A Glimpse Into Discrete Differential Geometry

A Glimpse into Discrete Differential Geometry

Keenan Crane, Max Wardetzky *

Communicated by Joel Hass

Note from Editor: The organizers of the 2018 Joint Mathematics Meetings Short Course on Discrete Dif- ferential Geometry have kindly agreed to provide this introduction to the subject. See p. XXX for more information on the JMM 2018 Short Course.

The emerging field of discrete differential geometry (DDG) studies discrete analogues of smooth geometric objects, providing an essential link between analytical descriptions and computation. In recent years it has un- Figure 1: Discrete differential geometry re-imagines earthed a rich variety of new perspectives on applied classical ideas from differential geometry without ref- problems in computational anatomy/biology, computa- erence to differential calculus. For instance, sur- tional mechanics, industrial design, computational archi- faces parameterized by principal curvature lines are tecture, and digital geometry processing at large. The ba- replaced by meshes made of circular quadrilaterals sic philosophy of discrete differential geometry is that (top left), the maximum principle obeyed by harmonic a discrete object like a polyhedron is not merely an ap- functions is expressed via conditions on the geome- proximation of a smooth one, but rather a differential ge- try of a triangulation (top right), and complex-analytic ometric object in its own right. In contrast to traditional functions can be replaced by so-called circle packings numerical analysis which focuses on eliminating approx- that preserve tangency relationships (bottom). These imation error in the limit of refinement (e.g., by taking discrete surrogates provide a bridge between geome- smaller and smaller finite differences), DDG places an try and computation, while at the same time preserv- emphasis on the so-called “mimetic” viewpoint, where ing important structural properties and theorems. key properties of a system are preserved exactly, independent of how large or small the elements of a mesh might be. Just as algorithms for simulating mechani- Most often, none of the resulting discrete objects pre- cal systems might seek to exactly preserve physical in- serve all the properties of the original smooth one—a so- variants such as total energy or momentum, structure- called no free lunch scenario. Nonetheless, the proper- preserving models of discrete geometry seek to exactly ties that are preserved often prove invaluable for partic- preserve global geometric invariants such as total curva- ular applications and algorithms. Moreover, this activity ture. More broadly, DDG focuses on the discretization of yields some beautiful and unexpected consequences— objects that do not naturally fall under the umbrella of such as a connection between conformal geometry traditional numerical analysis. This article provides an and pure combinatorics, or a description of constant- overview of some of the themes in DDG. curvature surfaces that requires no definition of curvature! To highlight some of the challenges and themes commonly encountered in DDG, we first consider the The Game. Our article is organized around a “game” of- simple example of the curvature of a plane curve. ten played in discrete differential geometry in order to come up with a discrete analogue of a given smooth object or theory: Discrete Curvature of Planar Curves. How do you define the curvature for a discrete curve? For a smooth arc- 1. Write down several equivalent definitions in the length parameterized curve 훾(푠) ∶ [0, 퐿] → ℝ2, curva- smooth setting. ture 휅 is classically expressed in terms of second deriva- 푑 tives. In particular, if 훾 has unit tangent 푇 ∶= 푑푠 훾 and 2. Apply each smooth definition to an object in the dis- unit normal 푁 (obtained by rotating 푇 a quarter turn in crete setting. the counter-clockwise direction), then

푑2 푑 3. Analyze trade-oﬀs among the resulting discrete def- 휅 ∶= ⟨푁, 푑푠2 훾⟩ = ⟨푁, 푑푠 푇⟩ . (1) initions, which are invariably inequivalent. Suppose instead we have a polygonal curve with vertices *Keenan Crane is assistant professor of computer science at Carnegie 2 훾1, … , 훾푛 ∈ ℝ , as often used for numerical computa- Mellon University. His e-mail address is [email protected]. Max Wardetzky is professor of mathematics at University of Göttin- tion (See Figure 2, right). Here we hit upon the most el- gen. His e-mail address is [email protected]. ementary problem of discrete diﬀerential geometry: dis-

1 crete geometric objects are often not sufficiently differentiable (in the classical sense) for standard definitions to apply. For instance, our curvature definition (Equa- tion 1) causes trouble, since at vertices our discrete curve is not twice differentiable, nor does it have well defined normals. The basic approach of DDG is to find alternative characterizations in the smooth setting that can be applied to discrete geometry in a natural way. With curvature, for instance, we can apply the fundamental theorem of calculus to Equation 1 to acquire a different state- ment: if 휑 is the angle from the horizontal to 푇, then Figure 2: A given geometric quantity from the smooth 푏 setting, like curvature 휅, may have several reasonable ∫ 휅 푑푠 = 휑(푏) − 휑(푎) mod 2휋. definitions in the discrete setting. Discrete differential 푎 geometry seeks definitions that exactly replicate prop- Put more simply: curvature is the rate at which the tan- erties of their smooth counterparts. gent turns. This characterization can be applied naturally to our polygonal curve: along any edge the change in angle is clearly zero. At a vertex it is simply the turn- Mirroring the observation in the smooth setting, we can ing angle 휃푖 ∶= 휑푖,푖+1 − 휑푖−1,푖 between the directions now say that whatever change we observe in the length 휑푖−1,푖, 휑푖,푖+1 of the two incident edges, yielding our first provides a definition for discrete curvature. The first two notion of discrete curvature: are the same as ones we have seen already: the circu- 퐴 lar arc yielding the expression from Equation 2, and the 휅 ∶= 휃푖 ∈ (−휋, 휋). (2) 푖 straight line corresponding to Equation 4. The third one Are there other characterizations that also lead natu- provides yet another notion of discrete curvature rally to a discrete formulation? Yes: for instance we 퐶 can consider the motion of 훾 that most quickly reduces 휅푖 ∶= 2 tan(휃푖/2). its length. In the smooth case it is well known that the change in length with respect to a smooth variation Finally, in the smooth case it is also well known that cur- 휂(푠) ∶ [0, 퐿] → ℝ2 that vanishes at endpoints is given by vature has magnitude equal to the inverse of the radius integration against curvature: of the so-called osculating circle, which agrees with the curve up to second order. A natural way to define an 푑 퐿 | length(훾 + 휀휂) = − ∫ ⟨휂(푠), 휅(푠)푁(푠)⟩ 푑푠. osculating circle for a polygon is to take the circle pass- 푑휀 휀=0 0 ing through a vertex and its two neighbors. From the Hence, the velocity that most quickly reduces length is formula for the radius 푅푖 of a circumcircle in terms of 휅푁. For a polygonal curve, we can simply differentiate the side lengths of the corresponding triangle, one eas- 푛−1 ily gets a discrete curvature that is different from the one the sum of the edge lengths 퐿 ∶= ∑푖=1 |훾푖+1 − 훾푖| with respect to any vertex position. At a vertex 푖 we obtain we saw before:

훾푖 − 훾푖−1 훾푖+1 − 훾푖 퐷 휅푖 ∶= 1/푅푖 = 2 sin(휃푖)/푤푖, (6) 휕훾푖 퐿 = − =∶ 푇푖−1,푖 − 푇푖,푖+1, (3) |훾푖 − 훾푖−1| |훾푖+1 − 훾푖| where 푤 ∶= |훾 − 훾 |. Apart from merely being i.e., just a difference of unit tangent vectors 푇 along 푖 푖+1 푖−1 푖,푖+1 different expressions, we can notice that 휅퐴, 휅퐵 and 휅퐶 consecutive edges. If 푁 ∈ ℝ2 is the unit angle bisector 푖 are all invariant under a uniform scaling of the curve, at vertex 푖, this difference can also be expressed as whereas 휅퐷 scales like the smooth curvature 휅. This sit- 퐵 휅푖 푁푖 ∶= 2 sin(휃푖/2)푁푖, (4) uation demonstrates another common phenomenon in discrete differential geometry, namely that depending providing a discretization of the curvature normal 휅푁. on which smooth characterization is used as a starting A closely-related idea is to consider how the length of point, one may end up with pointwise or integrated quan- a curve changes if we displace it by a small constant tities in the discrete case. amount in the normal direction. As observed by Steiner, the new length can be expressed as 퐿 length(훾 + 휀푁) = length(훾) − 휀 ∫ 휅(푠) 푑푠. (5) 0

Since this formula holds for any small piece of the curve, it can be used to obtain a notion of curvature at each point. How do we define normal offsets in the polygonal case? At vertices we again encounter the issue that we have no notion of normals. One idea is to break the curve into individual edges which can then be translated by 휀 along their respective normal directions. We can then close the gaps between edges in a variety of ways: using (A) a circular arc of radius 휀, (B) a straight line, or by (C) extending the edges until they intersect (see Figure 3). If we then calculate the lengths for these new curves, we get Figure 3: Different characterizations of curvature in 푛−1 the smooth setting naturally lead to different notions length퐴 = length(훾) − 휀 ∑푖=2 휃푖, 푛−1 of discrete curvature. (Here we abbreviate 푇푖,푖+1 and length퐵 = length(훾) − 휀 ∑푖=2 2 sin(휃푖/2), 푛−1 푇푖−1,푖 by 푢 and 푣, respectively.) length퐶 = length(훾) − 휀 ∑푖=2 2 tan(휃푖/2).

2 curvature normal 휅푁. A smooth, simple curve evolving under this flow exhibits several basic properties: it has at all times total curvature 2휋, its center of mass remains fixed, it tends toward a circle of vanishing radius, and remains embedded for all time, i.e., no self-crossings arise (Gage-Grayson-Hamilton). Do our discrete curvatures furnish these same properties? A numerical exper- iment is shown in Figure 4. Here we evolve our polygon by a simple time-discrete flow 훾푖 ← 훾푖 + 휏휅푖푁푖 with a 퐷 fixed time step 휏 > 0. For 휅 , 푁푖 is the unit vector along the circumradius; otherwise it is the unit angle bisector. Not surprisingly, 휅퐴 preserves total curvature (due to the fundamental theorem of calculus); 휅퐵 does not drift (consider summing Equation 3 over all vertices); and 휅퐷 Figure 4: Typically, not all properties of a smooth ob- has circular polygons as limit points (since all velocities ject can be preserved exactly at the discrete level. For point toward the center of a common circle). However, curve-shortening flow, for example, 휅퐴 exactly pre- no discrete curvature satisfies all three properties simul- serves the total curvature, 휅퐵 exactly preserves the taneously. Moreover, for a constant time step 휏 no such center of mass, and with 휅퐶 the flow remains station- flow can guarantee that new crossings do not occur. This ary (up to rescaling) for any circular solution. How- situation illustrates the no free lunch idea: no matter ever, no local definition of discrete curvature can pro- how hard we try, we cannot find a single discrete object vide all three properties simultaneously. that preserves all the properties of its smooth counter- part. Instead, we have to pick and choose the properties best suited to the task at hand. As one might imagine, there are many other possible starting points for obtaining a discrete analogue of cur- Suppose that instead of curvature flow, we consider vature. Eventually, however, all starting points end up two other beautiful topics in the geometry of plane leading back to the same definitions, suggesting that curves: the Whitney–Graustein theorem, which classifies there may be only so many possibilities. For example, regular homotopy classes of curves by their total curva- if 휙 ∶ ℝ2 → ℝ is the signed distance from a smooth ture, and Kirchhoff’s famous analogy between motions closed curve 훾, then applying the Laplacian Δ yields the of a spherical pendulum and elastic curves, i.e., curves 퐿 2 curvature of its level curves; in particular, Δ휙|휙=0 yields that extremize the bending energy ∫0 휅 푑푠 subject to the curvature of 훾. Likewise, if we apply the Laplacian boundary conditions. Among the curvatures discussed to the signed distance function for a discrete curve, we above, only 휅퐴 provides a discrete version of Whitney– recover 휅퐴 on one side and 휅퐵 on the other. Yet another Graustein, but does not provide an exact discrete ana- approach is the theory of normal cycles (as discussed by logue of Kirchhoff. Likewise, 휅퐶 preserves the struc- Morvan), related to the Steiner formula from Equation (5). ture of the Kirchhoff analogy, but not Whitney–Graustein. Here, rather than settle on a single normal 푁푖 at each ver- This kind of no free lunch situation is a characteristic fea- tex we consider all unit vectors between the unit normals ture of DDG. A similar obstacle is encountered in the the- of the two incident edges, ultimately leading back to the ory of ordinary differential equations, where it is known first discrete curvature 휅퐴. This theory applies equally that there are no numerical integrators for Hamiltonian well to both smooth and polygonal curves, again rein- systems that simultaneously conserve energy, momen- forcing the perspective that the fundamental behavior of tum, and the symplectic form. From a computational geometry is neither inherently smooth nor discrete, but point of view, making judicious choices about which can be well captured in both settings by picking the ap- quantities to preserve for which applications goes hand- propriate ansatz. More broadly, the fact that equivalent in-hand with providing formal guarantees on the reliabil- characterizations in the smooth setting lead to different ity and robustness of algorithms. inequivalent definitions in the discrete setting is not special to the case of curves, but is one of the central themes We now give a few glimpses into recent topics and in discrete differential geometry. trends in DDG.

From here, a natural question arises: which discrete curvature is “best”? A traditional criterion for discrimi- Discrete Conformal Geometry A conformal map is, nating among different discrete versions is the question roughly speaking, a map that preserves angles (see Fig- of convergence: if we consider finer and finer approx- ure 1, bottom left). A good example is Mercator’s pro- imating polygons, will our discrete curvatures converge jection of the globe: even though area gets stretched to the classical smooth one? However, convergence does out badly—making Greenland look much bigger than not always single out a best version: treated appropri- Australia!—the directions “north” and “east” remain at ately, all four of our discrete curvatures will converge. right angles, which is very helpful if you’re trying to nav- We must therefore look beyond convergence, toward ex- igate the sea. A beautiful fact about conformal maps act preservation of properties and relationships from the is that any surface can be conformally mapped to a smooth setting. Which properties should we try to pre- space of constant curvature (“uniformization”), provid- serve? The answer of course depends on what we aim to ing it with a canonical geometry. This fact, plus the use these curvatures for. fact that conformal maps can be efficiently computed (e.g., by solving sparse linear systems), have led in recent As a toy example, consider the curve-shortening flow years to widespread development of conformal mapping (depicted in Figure 4, top left), where a curve evolves algorithms as a basic building block for digital geometry according to the velocity that most quickly reduces its processing algorithms. In applications, discrete confor- length. As discussed above, this velocity is equal to the mal maps are used for everything from sensor network

3 ing a region 푈 ⊂ ℂ, Thurston noticed that this map ap- proximates a smooth conformal map 푓 ∶ 푈 → 퐷2 as the region is filled by smaller and smaller circles (see Fig- ure 1, bottom), as later proved by Rodin and Sullivan. Unlike a traditional finite element discretization, these so-called circle packings also preserve many of the basic structural properties of conformal maps. For instance, composition with a Möbius transformation of the disk yields another uniformization map, as in the smooth setting. More broadly, circle packings provide an unexpected bridge between geometry and combinatorics, Figure 5: What is the simplicial analogue of a confor- since the geometry of a map is determined entirely by mal map? Requiring all angles to be preserved is too incidence relationships 1. On the flip side, this means a rigid, forcing a global similarity (left). Asking only for different theory is needed to account for the geometry preservation of so-called length cross ratios provides of irregular triangulations, as more commonly used in just the right amount of flexibility, maintaining much applications. of the structure found in the smooth setting such as invariance under Möbius transformations (right) An alternative theory starts from the idea that under a conformal map the Riemannian metric 푔 experiences a 2푢 layout to comparative analysis of medical or anatomical uniform scaling at each point: 푔̃ = 푒 푔. In other words, data. Of course, to process real data one must be able to vectors tangent to a given point 푝 ∈ 푀 shrink or grow by 푢 compute conformal maps on discrete geometry. a positive factor 푒 . In the simplicial setting 푔 is replaced by a piecewise Euclidean metric, i.e., a collection of pos- What does it mean for a discrete map to be conformal? itive edge lengths ℓ ∶ 퐸 → ℝ>0 that satisfy the triangle ̃ As with curvature, one can play the game of enumerat- inequality in each face. Two such metrics ℓ, ℓ are then ing several equivalent characterizations in the smooth said to be discretely conformally equivalent if they are re- 2 ̃ (푢푖+푢푗)/2 setting. Consider for instance a map 푓 ∶ 푀 → 퐷 ⊂ ℂ lated by ℓ푖푗 = 푒 ℓ푖푗 for any collection of discrete from a disk-like surface 푀 with Riemannian metric 푔 to scale factors 푢 ∶ 푉 → ℝ. Though at first glance this re- the unit disk 퐷2 in the complex plane. This map is con- lationship looks like a simple numerical approximation, formal if it preserves angles, if it preserves infinitesimal it turns out to provide a complete discrete theory that circles, if it can be expressed as a pair of real conjugate preserves much of the structure found in the smooth harmonic functions 푓 = 푎 + 푏푖, if it is a critical point of setting, with close ties to theories based on circles. An 2 the Dirichlet energy ∫푀 |푑푓| 푑퐴, or if it induces a new equivalent characterization is the preservation of length metric 푔̃ ∶= 푑푓 ⊗ 푑푓 that at each point is a positive cross ratios 픠푖푗푘푙 ∶= ℓ푖푗ℓ푘푙/ℓ푗푘ℓ푙푖 associated with each rescaling of the original one: 푔̃ = 푒2푢푔. Each starting edge 푖푗 ∈ 퐸; for a mesh embedded in ℝ푛 these ratios point leads down a path toward different consequences are invariant under Möbius transformations, again mim- in the discrete setting, and to algorithms with different icking the smooth theory. This theory also leads to effi- computational tradeoffs. cient, convex algorithms for discrete Ricci flow, which is a starting point for many applications in digital geome- Oddly enough, the most elementary characterization try processing. of conformal maps, angle preservation, does not translate very well to the discrete setting (see Figure 5). Con- More broadly, discrete conformal geometry and dis- sider for instance a simplicial map that takes a triangu- crete complex analysis is an active area of research, with lated disk 퐾 = (푉, 퐸, 퐹) to a triangulation in the plane. elegant theories not only for triangulations but also for Any map that preserves interior angles will be a similar- lattice-based discretizations, which make contact with ity on each triangle, i.e., it can only rigidly rotate and the topic of (discrete) integrable systems, discussed be- scale. But since adjacent triangles share edges, the scale low. Yet basic questions about properties like conver- factor for all triangles must be identical. Hence, the only gence, or descriptions that are compatible with extrinsic discrete surfaces that can be conformally flattened in geometry, are still only starting to be understood. this sense are those that are (up to global scale) devel- opable, i.e., that can be rigidly unfolded into the plane. This outcome is in stark contrast to the smooth setting, where any disk can be conformally flattened. This sit- Discrete Differential Operators Differential geometry uation reflects a common scenario in DDG: rigidity, or and in particular Riemannian manifolds can be studied what in finite element analysis is sometimes called lock- from many different perspectives. In contrast to the ing. There are simply too few degrees of freedom relative purely geometric perspective (based on, say, notions of to the number of constraints: we want to match angles distance or curvature), differential operators provide a at all 3퐹 corners, but have only 2푉 < 3퐹 degrees of free- very different point of view. One of the most funda- dom. Hence, if we insist on angle preservation we have mental operators in both physics and geometry is the no chance of capturing the flexibility of smooth confor- Laplace–Beltrami operator Δ (or Laplacian for short) act- mal maps. ing on differential 푘-forms. It describes, for example, heat diffusion, wave propagation, steady state fluid flow, Other characterizations provide greater flexibility. and is key to the Schrödinger equation in quantum me- One idea is to associate each vertex of our discrete disk chanics. It also provides a link between analytical and 퐾 with a circle in the plane. A theorem of Koebe implies topological information: for instance, on closed Rieman- that one can always arrange these circles such that two nian manifolds the dimension of harmonic 푘-forms (i.e., circles are tangent if they belong to a shared edge and all those in the kernel of Δ) equals the dimension of the boundary circles are tangent to a common circle bound- 1See “Circle Packing” in the December 2003 Notices ing the rest. For a regular triangular lattice approximat- http://www.ams.org/notices/200311/fea-stephenson.pdf

4 푘th cohomology—a purely topological property. The spectrum of the Laplacian (i.e., the list of eigenvalues) likewise reveals a great deal about the geometry of the manifold. For example, the first nonzero eigenvalue of the 0-form Laplacian provides an upper and a lower bound on optimally cutting a compact Riemannian manifold 푀 into two disjoint pieces of, loosely speaking, maximal volume and minimal perimeter (Cheeger-Buser). These so-called Cheeger cuts have a wide range of applications across machine learning and computer vision; more broadly, eigenvalues and eigenfunctions of Δ help Figure 6: Left: two discrete parameterizations of a to generalize traditional Fourier-based simulation and pseudosphere (constant Gauß curvature 퐾 = −1), one signal processing to more general manifolds. with a Chebychev net along asymptotic directions (left) and another along principal curvature lines (right). These observations motivate the study of discrete Right: a discrete Chebyshev net on a surface of vary- Laplacians, which can be defined even in the purely com- ing curvature, resembling the weft and warp direc- binatorial setting of graphs. Here we briefly outline their tions of a woven material. definition for orientable finite simplicial 푛-manifolds, such as polyhedral surfaces, without boundary. Our ex- discrete Laplacians with all of these properties. How- position is similar to what has become known as discrete ever, certain types of meshes (such as Delaunay triangu- exterior calculus. To this end, consider the simplicial lations) do indeed allow for “perfect” discrete Laplacians, boundary operators 휕 ∶ 퐶 → 퐶 acting on 푘-chains 푘 푘 푘−1 offering a connection between geometry and (discrete) (i.e., formal linear combinations of 푘-simplices). The cor- 푘 differential operators. responding dual spaces (cochains) 퐶 ∶= Hom(퐶푘, ℝ) 푘 푘+1 and respective dual operators 훿푘 ∶ 퐶 → 퐶 give rise to the chain complex Discrete Integrable Systems Another topic that has pro- 0 1 푛 {0} → 퐶 → 퐶 → … → 퐶 → {0}. vided inspiration for many ideas in DDG is parameterized surface theory. Consider for instance the problem The chain property says that 훿 ∘훿 = 0, and one hence 푘 푘−1 of dressing a given surface by a fishnet stocking, i.e., a obtains simplicial cohomology 퐻푘 ∶= ker(훿 )/im(훿 ). 푘 푘−1 woven material composed of inextensible yarns follow- To define a Laplacian in this setting, we equip each 퐶푘 ing transversal “warp” and “weft” directions (see Figure with a positive definite inner product (⋅, ⋅) , and let 훿∗ be 푘 푘 6, right). This task corresponds to decorating a surface the adjoint operator with respect to these inner products, with a tiling where each vertex is incident to four par- i.e., (훿 훼, 훽) = (훼, 훿∗훽) for all 훼, 훽. The Laplacian 푘 푘+1 푘 푘 allelograms. Infinitesimally, such a tiling is known as a on 푘-cochains is then defined as weak Chebyshev net (Chebyshev 1878), and locally corre- Δ ∶= 훿∗훿 + 훿 훿∗ . sponds to a regularly parameterized surface 푓 ∶ 푈 ⊂ 푘 푘 푘 푘−1 푘−1 2 3 ℝ → ℝ where the directional derivatives 푓푢 and 푓푣 The resulting space of harmonic 푘-chains, {훼 ∈ along the coordinate directions satisfy |푓푢|푣 = |푓푣|푢 = 0, 푘 푘 퐶 |Δ푘훼 = 0} is then isomorphic to 퐻 —just as in the i.e., partial derivatives with respect to one parameter smooth setting. This fact is independent of the choice have constant length along the parameter lines of the of inner product, mirroring the fact that cohomology de- other parameter. The special case of rhombic tilings pends only on topological structure. Likewise, for any (|푓푢| = |푓푣| = 1) are known as (strong) Chebyshev nets. inner product one obtains a discrete Hodge decomposi- Can every smooth surface be wrapped in a stocking? Lo- tion cally (i.e., in a small patch around any given point) the 푘 ∗ 퐶 = ker(Δ푘) ⊕ im(훿푘−1) ⊕ im(훿푘 ), answer is “yes”. Globally, however, there are severe ob- structions to doing so, which provide some fascinating where here the subspaces do depend on the choice of connections to physics. inner product. Consider for instance the special case of so-called K- At this point we return again to the game of DDG: surfaces, characterized by constant Gauß curvature 퐾 = which choice of inner product is best? A trivial dot −1. Every K-surface admits a parameterization 푓 ∶ 푈 ⊂ product leads to purely combinatorial graph Laplacians, ℝ2 → ℝ3 aligned with the two transversal asymptotic di- which do not (in general) converge to their smooth coun- rections along which normal curvature vanishes. Hence, terparts (e.g., when approximating a smooth manifold if 푁 is the unit surface normal then by a polyhedral one). Another choice is to consider linear interpolation of 푘-cochains over 푛-dimensional sim- ⟨푓푢푢, 푁⟩ = ⟨푓푣푣, 푁⟩ = 0. (7) plices, resulting in what are known as Whitney elements. For 푛 = 2, we get the so-called cotan Laplacian (Pinkall Asymptotic parameterizations are weak Chebyshev nets and Polthier), which is widely used in digital geometry since processing. Though other choices are possible, we again encounter a no free lunch situation: no choice of inner 푎 ∶= |푓푢|, 푏 ∶= |푓푣| satisfy 푎푣 = 푏푢 = 0. (8) product can preserve all the properties of the smooth Moreover, one can show that the angle 휙 between asymp- Laplacian. Which properties do we care about? Be- totic lines satisfies the sine-Gordon equation yond convergence, perhaps the most desirable properties are the maximum principle (which ensures, for in- 휙푢푣 − 푎푏 sin 휙 = 0, (9) stance, proper behavior for heat flow), and the property that, for flat domains, linear functions are in the ker- and conversely, every solution to the sine-Gordon equa- nel (leading to a proper definition of barycentric coor- tion describes a parameterized K-surface. Hilbert used dinates). For general unstructured meshes there are no this equation (and Chebyshev nets) to prove that the

5 sets of constant width, making them attractive for the construction of (for instance) glass-paneled structures, as in Figure 7.

For further reading, see “Discrete Diﬀerential Geome- try” (2008, Alexander Bobenko ed.).

ABOUT THE AUTHORS Figure 7: Discrete parameterized surfaces play a role in Keenan Crane works at the architectural geometry, where special incidence rela- interface between differential tionships on quadrilaterals translate to manufacturing geometry and geometric algo- constraints like zero nodal torsion, or offset surfaces rithms. His scientific career of constant thickness. Here a curvature line parame- started out studying Pluto— terized surface discretized by a conical net is used in back when it was still a planet! the design of a railway station (courtesy B. Schneider). complete hyperbolic plane cannot be embedded isomet- rically into ℝ3. More generally, the sine-Gordon equation has attracted much interest both in mathematics as an example of an infinite-dimensional integrable system, and in physics as an example of a system that admits remarkably stable soliton solutions, akin to waves that Max Wardetzky’s interest in travel uninterrupted all the way across the ocean. An- discrete differential geometry other key property of the sine-Gordon equation is the comes from his outstanding existence of a so-called spectral parameter 휆 > 0: Equa- and truly inspiring teachers in tion (9) is invariant under a rescaling 푎 → 휆푎 and 푏 → differential geometry, and his 휆−1푏, giving rise to a one-parameter associated family of exposure to marvelous people K-surfaces. Geometrically, the parameter 휆 rescales the in computer graphics. edges of parallelograms while preserving the angle between asymptotic lines.

Do these properties depend critically on the smooth nature of the solutions, or can they also be faithfully captured in the discrete setting? Hirota derived such a discrete version without any reference to geometry. Later Bobenko and Pinkall suggested a geometric definition of discrete K-surfaces that recovers Hirota’s equation. In their setting, discrete K-surfaces are defined as discrete (weak) Chebyshev nets with the additional property that all four edges incident to any vertex lie in a common plane. The last requirement is a natural discrete analogue of Equation (7). This definition of discrete K- surfaces also comes with a spectral parameter 휆 and results in Hirota’s discrete sine-Gordon equation—without requiring any notion of discrete Gauß curvature. Only re- cently has a discrete version of Gauß curvature been suggested that results in discrete K-surfaces indeed having constant negative Gauß curvature.

For discrete K-surfaces with all equal edge lengths (i.e., the rhombic case) the four neighboring vertices of a given vertex must lie on a common circle. By consider- ing a subset of the diagonals of the quadrilaterals, one obtains another quad mesh with the property that all quads have a circumscribed circle, resulting in so-called cK-nets (see Figure 6, left). In the discrete setting, regular networks of circular quadrilaterals play the role of curvature line parameterized surfaces (as in Figure 1, top left). This transformation therefore mimics the smooth setting, where the angle bisectors of asymptotic lines are lines of principal curvature. More broadly, the theory of quad nets with special incidence relationships is closely linked to physical manufacturing considerations in the ﬁeld of architectural geometry. For example, a quad net is conical if the four quads around each vertex are tangent to a common cone—such surfaces admit face oﬀ-