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, inde- pendent 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 curva- ture! 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 ob- ject 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 differ- entiable (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 alterna- tive characterizations in the smooth setting that can be applied to discrete geometry in a natural way. With cur- vature, for instance, we can apply the fundamental theo- rem 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 natu- rally 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 polyg- onal 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).