Discrete Differential Geometry: An Applied Introduction SIGGRAPH 2006 LECTURERS COURSE NOTES Mathieu Desbrun Konrad Polthier ORGANIZER Peter Schröder Eitan Grinspun Ari Stern Preface The behavior of physical systems is typically described by a set of continuous equations using tools such as geometric mechanics and differential geometry to analyze and capture their properties. For purposes of computation one must derive discrete (in space and time) representations of the underlying equations. Researchers in a variety of areas have discovered that theories, which are discrete from the start, and have key geometric properties built into their discrete description can often more readily yield robust numerical simulations which are true to the underlying continuous systems: they exactly preserve invariants of the continuous systems in the discrete computational realm. This volume documents the full day course Discrete Differential Ge- ometry: An Applied Introduction at SIGGRAPH ’06 on 30 July 2006. These notes supplement the lectures given by Mathieu Des- brun, Eitan Grinspun, Konrad Polthier, Peter Schroder,¨ and Ari Stern. These notes include contributions by Miklos Bergou, Alexan- der I. Bobenko, Sharif Elcott, Matthew Fisher, David Harmon, Eva Kanso, Markus Schmies, Adrian Secord, Boris Springborn, Ari Stern, John M. Sullivan, Yiying Tong, Max Wardetzky, and Denis Zorin, and build on the ideas of many others. A chapter-by-chapter synopsis The course notes are organized similarly to the lectures. We in- troduce discrete differential geometry in the context of discrete curves and curvature (Chapter 1). The overarching themes intro- duced here, convergence and structure preservation, make repeated appearances throughout the entire volume. We ask the question of which quantities one should measure on a discrete object such as a triangle mesh, and how one should define such measurements (Chapter 2). This exploration yields a host of measurements such as length, area, mean curvature, etc., and these in turn form the basis for various applications described later on. We conclude the intro- duction with a summary of curvature measures for discrete surfaces (Chapter 3). The discussion of immersed surfaces paves the way for a discrete treatment of thin-shell mechanics (Chapter 4), and more generally of isometric curvature energies, with applications to fast cloth sim- ulation and Willmore flow (Chapter 5). Continuing with the theme of discrete surfaces, we define straightest geodesics on polyhedral surfaces with applications to integration of vector fields (Chapter 6). At this point we shift down to explore the low-level approach of discrete exterior calculus: after an overview of the field (Chapter 7), we lay out the simple tools for implementing DEC (Chapter 8). With this in place, numerically robust and efficient simulations of the Navier-Stokes equations of fluids become possible (Chapter 9). Simulations of thin-shells, cloth, and fluids, and geometric mod- eling problems such as fairing and parametrization, require robust numerical treatment in both space and time: we describe the in- trinsic Laplace-Beltrami operator, which satisfies a local maximum principle and leads to better conditioned linear systems (Chapter 10), and we overview the variational time integrators offered by the geometric mechanics paradigm (Chapter 11). Eitan Grinspun with Mathieu Desbrun, Konrad Polthier, and Peter Schroder¨ 11 May 2006 i Contents Chapter 1: Introduction to Discrete Differential Geometry: The Geometry of Plane Curves ........................1 Eitan Grinspun and Adrian Secord Chapter 2: What can we measure?...............................................................................................................5 Peter Schröder Chapter 3: Curvature measures for discrete surfaces.................................................................................. 10 John M . Sullivan Chapter 4: A discrete model for thin shells................................................................................................ 14 Eitan Grinspun Chapter 5: Discrete Quadratic Curvature Energies..................................................................................... 20 Miklos Bergou, Max Wardetzky, David Harmon, Denis Zorin, and Eitan Grinspun Chapter 6: Straightest Geodesics on Polyhedral Surface............................................................................ 30 Konrad Polthier and Markus Schmies Chapter 7: Discrete Differential Forms for Computational Modeling......................................................... 39 Mathieu Desbrun, Eva Kanso, and Yiying Tong Chapter 8: Building Your Own DEC at Home........................................................................................... 55 Sharif Elcott and Peter Schröder Chapter 9: Stable, Circulation-Preserving, Simplicial Fluids ..................................................................... 60 Sharif Elcott, Yiying Tong, Eva Kanso, Peter Schröder, and Mathieu Desbrun Chapter 10: An Algorithm for the Construction of Intrinsic Delaunay Triangulations with Applications to Digital Geometry Processing .................................................................................... 69 Matthew Fisher, Boris Springborn, Alexander I. Bobenko, Peter Schröder Chapter 11: Discrete Geometric Mechanics for Variational Time Integrators ............................................ 75 Ari Stern and Mathieu Desbrun ii Discrete Differential Geometry: An Applied Introduction SIGGRAPH 2006 Chapter 1: Introduction to Discrete Differential Geometry: The Geometry of Plane Curves Eitan Grinspun Adrian Secord Columbia University New York University 1 Introduction The nascent field of discrete differential geometry deals with discrete geometric objects (such as polygons) which act as analogues to continuous geometric objects (such as curves). The discrete objects can be measured (length, area) and can interact with other discrete objects (collision/response). From a computational standpoint, the discrete objects are attractive, because they have been designed from the ground up with data-structures and algorithms in mind. From a mathematical standpoint, they present a great challenge: the discrete objects should have properties which are ana- logues of the properties of continuous objects. One impor- P tant property of curves and surfaces is their curvature, which plays a significant role in many application areas (see, e.g., Chapters 4 and 5). In the continuous domain there are re- Figure 1: The family of tangent circles to the curve at point markable theorems dealing with curvature; a key require- P . The circle of curvature is the only one crossing the curve ment for a discrete curve with discrete curvature is that it at P . satisfies analogous theorems. In this chapter we examine the curvature of continuous and discrete curves on the plane. The notes in this chapter draw from a lecture given by P . If the curve is sufficiently smooth (“curvature-continuous John Sullivan in May 2004 at Oberwolfach, and from the at P ”) then the circle thus approaches a definite position writings of David Hilbert in his book Geometry and the known as the circle of curvature or osculating circle; the Imagination. center and radius of the osculating circle are the center of curvature and radius of curvature associated to point P on the curve. The inverse of the radius is κ, the curvature of 2 Geometry of the Plane Curve the curve at P . Consider a plane curve, in particular a small piece If we also consider a sense of traversal along the curve of curve which does not cross itself (a simple curve). segment (think of adding an arrowhead at one end of the Choose two points, P and Q, segment) then we may measure the signed curvature, iden- on this curve and connect them tical in magnitude to the curvature, but negative in sign P with a straight line: a secant. whenever the curve is turning clockwise (think of riding a Fixing P as the “hinge,” ro- bicycle along the curve: when we turn to the right, it is tate the secant about P so because the center of curvature lies to the right, and the Q that Q slides along the curve curvature is negative). toward Q. If the curve is Another way to define the circle of curvature is by con- sufficiently smooth (“tangent- sidering the infinite family of circles which are tangent to continuous at P ”) then the se- the curve at P (see Figure 1). Every point on the normal cant approaches a definite line: the tangent. Of all the to the curve at P serves as the center for one circle in this straight lines passing through P , the tangent is the best family. In a small neighborhood around P the curve divides approximation to the curve. Consequently we define the di- the plane into two sides. Every circle (but one!) in our fam- rection of the curve at P to be the direction of the tangent, ily lies entirely in one side or the other. Only the circle of so that if two curves intersect at a point P their angle of curvature however spans both sides, crossing the curve at P . intersection is given by the angle formed by their tangents It divides the family of tangent circles into two sets: those at P . If both curves have identical tangents at P then we with radius smaller than the radius of curvature lying on one say “the curves are tangent at P .” Returning to our sin- side, and those with greater radius lying on the other side. gle curve, the line perpendicular to the tangent and passing There may exist special points on the curve at which the through P is called the normal to the curve at P . Together