Slides—\(k\)-Forms

Today we continue our journey toward building up (discrete) exterior calculus by talking about how to measure little k-dimensional volumes. Just like rulers measure length, and cups measure volume, k-forms will be used to take measurements of the little k-dimensional volumes or k-vectors that we built up using exterior algebra in our previous lecture. Such measurements will ultimately allow us to talk about integration over curved spaces; in the discrete setting, these measurements will be the basic data we associate with the elements of mesh.

Slides—Exterior Algebra

Today’s lecture will cover one of the basic tools we’ll use throughout the rest of the course: exterior algebra. The basic idea is to add a couple new operations to our usual list of vector operations (dot product, cross product, etc.) that make it easy to talk about volumes rather than just vectors. If you felt ok working with things like the cross product and the determinant in your linear algebra/vector calculus courses, this shouldn’t be too big of a leap. (If not, could be a good moment for a review!)

Assignment 0 (Coding): Combinatorial Surfaces — due 2/8

For the coding portion of your first assignment, you will implement some operations on simplicial complexes which were discussed in class and in Chapter 2 of the course notes. Once implemented, you will be able to select simplices and apply these operations to them by clicking the appropriate buttons in the viewer (shown above).

Getting Started

  • Please download or clone the files in this repository. It contains a fast and flexible framework for 3D geometry processing implemented in Javascript. Over the course of the semester, you will implement all of your coding assignments here.
  • For this assignment, you will need to implement the following routines in projects/simplicial-complex-operators/simplicial-complex-operators.js:
    • assignElementIndices
    • buildVertexEdgeAdjacencymatrix
    • buildEdgeFaceAdjacencymatrix
    • buildVertexVector
    • buildEdgeVector
    • buildFaceVector
    • star
    • closure
    • link
    • isComplex
    • isPureComplex
    • boundary

Notes

  • This assignment comes with a viewer projects/simplicial-complex-operators/index.html which lets you apply your operators to simplices of meshes and visualize the results.
  • Pay close attention to the course notes! Some routines really must be implemented with sparse matrices, not directly with the halfedge mesh data structure.
  • Selecting simplices will not work until you fill in the assignElementIndices function.
  • The assignment also comes with a test script tests/simplicial-complex-operators/text.html which you can use to verify the correctness of your operators.
  • The code framework is implemented in Javascript, which means no compilation or installation is necessary on any platform. You can simply get started by opening the index.html file in projects/discrete-exterior-calculus/ in a web browser. We recommend using Chrome or Firefox. Safari has poor WebGL performance.
  • If you do not have prior experience with Javascript, do not worry! You should be able to get a handle on Javascript syntax by reading through some of the code in the framework (a good place to start might be core/geometry.js). The framework also contains extensive documentation (see docs/index.html).
  • All browsers come with tools for debugging (for instance the JavaScript Console in Chrome).

Submission Instructions

Important: Please rename your simplicial-complex-operators.js file to simplicial-complex-operators.txt, so that your email does not get filtered out by the email server.

The assignment is due on February 8, 2020 at 5:59:59pm Eastern (not at midnight!). Remember to turn in the whole assignment via a single email including both the written exercises (as a PDF file) and the code (in a ZIP file). Further hand-in instructions can be found on this page.

Assignment 0 (Written): Combinatorial Surfaces — due 2/8

For the written part of your first homework you will do some exercises that will help familiarize you with basic descriptions and representations of combinatorial surfaces (simplicial surfaces, adjacency matrices, halfedge meshes), which will help prepare you to work with such surfaces as we continue through the course. (If any of this stuff seems abstract right now, don’t worry: we’ll use it over and over again to implement “real” algorithms starting in just a couple weeks!)

You must complete 8 out of 15 exercises in the Written Exercises section of Chapter 2 of the course notes. You may choose any set of 8 exercises you like, but if you do more than 8, please mark clearly on your submission which ones you would like us to grade.

The assignment is due on February 8, 2020 at 5:59:59pm Eastern (not at midnight!). Remember to turn in the whole assignment via a single email including both the written exercises (as a PDF file) and the code (in a ZIP file). Further hand-in instructions can be found on this page.

Reading 2: Combinatorial Surfaces (Due 2/4)

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 4, 2020.  Handin instructions can be found on the assignment page.

Reading 1: Overview of DDG (Due 1/21)

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 (September 7, 2017).

Enjoy!

Assignment -1: Favorite Formula

Part of your course grade is determined by participation, which can include both in-class participation as well as discussion here on the course webpage.  Therefore, your first assignment is to:

  1. create an account, and
  2. leave a comment on this post containing your favorite mathematical formula (see below).
To make things interesting, your comment should include a description of your favorite mathematical formula typeset in $\LaTeX$.  If you don’t know how to use $\LaTeX$ this is a great opportunity to learn — a very basic introduction can be found here.  (And if you don’t have a favorite mathematical formula, this is a great time to pick one!)
 
(P.S. Anyone interested in hearing about some cool “favorite theorems” should check out this podcast.)

Welcome to Discrete Differential Geometry! (Spring 2020)

Welcome to the website for 15-458/858 (Discrete Differential Geometry).  Here you’ll find course notes, lecture slides, and homework (see links on the right).

If you are a student in the class, register now by clicking here!

To participate in the class, you must register using your Andrew (CMU) email address.

A few things to note:

  • You will be subscribed to receive email notification about new posts, comments, etc.
  • You can ask questions by leaving a comment on a post.  If you’re apprehensive about asking questions in public, feel free to register under a pseudonym.
  • Otherwise, please associate a picture to your profile by registering your email address at Gravatar.com—doing so will help us better recognize you in class!
  • You can include mathematical notation in your questions using standard $\LaTeX$ syntax.  For instance, when enclosed in a pair of dollar signs, an expression like dollar\int_M K dA = 2\pi\chidollar gets typeset as $\int_M K dA = 2\pi\chi$.
If you encounter any problems while trying to use the website, please contact the TA (listed under the course description).