Inequalities for the Curvature of Curves and Surfaces ∗ David Cohen-Steiner Herbert Edelsbrunner Departent of Computer Science Department of Computer Science Duke University, Durham Duke University, Durham North Carolina, USA Raindrop Geomagic, RTP [email protected] North Carolina, USA [email protected] ABSTRACT While curvature is a priori defined only for smooth objects, total In this paper, we bound the difference between the total mean cur- mean curvature can be defined consistently for a large class of non- vatures of two closed surfaces in R3 in terms of their total absolute smooth, including piecewise-linear objects [15]. This is of practi- curvatures and the Frechet´ distance between the volumes they en- cal interest since smooth surfaces are often represented by approx- close. The proof relies on a combination of methods from algebraic imating meshes. From this perspective, an important question is topology and integral geometry. We also bound the difference be- whether the total mean curvature of a piecewise-linear surface is tween the lengths of two curves using the same methods. close to the one of approximated the smooth surface. More gen- erally, understanding the behavior of Lipschitz-Killing curvatures under approximations is an important unsolved theoretical ques- Categories and Subject Descriptors tion [12]. In this direction, a convergence result for the Lipschitz- F.2.2 [Analysis of Algorithms and Problem Complexity]: Non- Killing curvatures of increasingly fine triangulations inscribed in a numerical Algorithms and Problems—Geometrical problems and smooth submanifold was obtained in [11]. For surfaces, this result computations, Computations on discrete structures; G.2.1 [Discrete was strengthened in [5, 7] by extending it to curvature tensors and Mathematics]: Combinatorics—Counting problems by giving an explicit error bound. An essential requirement in the latter approximation result is that the normals to the mesh are close to the normals to the smooth surface. General Terms In this paper, we give an approximation result for the total mean Algorithms, Theory curvature of surfaces under weaker conditions, using a stability re- sult on topological persistence proved in [6]. We show that control- Keywords ling the Frechet´ distance between the smooth object and its approx- imation is sufficient to guarantee that their total mean curvatures are Integral geometry, curvature, Frechet´ distance, persistence diagrams, close, provided the approximation has bounded total absolute cur- bottleneck distance, approximation, stability. vature. Surprisingly, the closeness between the normals to the two surfaces is not explicitly required. We prove this using the integral- 1. INTRODUCTION geometric interpretations of the total absolute and the total mean Given an oriented hypersurface S smoothly embedded in Rd, curvatures [14]. Our method also yields a simple and apparently new geometric inequality for curves. More precisely, we obtain the mean curvature of S at p S is the average of the d 1 principal curvatures at p. The total∈ mean curvature is the integral− a bound on the difference between the lengths of two curves as a of the mean curvature over S. It is the second in a sequence of function of their Frechet´ distance and of their total curvature. This d quantities associated with a hypersurface, called Quermassinte- result can be viewed as a generalization of Fary’s´ theorem [10]. grale, Minkowski functionals,orLipschitz-Killing curvatures in the literature [3]. The total mean curvature and its relatives have many Outline. Section 2 recalls the necessary background on topological interesting properties and have been extensively studied by mathe- persistence and integral geometry. Section 3 states and proves our maticians during the last 150 years. results on surfaces and curves. Section 4 concludes the paper. ∗Both authors were partially supported by NSF under grant CCR- 2. BACKGROUND 00-86013 and by DARPA under grant HR0011-05-1-0007. Before we state and prove our results, we review the necessary background. We need to understand topological persistence, a topic in algebraic topology [13], and we need integral-geometric inter- Permission to make digital or hard copies of all or part of this work for pretations of curvature and length, concepts studied in differential personal or classroom use is granted without fee provided that copies are geometry [8]. not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to Topological persistence. Given a generic smooth function f on a republish, to post on servers or to redistribute to lists, requires prior specific manifold M, Morse theory shows that the changes in the topology permission and/or a fee. 1 of the sublevel sets f − ( ,x]are related to the critical points of SCG’05, June 6–8, 2005, Pisa, Italy. −∞ Copyright 2005 ACM 1-58113-991-8/05/0006 ...$5.00. f. More precisely, when the level x increases and passes a critical value, a handle is attached to the current sublevel set. This opera- k-TRIANGLE LEMMA [9]. The persistent Betti number βk(x, y) tion can have two kinds of consequences: either a new homology is the total number of points in Dk(f) contained in the quadrant class is created or some homology class is destroyed. Topological [ ,x] (y, + ]. persistence is a canonical way to pair up critical values that create −∞ × ∞ homology classes with critical values that destroy them [9]. Each Another important property of topological persistence is its stabil- such pair forms a half-open persistence interval [a1,a2) that can ity under perturbation. We measure the distance between points in ¯ 2 2 be interpreted as the life-span of a topological feature in the fil- the extended plane, R =[ , ] ,as a a0 = max a1 −∞ ∞ k − k∞ R¯{|2 − tration of M by sublevel sets. An example is shown in Figure 1. a10 , a2 a20 . Given two multisets of points A and A0 in ,we | | − |} Sweeping the curve from bottom to top, we encounter five critical define their bottleneck distance as dB(A, A0) = inf sup a γ(a) , γ a k − k∞ ∞ where a ranges over all points of A and γ ranges over all bijections from A to A0. STABILITY THEOREM [6]. If f and g are sufficiently regular real-valued functions defined over the same space, then dB(Dk(f), Dk(g)) sup f(p) g(p) , ≤ p | − | for all k 0. Figure 1: A one-dimensional real-valued function, its per- ≥ sistence intervals, and its persistence diagram. The dark, In other words, persistence diagrams are stable under possibly ir- medium, light regions corresponds to persistent Betti numbers regular perturbations of small amplitude, as illustrated in Figure 2. 3, 2, 1. values in sequence. The first three create a new component each. The fourth critical value merges the component created last with the component created just before. The life-span of the third com- ponent is thus the interval between the third critical value and the fourth, as indicated in the middle of the figure. Similarly, the fifth critical value merges the component created by the second critical value with the one created by the first, and is therefore paired with the second value. The component created first is never destroyed and its life-span is the half-infinite interval starting at the first crit- ical value. Because the function in Figure 1 is one-dimensional, the only interesting features in the sublevel sets are their connected Figure 2: Left: two close functions, one with many and the components, which correspond to classes in the 0-th homology other with just four critical values. Right: the persistence dia- group, but persistence intervals can be defined similarly for k-th grams of the two functions, and the bijection between them. homology groups, for arbitrary k. Also, they can be defined for a large class of functions, including height functions on simplicial complexes in addition to Morse functions on smooth manifolds [6]. Integral geometry for surfaces. Given a smoothly embedded sur- In this more general setting, critical values are replaced by homo- face S in R3, we recall that the two principal curvatures at a point logical critical values that correspond to levels at which the homol- p S are the maximum and minimum normal curvatures, denoted ∈ ogy groups of the sublevel sets change. as κ1(p) κ2(p). The total mean curvature is the integral of the mean curvature≥ and the total absolute curvature is the integral of Persistence diagrams. As shown in [4], the (multi-) set of all the absolute Gaussian curvature: persistence intervals of a function provides a rather complete de- Z 1 scription of the topological relationships between different sublevel H(S)= (κ1(p)+κ2(p)) dp, 1 2S sets. For example, the image of the map from Hk(f − ( ,x]) to Z 1 −∞ Hk(f − ( ,y]) induced by inclusion on k-th homology groups −∞ G(S)= κ1(p)κ2(p)dp. has dimension equal to the number of persistence intervals that S| | contain both x and y. These numbers are called persistent Betti The latter is also the total area swept by the normal vector and numbers and we denote them by βk(x, y). To visualize them, we measures the intuitive notion of bumpiness. For example, surfaces represent each persistence interval [a1,a2) by the point (a1,a2) with zero total absolute curvature are developable, that is, they are in the plane, allowing infinite coordinates for unbounded intervals. everywhere locally isometric to the plane. To prove our results, The (multi-) set of all such points, together with the diagonal, ∆= we will use the integral-geometric interpretation of total absolute (x, x) x R , is the persistence diagram of the function, which { | ∈ } and total mean curvature, which we now recall.