeding:

  • Uniform pointwise convergence of principal curvatures obtained by a -local construction from nets of curvature lines is somewhat surprising since in general -local polyhedral curvatures may not even converge in , even if the mesh vertices reside on the smooth limit surface (see Hildebrandt et al., 2006).

  • Our pointwise curvature approximations arise from dividing integrated “Steiner-type” curvatures by associated area terms. For example, the integrated mean curvature of an interior edge of a polyhedral surface may be defined as the product between the length of that edge and the signed angle between the normals of its adjacent faces – a definition that arises from Steiner’s view of considering offset surfaces. To obtain pointwise curvature approximations from discrete integrated curvatures, we divide by so-called circumcentric areas. While this approach is not new, see, e.g., Desbrun et al. (2005), our convergence result may be interpreted as a justification of this construction provided that the edges of a polyhedral surface well approximate the principal curvature directions of a smooth limit surface. We prove uniform lower and upper bounds for edge-based circumcentric areas that may be of interest in their own right.

  • For our result to hold we require the explicit knowledge of positions of the vertices of a net of curvature lines on a smooth surface as well as the combinatorics of this net. More precisely, our curvature approximations at a vertex require the position of and the positions of its direct neighbors (with respect to the combinatorics of the net). Note that we do not require the knowledge of the entire net, though. (Given such an entire smooth net, it would be trivial to compute the principal curvatures at its vertices.) It would be desirable to drop from our approach the requirement of the exact knowledge of vertex positions of a smooth net of curvature lines. Here, one avenue for further study might be to consider discrete analogues of curvature line nets: so-called principle contact element nets, see Bobenko and Suris (2008).

Our uniform convergence result given in Theorem 1 is a consequence of a corresponding local error estimate given in Theorem 2. This error estimate holds up to and including umbilical points, where singularities in the curvature line pattern arise. Using a refinement sequence for each of the three surfaces shown in Figure 2, we observed numerically that although the shape regularity of the net may blow up near umbilics, linear convergence with respect to the maximum edge length remains valid in these cases. Our experiments also indicate that linear convergence is optimal.

lemon star monstar
Figure 2: The three generic patterns of curvature lines near an umbilical point, called lemon, star, and monstar by Berry and Hannay (1977).

Alternative approaches

An alternative point of departure for establishing pointwise convergence of discrete curvatures is to give up -locality and to allow for as the mesh refinement increases. In fact, the above mentioned convergence results of (ii) and (iii) may be interpreted in this way: by decreasing the diameter of the domains over which discrete curvatures are integrated (measured), while simultaneously increasing the mesh refinement inside these domains at a sufficiently fast rate, one recovers classical pointwise notions of smooth curvatures in the limit. In a similar fashion, Belkin et al. (2008) proposed a discrete Laplace operator based on the heat kernel. This operator converges in a pointwise manner if the kernel is scaled down while the mesh resolution is increased sufficiently fast relative to the scaling of the kernel. In contrast to these works, which need to allow for to establish pointwise convergence, our result is obtained by working with the simplest and most local definition: .

2 Discrete curvatures from nets of curvature lines

In order to motivate our definition of discrete curvatures for nets of curvature lines, we recall some important notions of curvature for polygonal curves and polyhedral surfaces. For a similar discussion, we refer to Sullivan (2008).

2.1 Discrete curvatures of “Steiner-type”

Integrated curvatures for polygonal curves

Generalizations of classical smooth notions of curvature date at least back to Steiner (1840), who considered parallel offsets of convex hypersurfaces, relating integrated or total curvatures to changes in length, area, and enclosed volume. For example, for a convex curve , one of Steiner’s formulas reads

(1)

where is the length functional, denotes the curve’s curvature, and is the offset curve obtained by displacing along its normals by some constant amount .

Steiner’s offset formula can be extended to the non-smooth and non-convex case (Federer, 1959; Wintgen, 1982; Zähle, 1986). In particular, various notions for curvatures of polygonal curves may be interpreted using Steiner’s framework. Consider, e.g.,

(2)

where denotes an inner vertex of a polygonal curve, and is the turning angle between the two line segments incident to . These notions arise by applying (1) to the three different types of offsets depicted in Figure 3. Among these, the first notion is the one considered by Steiner, the second corresponds to a finite element discretization using piecewise linear functions, and the third also arises in the theory of discrete integrable systems (Bobenko and Suris, 1999; Hoffmann, 2008).

Figure 3: Applying Steiner’s formula (1) to the three depicted definitions of offset curves for a given polygonal curve lead to the three discrete curvatures in (2).

Integrated curvatures for polyhedral surfaces

By a polyhedral surface, we mean a piecewise linear immersion of a compact simplicial surface into . Extending the notions of discrete curvatures from polygonal curves to oriented polyhedral surfaces leads to the following edge-based definitions of integrated normal curvature:

(3)

Here is the signed angle between the normals of the two flat faces incident to the edge . Notice that measures curvature orthogonal to , since there is no curvature along itself. Integrated mean curvature is accordingly defined as .

In the planar limit (), the definitions in (3) agree up to second order in the angle variable. Therefore, as it turns out, it suffices for our purposes to prove convergence of one of these definitions in order to obtain convergence for all of them. Convergence of the first definition in (3) in the sense of measures was investigated in Fu (1993); Cohen-Steiner and Morvan (2006).

For completeness, we remark that the above edge-based definitions give rise to vertex-based notions of integrated mean curvatures by adding the mean curvatures over all edges emanating from a given vertex , i.e.,

(4)

The factor takes the meaning of distributing the normal curvature of each edge equally among its two adjacent vertices.

Finally, the scalar-valued definitions considered so far can be extended to corresponding vector-valued notions. In the edge-based case, we obtain normal curvature vectors by multiplying with the angle-bisecting unit normal vector at (see Figure 4, left), and similarly for mean curvatures. Analogously to (4), we then obtain vertex-based mean curvature vectors. We remark that for , the resulting mean curvature vector coincides with the surface area gradient at when restricting to piecewise linear surface variations (yielding the so-called cotangent formula, see  Pinkall and Polthier, 1993). Its convergence in the sense of Sobolev norms was studied in Hildebrandt et al. (2006).

From integrated to pointwise curvatures

Figure 4: Edge-based quantities. Left: dihedral angle and discrete curvature vector . Right: dual edge and circumcentric area .

In order to obtain pointwise curvatures, we divide the above integrated curvatures by corresponding area terms. Intuitively, these areas can be thought of as the domain of integration from which integrated curvatures were obtained. Whether or not one obtains convergent curvatures depends on a careful choice of these areas. It turns out that for triangulated polyhedral surfaces, one good choice are the so-called circumcentric areas, such as considered in Desbrun et al. (2005). For each edge we define

(5)

where denotes the intrinsic length of the circumcentric dual edge . This dual edge intrinsically connects the circumcenters and of the two triangles and incident to . Here, intrinsic means that one can think of and as being unfolded onto the plane (see Figure 4, right). The sign is positive if along the direction of the ray from through , triangle lies before , and negative otherwise. Note that (and therefore ) iff , where and are the angles opposite to in the triangulation (see Figure 4, right). Consequently, we require lower bounds that ensure positivity of circumcentric areas. For nets of curvature lines, we provide such bounds in Section 3.1.

Similar to vertex-based integrated curvatures, we obtain vertex-based circumcentric areas from the edge-based case via

(6)

where the sum is taken over all edges emanating from a given vertex . If all edges incident to a vertex are intrinsically Delaunay (compare Bobenko and Springborn (2007)), then coincides with the intrinsic Voronoi area of and is therefore positive. However, as pointed out in (Dyer and Schaefer, 2009), might become negative in general, and a bound similar to the edge-based case is not possible. Therefore, we will not treat vertex-based pointwise curvatures based on .

2.2 A local error estimate for discrete curvatures

In this section, we state our main local error estimate (Theorem 2), from which we derive our global uniform convergence result (Theorem 1).

Throughout we assume that is a smooth compact oriented surface without boundary immersed into . By a discrete net on we mean a cellular decomposition of such that all attaching maps are homeomorphisms and the intersection of any two cells is either empty or a single cell. As usual, we denote by the set of edges and by the set of vertices. We also assume that all edges are smoothly embedded. In a discrete net of curvature lines on , all edges are additionally required to be segments of curvature lines, non-umbilical vertices are required to have valence four, and umbilical vertices are required to have valence greater than two. In a completely umbilical region (such as ), any net in the above sense serves as a net of curvature lines for our purposes.

In order to be able to apply the concepts of discrete curvatures on polyhedral surfaces to nets of curvature lines, we require local polyhedral approximations of smooth curvature line nets.

Figure 5: The triangulated vertex star at a vertex is a polyhedral surface approximating the (curved) discrete net of curvature lines at .

Local polyhedral approximation

In the sequel, the letters e, f, and g will be reserved for (curved) edges in the edge set , incident to a common vertex , while corresponding bold face letters, , , and , will denote the straight edge vectors in obtained by connecting the endpoints of , , and , respectively, by straight lines (see Figure 5). We additionally assume that these edge vectors are oriented such that they point away from . For each disjoint edge pair incident to and contained in a common -cell, we consider the flat triangle spanned by and . The union of these triangles forms the triangulated vertex star of , denoted by . Whenever we consider the triple , we will always assume that the pairs and span two triangles in , such that becomes their common edge. Finally, as later justified by our sampling condition (10) and Corollary 9, we may assume that and , where denotes the normal of at .

Each triangulated vertex star thus yields the requisite local polyhedral approximation222Observe that we do not require that our local polyhedral approximations yield a consistent global one., which forms the basis for our curvature approximations. As outlined in the previous section, our definition of pointwise curvatures relies on the division by certain circumcentric areas, which may become zero or negative in general. This motivates, for a given principal direction, to choose the associated edge vector with maximal circumcentric area.

Definition 1 (area maximizing edge).

Consider a vertex in a discrete net of curvature lines. If is umbilical, we call an edge vector area maximizing if it maximizes the circumcentric area among all edges emanating from in the local polyhedral approximation. If is non-umbilical, let be the principal direction canonically associated with an edge vector . We call area maximizing if it maximizes the circumcentric area among the two edge vectors associated with .

We show in Section 3.1 that area maximizing edges always have circumcentric areas that are bounded away from zero.

Definition 2 (principal curvature approximations).

Consider a vertex in a discrete net of curvature lines and let be an area maximizing edge (associated with a principal direction if is non-umbilical). Then

defines the principal curvature approximation of , where refers to the principal curvature corresponding to if is non-umbilical and refers to the unique normal curvature if is umbilical. Here is one of the edge-based integrated polyhedral curvatures defined in (3).

The fact that in the above definition the edge vector is associated with while approximates is not an oversight: measures curvature orthogonal to .

Figure 6: A discrete net of curvature lines, with the two families of directions depicted as solid and dotted lines, respectively, together with the circumcentric areas (gray and white diamonds) per edge.

The intuitive reason for dividing by twice the circumcentric area in the above definition may (at least qualitatively) be explained as follows. Consider Figure 6, where one set of principal curvature directions, say those associated with , is depicted by solid lines, while the direction corresponding to is represented by dotted ones. Likewise, the circumcentric areas corresponding to -directions are drawn as gray diamonds, while the circumcentric areas corresponding to -directions are represented as withe diamonds. Roughly, the gray diamonds cover only half of the total surface. Hence, taking only the gray diamonds as regions of support for our (integrated) principal curvature approximations corresponding to , would mean to be roughly missing a factor of two. This motivates, for each edge along a given principal direction, to consider twice its circumcentric area as the domain of integration.

Global constants

For each , let denote the shape (or Weingarten) operator. Our estimates depend on both and its covariant derivative, . Accordingly, we define

(7)

where denotes the usual norm for linear operators. Note that is an upper bound for the normal curvatures of , whereas provides an upper bound for directional derivatives of principal curvatures .

Local constants

We also consider local constants – shape regularity and maximum edge length – that are specific for each vertex in the net of curvature lines. The reason for introducing local constants is that a high aspect ratio at one vertex should not affect the sampling condition (see below) at another vertex. In the following, we assume an arbitrary but fixed vertex .

We let denote the largest intrinsic edge length over all (curved) edges emanating from (denoted by ),

(8)

Notice that the length of every edge vector emanating from in is thus also bounded above by .

Our estimates also depend on shape regularity, or aspect ratio. We define to be the smallest number such that for all pairs of edge vectors emanating from and forming a triangle in , one has

(9)

The former inequality implies , while the latter means .

Sampling condition

In addition to the above definitions of maximum (local) edge length and (local) shape regularity, we assume the (local) sampling condition

(10)

In some of our estimates, it will suffice to work with weaker sampling conditions, such as or , both of which are implied by (10).

Theorem 2 (Local error estimate).

Let be a smooth compact oriented surface without boundary immersed into . Consider a vertex in a discrete net of curvature lines on and assume the sampling condition (10). Let denote the approximations of the smooth principal curvature as in Definition 2. Then

(11)

The constant depends only on the curvature bounds (7) and the shape regularity (9).

Remark.

Equivalent estimates can be obtained when replacing the scalar-valued quantities in (11) by their corresponding vector-valued counterparts (for definitions, see Section 2). Proofs remain nearly identical.

Our global uniform convergence theorem stated in the introduction is a direct consequence of our local error estimate.

Proof of Theorem 1.

Let be defined by mapping each to its nearest point in the vertex set , i.e., , where denotes the intrinsic distance on . We extend our curvature approximations (initially only defined on V) to functions via . By the assumptions of Theorem 1, was globally chosen such that for all . This implies, by connecting and by a shortest geodesic arc, that

Furthermore, was globally chosen such that it maximizes global edge length. Hence Theorem 2 implies that there exists a constant such that

for all . Additionally, note that the constant in Theorem 2, besides depending on and , is monotonically increasing with respect to shape regularity. (This will become evident in the proof of Theorem 2.) Hence, depends only on , , and the largest local shape regularity constant . This together with an application of the triangle inequality,

implies the claim. ∎

3 Proof of local error estimate

The proof of Theorem 2 proceeds in several steps. First, we provide uniform lower and upper bounds for the edge-based circumcentric areas by which we divide integrated curvatures to obtain pointwise notions (Section 3.1). In a second step, we provide estimates for edge-based integrated curvatures by using their corresponding discrete curvature vector. We establish that for each vertex in a net of curvature lines, the projection of these vectors onto the tangent plane is negligible. Furthermore, we show that the remaining normal component leads to the error estimate in Theorem 2 up to a certain error term (Section 3.2). While for general meshes the resulting error term cannot be controlled (and indeed causes failure of pointwise convergence), we provide bounds for this error term for the specific case of nets of curvature lines (Section 3.3).

Basic assumptions

In order to avoid excessive repetition, we summarize our basic assumptions and notations. We write for a non-boundary vertex of a polyhedral surface, with vertex star denoted by . We assume that is an upper bound for the length of the (straight) edges emanating from , and we let denote the shape regularity as defined in (9). If arises from the local polyhedral approximation of a net of curvature lines, then is defined by (8), i.e., as the maximum edge length of the curved edges emanating from . Throughout, we assume the sampling condition (10). As before, we denote curved edges by , , and , and their corresponding straight edge vectors by , , and .

3.1 Uniform bounds for circumcentric areas

In this section we prove upper and lower bounds for the circumcentric areas of area maximizing edges in the sense of Definition 1.

Proposition 3.

Consider a vertex in a discrete net of curvature lines and let be an area maximizing edge vector at . Then our basic assumptions imply the existence of some such that

where only depends on the shape regularity constant .

Note that for non-umbilical vertices, this result implies the existence of an edge with positive circumcentric area for each of the two principal curvature directions.

The remainder of this section is concerned with proving Proposition 3. First observe that

where and are the angles opposing in the two triangles meeting at , respectively. The requisite upper bound on is relatively straightforward to obtain.

Lemma 4 (upper bound).

Let be a vertex in a discrete net of curvature lines on . Then our basic assumptions imply that each edge emanating from satisfies

Proof.

Clearly, we have

Let be the edge vector emanating from such that belongs to the triangle formed by and . Then the definition of shape regularity (9) implies

(12)

A similar estimate holds for . Hence, . ∎

Similarly, we obtain a lower bound for at least one edge emanating from .

Lemma 5 (lower bound).

Let be a vertex in a discrete net of curvature lines on . Then our basic assumptions imply that there exists an edge vector emanating from such that

Proof.

Let be the shortest edge emanating from , let be the straight edge emanating from such that belongs to the triangle formed by and , and define . Since , it follows that . Since in particular , it follows that . By (9) we have , and hence

Applying similar arguments, we obtain . Together, this yields

Lower bounds for vertices of valence four

The above lower bound on the circumcentric area of at least one edge suffices for umbilical vertices. However, it does not suffice at non-umbilical ones, since we require lower bounds for edges associated with each of the two principal directions. We achieve this by showing that for each vertex of valence four, there are at least three edges that satisfy the required lower bound. Hence, for each vertex in a net of curvature lines, we have at least one good edge per principal direction.

We note that some of the following results are also valid for vertices of valence different from four, such as, in particular, Corollary 9.

As before, we let and denote the angles opposing the straight edge in the two triangles meeting at , respectively. If , , and for some , then

(13)

which provides a useful lower bound if can be bounded away from zero. Accordingly, we introduce the notion of -Delaunay edges, a nomenclature that is borrowed from the classical case of Delaunay triangulations (corresponding to ).

Definition 3 (-Delaunay).

Let and be the angles opposing an edge in the two triangles meeting at , respectively. Then is called -Delaunay if there exists such that , , and .

Assume for a moment that has valence four and that is planar. Assume further that all of the eight angles opposing the four edges emanating from are bounded from below by . Then it is straightforward to verify that at least three among the four edges incident to are -Delaunay. In general, however, is not planar, and we have to account for Gaussian curvature. As usual, we define discrete Gauss curvature at a vertex of a polyhedral surface as the angle defect, i.e., by , where are the intrinsic angles meeting at . We obtain:

Lemma 6.

Let be a non-boundary vertex of valence four on a triangulated polyhedral surface, and let denote the pairs of angles opposing the four edges emanating from . Assume that and for all , as well as . Then at least three among the four edges meeting at are -Delaunay.

Proof.

Using , the result follows from a straightforward calculation. ∎

We now show that our basic assumptions imply the assumptions of Lemma 6 when setting . To see this, first observe that with the above notations, by (12), and analogously for . It remains to check the condition for the discrete Gauss curvature. The requisite bound will be established in Lemmas 7 and 8. The resulting consequence for the existence of -Delaunay edges is summarized in Lemma 10. Finally, Lemma 11 establishes the lower bound for .

Lemma 7.

Let be a non-boundary vertex of an oriented polyhedral surface. Assume that the normals of the triangles incident to make an angle no greater than with some fixed direction in . Then the discrete Gauss curvature associated with satisfies .

Proof.

The lemma is a consequence of the Gauss-Bonnet theorem. Let denote the unit normals of the triangles incident to , ordered according to the orientation of the polyhedral surface. Each represents a point on . Connecting consecutive pairs by geodesic arcs on yields a spherical polygon (possibly with intersecting edges). For each , the exterior angle of at is then equal to the interior angle of the Euclidean triangle at . In particular, the sum of the exterior angles of satisfies .

Moreover, by assumption we have , so all lie on the same hemisphere. We can hence consider the spherical convex hull of , i.e., the smallest spherical polygon that contains all and that is convex with respect to shortest geodesic arcs. Let denote the sum of exterior angles of . It is easy to verify that . Hence , where the last equality follows from the Gauss-Bonnet theorem. Since , the polygon is contained in a geodesic disk of radius , the area of which is . This proves the claim. ∎

In order to make use of the previous lemma, we seek a bound on the angle between the surface normal at and the normals to the triangles incident to . Note that our basic assumptions, and in particular our sampling condition, imply , where is our curvature bound. Hence we can infer from Morvan and Thibert (2004, Section 3, Corollary 1):

Lemma 8.

Let be a vertex (umbilical or not) in a net of curvature lines on . Given our basic assumptions, can be oriented such that the maximum angle between the surface normal at and the normals to the triangles in satisfies , where is the shape regularity. In particular, .

Corollary 9.

Under the assumptions of Lemma 8, can be oriented such that the orthogonal projection of onto is injective and orientation-preserving.

Note that our assumptions are slightly different from those used in Morvan and Thibert (2004); in fact, our assumptions are stricter. While their sampling condition bounds the maximal extrinsic distance between two neighboring vertices, we consider the intrinsic length on the smooth surface, which is always larger. Additionally, Morvan and Thibert (2004) require that the distance between the discrete and the smooth surface is less than the reach of the smooth one. This requirement is implicitly fulfilled locally by the sampling condition , since the reach of a surface patch formed by an intrinsic -disk around is nothing but the minimal radius of curvature of that surface patch .

Lemma 10.

In addition to the assumptions of Lemma 8, assume that is of valence four. Let , where is the shape regularity. Then at least three edges incident to  are -Delaunay.

Proof.

Using that for , Lemmas 7 and 8 show that the discrete Gauss curvature at satisfies

Setting , the sampling condition (10) implies . Moreover, (12) implies and for all pairs of angles opposing the four straight edges emanating from . Finally, Lemma 6 implies the claim. ∎

Lemma 11 (lower bound for valence four).

Under the assumptions of Lemma 10, there exist at least three straight edges among the four edges emanating from , such that

for each edge among these three.

Proof.

Observe that (13) implies for all -Delaunay edges. Applying Lemma 10 and using gives

Proof of Proposition 3.

Lemma 4 provides an upper bound for both umbilical and non-umbilical vertices. Lemma 5 provides the requisite lower bound for umbilical vertices. Finally, Lemma 11 provides the lower bound for non-umbilical ones, since it implies the existence of at least one edge per principal direction with circumcentric area bounded from below. ∎

3.2 Estimates for discrete integrated curvatures

In this section, we establish a bound for the difference between the edge-based integrated curvatures and (an appropriately scaled version of) the smooth principal curvatures of . Specifically, for a given edge in our polyhedral approximation, we work with the discrete integrated curvature introduced in Section 2.1.

Darboux frames

For our purposes, it turns out to be useful to express vectors in frames that are locally adapted to the geometry of the surface . Specifically, a Darboux frame at a non-umbilical point is an adapted frame given by , where and are (normalized) principal directions of at , and is the surface normal (induced by the orientation of ). For umbilical points, any adapted (i.e., ) and orthonormal (i.e., and ) frame may be considered a Darboux frame. Throughout, we employ the notation

to represent a vector in the coordinates given by a Darboux frame. In the sequel, we assume a fixed Darboux frame at every vertex of our discrete net of curvature lines. If is non-umbilical then each edge vector emanating from is canonically associated to exactly one of the principal directions or . If is umbilical, we additionally require an explicit association of each edge vector with one of either or . In order to state the main result of this section, we require the notion of tangential deviation of an edge vector with respect to a Darboux frame.

Figure 7: The tangential deviation  of a straight edge .
Definition 4 (tangential deviation).

Let be a vertex in a discrete net of curvature lines on and let be the Darboux frame at . If an edge vector is associated with the principal direction , we call its tangential deviation, see Figure 7. Likewise, if is associated with , we let .

The following proposition summarizes the main result of this section.

Proposition 12.

Using the notations of Definition 4, let the edge vector emanating from vertex be associated with , and let its directly neighboring edges and be associated with . Then our basic assumptions imply that

where denotes the principal curvature of in direction , (with in the umbilical case), and .

Remark.

We point out that the estimates in this section are not entirely specific for nets of curvature lines. In fact, they hold in the more general setting of arbitrary smooth nets embedded in , as long as we require bounded shape regularity and the sampling condition , where is the maximum intrinsic edge length of the net and denotes our usual curvature bound. As a consequence, the estimates of the current section do not suffice to establish our main error estimate. Indeed, to obtain uniform convergence, we require that the term appearing in Proposition 12 is of order . However, this is false for general nets (and causes failure of uniform convergence), but is true for nets of curvature lines as we will show in Section 3.3. For clarity’s sake, though, we have decided to restrict the discussion of the current section to nets of curvature lines and rely on our (rather strong) basic assumptions set forth in the beginning of Section 3.

Proposition 12 is proven in several steps. We commence by estimating the normal component of the straight edge vector (Lemma 13 and Corollary 14). Then we switch to the edge-based curvature vector corresponding to . We show that its tangential component is negligible (Lemma 15). The final proof of Proposition 12 is given at the end of this section, where we show that the normal component of yields the desired estimate.

Lemma 13.

Let be a vertex of a discrete net of curvature lines on . Consider an edge emanating from with corresponding straight edge vector . Writing with respect to a Darboux frame centered at , our basic assumptions imply that

Furthermore, if denotes the osculating paraboloid, then

where only depends on our global curvature bound and the bound on curvature derivatives .

Proof.

Using a Darboux frame at , the surface can locally be parameterized by a height function over the tangent plane . In the coordinates of the Darboux frame, we have , with . We let , where throughout this proof denotes the Euclidean norm in the parameter domain. Furthermore, we consider the constant vector field in the parameter domain.

Let denote the th iteration of the directional derivative of along , i.e., and , and let denote the th total derivative with respect to the standard Euclidean metric in the parameter domain. Observe that , and hence

(14)

Consequently, in order to prove the first part of the lemma, we seek an upper bound on in terms of .

Let denote the shape operator and let and be the first and second fundamental form of , respectively, with respect to the local parameterization induced by . From

(15)

and

(16)

we obtain the estimate

(17)

In order to bound , first observe that our sampling condition implies . This, in turn, provides a bound on the (positive) angle between the surface normal at and the surface normal at any point on the (curved) edge incident to in the given net of curvature lines on . To see this, consider an arc-length parametrized curve with and . Notice that our basic assumptions imply that can be chosen such that . The Gauss image of is a curve on the unit sphere. The length of is therefore bounded from below by , i.e., the length of the minimizing geodesic joining and on . The tangent vector of at a point , , is given by , and the norm of this vector is therefore bounded above by . Hence,

Writing with respect to the Darboux frame centered at , it follows that

(18)

Plugging this into (17), we find that , which, together with (14), yields the first part of the lemma.

In order to prove the second part, we first note that and that is constant. Hence

(19)

We consequently seek a bound for in terms of and . Considering (16) and taking another derivative with respect to yields

(20)

We bound the terms appearing on the right hand side one by one. To treat the first term, we use (16) and , where denotes covariant differentiation with respect to the metric induced by , to derive

(21)

Let be a constant vector field in the parameter domain. Using the Koszul formula for the Levi-Civita connection and applying (15) yields

The last equality only depends on the value of at the point and therefore holds for any field in the parameter domain. From (17), (18), and , we obtain

with . This can be used in (21), together with our bounds on the norms of and in terms of and , respectively, to obtain an upper bound for the first term in (20) by . Bounds for the remaining two terms in (20) can be obtained in a similar fashion using (15) and (16), proving the estimate . Using (19) and then implies the claim of the second part in the statement of the lemma. ∎

Corollary 14.

With the same assumptions as in Lemma 13 and by defining , we obtain

with .

Proof.

From the second part of Lemma 13 and simple algebraic manipulations, we deduce that

The last equality follows from Lemma 13 and the sampling condition (which is implied by our basic assumptions), which together ensure that . The second equation in the statement of the corollary follows analogously. ∎

Figure 8: The discrete curvature vector is the sum of (the rotation of by about the axis ) and (the rotation of by about the axis ).

For the following discussion, it will be useful to work with the curvature vector corresponding to the discrete integrated curvature (see Section 2.1). With our usual notions, consider the two edge vectors and directly neighboring in (see Figure 8). Recall from Corollary 9 that we may assume that and , where is the normal of at . A straightforward calculation reveals that we can express the discrete curvature vector by

(22)

where and denote rotations by around the axes and , respectively (see Figure 8). Consider now the splitting

of into its tangential and normal component with respect to . We show that the tangential component is negligible:

Lemma 15.

Under the same assumptions as in Lemma 13, the projection of the discrete curvature vector onto the tangent plane satisfies .

Proof.

With the notions and assumption of the preceding discussion, we first note that a straightforward calculation reveals that

(23)

and analogously for .

With respect to , let , , and denote the tangential components of the vectors , , and , respectively. Then, by assumption, we have and . The tangential part of the discrete curvature vector is given by , where we deduce from (23), using the coordinates of our fixed Darboux frame at , that

and similarly

Using the first part of Lemma 13 and the definition of shape regularity (9), we obtain