Category: Readings

There are way more topics and ideas in Discrete Differential Geometry than we could ever hope to cover in this course.  For this final reading assignment, you can choose from one of several options that we’ll cover in the remainder of our course:

• Intrinsic Triangulations. In the next couple lectures we’ll revisit polyhedral surfaces through the powerful intrinsic perspective from differential geometry, where we no longer think about how a shape sits in space (i.e., no vertex positions) but instead only have measurements of local quantities like edge lengths and corner angles.  This simple twist provides a huge amount of flexibility for computation and algorithms, since there are infinitely many ways to intrinsically triangulate the same polyhedral surface without corrupting its geometry.  For this reading, you should read Sharp et al, “Geometry Processing with Intrinsic Triangulations”, pp. 1–29, plus one other chapter of your choice.  Summarize what you read, and say why you chose the chapter you did.
• Repulsive Geometry. In our final lecture we’ll discuss a recent perspective on drawing/embedding/optimizing shapes using so-called repulsive energies. The basic starting point is to think about charged particles (like electrons) that try to repel each-other, often producing a nice, uniform distribution of particles.  Now imagine every point of a curve or surface is charged, and let it evolve to the most “self-avoiding” configuration… some very beautiful things emerge!  For this reading you should read Yu et al, “Repulsive Surfaces”, pp. 1–17.  Since this is a research paper, you shouldn’t be worried about understanding 100% of the details.  Rather, you should just get a general feeling for the problem motivation, the approach to the solution, and the potential applications of the method.
• Tangent Direction Fields. Related to the final assignment (A6), you can choose to read one of two surveys on methods for tangent vector field processing—which should give you some additional perspective/context for this assignment.  Tangent vector fields, and more generally, symmetric n-direction fields are everywhere in physical and geometric applications, and a long-running question in DDG is: how should we represent tangent-valued data on discrete surfaces?  For this reading you can read either de Goes et al, “Vector Field Processing on Triangle Meshes”, pp. 5-21, and then choose two out of three of the remaining chapters: face-based, edge-based, and vertex-based vector fields.  Your writeup should compare and contrast these two choices: what are the basic pros and cons?  Alternatively, you can read Vaxman et al, “Directional Field Design, Synthesis, and Processing”, pp. 1–24.  Here again, your writeup should focus on the trade offs between different representations.

This reading complements our lecture on algorithms for computing geodesic paths with an overview of algorithms for computing geodesic distances:

• Crane, Livesu, Puppo, Qin, “A Survey of Algorithms for Geodesic Paths and Distances” (2020), pp. 1–20.

As with many of our readings, the point here is to just get a broader perspective of the material covered in lecture—you are not responsible for knowing every little detail. The algorithms discussed in Section 3 especially are well-connected to the perspective & tools we’ve been developing throughout the semester (e.g., the discrete Laplacian), and will help get you prepared for the assignment on computing geodesic distance.  For this reading you should summarize the high-level ideas from the first part of the survey, and any questions you might have.

Your next reading will take a deep dive into conformal geometry and the many ways to discretize and compute conformal maps. This subject makes some beautiful and unexpected connections to other areas of mathematics (such as circle packings, and hyperbolic geometry), and is in some sense one of the biggest “success stories” of DDG, since there is now a complete uniformization theorem that mirrors the one on the smooth side. You’ll find out more about what this all means in the reading! The reading comes from the note, “Conformal Geometry of Simplicial Surface”:

For your assignment you will need to read the Overview (1.1) and Preliminaries (1.2); you must also pick one of Part I, Part II, or Part III to read, each of which covers a different perspective on discrete conformal maps. The most interesting subject, perhaps, is the connections to hyperbolic geometry in Part IV, which you can read for your own enjoyment! 🙂

Your next reading covers one of the most fundamental objects in differential geometry, and one of the most useful objects in practical geometry processing: the Laplace-Beltrami operator \(\Delta\), which we’ll often refer to as just the “Laplacian”. This operator generalizes the familiar Laplace operator \(\Delta = \frac{\partial^2}{\partial x_1^2} + \cdots + \frac{\partial^2}{\partial x_n^2}\) from Euclidean \(\mathbb{R}^n\) to general curved manifolds. Like the ordinary Laplacian, at a very basic level Laplace-Beltrami provides information about the “curvature” of a function. It also shows up in an enormous number of physical and geometric equations, and for this reason there has been intense study of different ways to discretize the Laplacian (not only for simplicial meshes, but also point clouds and other discrete surface representations).

The reading will expose you to some of the key issues to think about when designing a discrete Laplacian. For this reading, you can choose either of the following two papers:

• Wardetzky, Mathur, Kälberer, Grinspun, “Discrete Laplace Operators: No Free Lunch” (2007)
• Bobenko & Springborn, “A Discrete Laplace-Beltrami Operator for Simplicial Surfaces” (2007)

You should not worry about deeply understanding all of the mathematical details in these papers; the point is just to get a sense of the issues at stake, and how these considerations translate into practical definitions of discrete Laplace matrices. The first paper, by Wardetzky et al, considers a “No Free Lunch” theorem for discrete Laplacians that continues our story of “The Game” played in discrete differential geometry. The second paper, by Bobenko & Springborn considers the important perspective of intrinsic triangulations of polyhedral surfaces, and uses this perspective to develop a Laplace operator that is well-behaved even for very poor quality triangulations. You should simply summarize the high-level ideas in these papers, and any questions you might have.

The reading is due on Thursday, March 30 at 10am Eastern time. Hand-in instructions are as usual described on the assignments page.

Your next reading complements our in-class discussion of the geometry of curves and surfaces. In particular, you should read Chapter 3 of the course notes, pages 28–44. This reading is due Thursday, March 16.

Handin instructions are described on the Assignments Page.

Since these notes just barely scratch the surface (literally), I am often asked for recommendations on books that provide a deeper discussion of surfaces. The honest answer is, “I don’t know; I mostly didn’t learn it from a book.” But there are a couple fairly standard references (other) people seem to like, both of which should be available digitally from the CMU library:

• O’Neill, “Elementary Differential Geometry”
• do Carmo, “Differential Geometry of Curves and Surfaces”

The next reading assignment will wrap up our discussion of exterior calculus, both smooth and discrete. In particular, it will explore how to differentiate and integrate \(k\)-forms, and how an important relationship between differentiation and integration (Stokes’ theorem) enables us to turn derivatives into discrete operations on meshes. In particular, the basic data we will work with in the computational setting are “integrals of derivatives,” which amount to simple scalar quantities we can associate with the vertices, edges, faces, etc. of a simplicial mesh. These tools will provide the basis for the algorithms we’ll explore throughout the rest of the semester.

The reading is the remainder of Chapter 4 from the course notes, “A Quick and Dirty Introduction to Exterior Calculus”, Sections 4.6 through 4.8 (pages 67–83). Note that you just have to read these sections; you do not have to do the written exercises; a different set of written problems will be posted later on. The reading is due Thursday, February 16 at 10am. See the assignments page for handin instructions.

Your next reading assignment will help you review the concepts we’ve been discussing in class: describing “little volumes” or \(k\)-vectors using the wedge product and the Hodge star, and measuring these volumes using “dual” volumes called \(k\)-forms. These objects will ultimately let us integrate quantities over curved domains, which will also be our main tool for turning smooth equations from geometry and physics into discrete equations that we can actually solve on a computer.

The reading is Chapter 4, “A Quick and Dirty Introduction to Exterior Calculus”, up through section 4.5.1 (pages 45–65). It will be due Thursday, February 9 at 10am Eastern time. See the assignments page for handin instructions.

Your next homework will give you some hands-on practice with differential forms; just take this time to get familiar with the basic concepts.

Your next reading will take a dive into purely combinatorial descriptions of surfaces, i.e., those that capture connectivity, but not geometry.  These descriptions and data structures will provide the foundation for all the geometry and algorithms we’ll build up in this class.  (The reading also provides the essential background for your first written and coding assignments!)

The reading is Chapter 2, pages 7–20 of our course notes, which can always be accessed from the link above.

Your short 2-3 sentence summary is due by 10am Eastern on February 2, 2023.  Handin instructions can be found on the assignment page.

Your first reading assignment will be to read an overview article on Discrete Differential Geometry. Since we know we have a diverse mix of participants in the class, you have several options (pick one):

1. (pages 1–3) Crane & Wardetzky, “A Glimpse into Discrete Differential Geometry”.
   This article discusses the “no free lunch” story about curvature we looked at in class, plus a broader overview of the field.
2. (pages 1–5) Pottman et al, “Architectural Geometry”.
   This article discusses the beautiful tale of how discrete differential geometry is linked to modern approaches to computational design for architecture, as well as fabrication and “rationalization” of free-form designs.
3. (pages 5–9) Bobenko & Suris, “Discrete Differential Geometry: Consistency As Integrability”.
   This article provides another overview of discrete differential geometry, with an emphasis on nets and their connection to the notion of integrability in geometry and physics.

Though written for a broad audience, be warned that all of these articles are somewhat advanced—the goal here is not to understand every little detail, but rather just get a high-level sense of what DDG is all about.

Assignment: Pick one of the readings above, and write 2–3 sentences summarizing what you read, plus at least one question about something you didn’t understand, or some thought/idea that occurred to you while reading the article.  For this first assignment, we are also very interested to know a little bit about YOU! E.g., why are you taking this course?  What’s your background?  What do you hope to get out of this course?  What are your biggest fears about the course?  Etc.

Handin instructions can be found in the “Readings” section of the Assignments page.  Note that you must send your summary in no later than 10am Eastern on the day of the next lecture (January 24, 2023).

Enjoy!