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Course Notes Are Organized Similarly to the Lectures Preface This volume documents the full day course Discrete Differential Geometry: An Applied Introduction pre- sented at SIGGRAPH ’05 on 31 July 2005. These notes supplement the lectures given by Mathieu Desbrun, Eitan Grinspun, and Peter Schr¨oder,compiling contributions from: Pierre Alliez, Alexander Bobenko, David Cohen-Steiner, Sharif Elcott, Eva Kanso, Liliya Kharevych, Adrian Secord, John M. Sullivan, Yiy- ing Tong, Mariette Yvinec. 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. A chapter-by-chapter synopsis The course notes are organized similarly to the lectures. Chapter 1 presents an introduction to discrete differential geometry in the context of a discussion of curves and cur- vature. The overarching themes introduced there, convergence and structure preservation, make repeated appearances throughout the entire volume. Chapter 2 addresses the question of which quantities one should measure on a discrete object such as a triangle mesh, and how one should define such measurements. 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. Chapter 3 gives a concise summarization of curvature measures for discrete surfaces, paving the way for the discrete treatment of thin shell mechanics developed in Chapter 4. Continuing with the theme of discrete surfaces, Chapter 5 describes a discrete Willmore energy for fairing applications, this time preferring a discrete surface made up of linked circles instead of triangles. Such circle patterns are also key to a discrete formulation of conformal parameteri- zation, explored in Chapter 6. At this point we shift down to explore the low-level approach of discrete exterior calculus: Chapter 7 overviews this exciting field, and Chapter 8 details a layman’s approach to implementing DEC. With this in place, numerically robust and efficient simulations of the Navier-Stokes equations of fluids become possible, as described in Chapter 9. Unlike many graphics simulations of fluids which require regular grids, these fluid simulations are adept for arbitrary meshes around boundaries with complex shapes. The generation of such meshes is the subject of Chapter 10. 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 2 Geometry of the Plane Curve the curve. The inverse of the radius is κ, the curvature of 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 circle of curvature does not locally cross the curve, and in the tangent and normal form the axes of a local rectangular general these are finite and isolated points where the curve coordinate system. In addition, the tangent can be thought has a (local) axis of symmetry (there are four such points on of as a local approximation to the curve at P . an ellipse). However on a circle, or a circular arc, the special A better approximation than the tangent is the circle of points are infinitely many and not isolated. curvature: consider a circle through P and two neighboring That the circle of curvature crosses the curve may be rea- points on the curve, and slide the neighboring points towards soned by various arguments. As we traverse the curve past +1 -1 +2 0 Figure 2: The Gauss map assigns to every point on the curve Figure 3: Turning numbers of various closed curves. Top a corresponding point on the unit circle. row: Two simple curves with opposite sense of traversal, and two self-intersecting curves, one of which “undoes” the turn. Bottom row: Gaussian image of the curves, and the point P , the curvature is typically either increasing or de- associated turning numbers. creasing, so that in the local neighborhood of P , so that the osculating circle in comparison to the curve will have a higher curvature on one side and lower on the other. An curve: the image is always the unit circle. If we allow the alternative argument considers our three point construction. closed curve to intersect itself, we can count how many times Trace along a circle passing through three consecutive points the image completely “wraps around” the unit circle (and in on the curve to observe that the circle must pass from side which sense): this is the turning number or the index of A to side B on the first point, B to A on the second, and A rotation, denoted k. It is unity for a simple closed curve to B on the third. Similar reasoning of our two-point con- traversed counterclockwise. It is zero or ±2 for curve that struction shows that in general the tangent does not cross self-intersects once, depending on the sense of traversal and the curve—the isolated exceptions are the points of inflec- on whether or not the winding is “undone.” tion, where the radius of curvature is infinite and the circle Turning Number Theorem. An old and well-known of curvature is identical to the tangent. fact about curves is that the integral of signed curvature over Informally we say that P , the tangent at P , and the oscu- a closed curve, Ω, is dependent only on the turning number: lating circle at P have one, two, and three coincident points Z in common with the curve, respectively.
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