Slides—Exterior Calculus in $R^n$

Later this week we’ll start talking about exterior calculus, which is a modern language used across differential geometry, mathematical physics, geometric computation… and the rest of our class! :-). Initially this language can look a bit daunting, but by making some connections with familiar ideas from vector calculus (like grad, div, and curl), we’ll see that it’s actually not so bad once you get down to concrete calculations. Slides here:

Assignment 1 (Coding): Discrete Exterior Calculus

For the coding portion of your first assignment, you will implement the discrete exterior calculus (DEC) operators $\star_0$, $\star_1$, $\star_2$, $d_0$ and $d_1$. Once implemented, you will be able to apply these operators to a scalar function (as depicted above) by pressing the “*” and “d” button in the viewer. The diagram shown above will be updated to indicate what kind of differential k-form is currently displayed. These basic operations will be the starting point for many of the algorithms we will implement throughout the rest of the class; the visualization (and implementation!) should help you build further intuition about what these operators mean and how they work.

Getting Started

• Please clone 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. Please note: If you already cloned the repository during recitation, clone again!
• For this assignment, you need to implement the following routines:
1. in core/geometry.js
1. cotan
2. barycentricDualArea
2. in core/discrete-exterior-calculus.js
1. buildHodgeStar0Form
2. buildHodgeStar1Form
3. buildHodgeStar2Form
4. buildExteriorDerivative0Form
5. buildExteriorDerivative1Form

In practice, a simple and efficient way to compute the cotangent of the angle $\theta$ between two vectors $u$ and $v$ is to use the cross product and the dot product rather than calling any trigonometric functions directly; we ask that you implement your solution this way. (Hint: how are the dot and cross product of two vectors related to the cosine and sine of the angle between them?)

In case we have not yet covered it in class, the barycentric dual area associated with a vertex $i$ is equal to one-third the area of all triangles $ijk$ touching $i$.

The discrete Hodge star and discrete exterior derivatives are introduced in Section 3.8 of the course notes; the matrix representation of these operators (which you need to implement!) will be discussed in class. They were also basically covered already in our discussion of signed incidence matrices, in the lecture on the simplicial complex.

Notes

• This assignment comes with a viewer (projects/discrete-exterior-calculus/index.html) which lets you apply your operators on random k-forms and visualize the results.
• This assignment also comes with a grading script (tests/discrete-exterior-calculus/test.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) with examples on how to use the halfedge data structure and the linear algebra classes.
• All browsers come with tools for debugging (for instance the JavaScript Console in Chrome).

Submission Instructions

Please rename your geometry.js and discrete-exterior-calculus.js files to geometry.txt and discrete-exterior-calculus.txt (respectively) and submit them in a single zip file called solution.zip by email to Geometry.Collective@gmail.com.

Slides—Differential Forms in $R^n$

Following our lecture on exterior algebra, we will start building up differential forms, which is the next step on our journey toward doing computation on meshes with discrete exterior calculus. This material may be helpful for those of you working through the second part of the written homework:

Slides—Exterior Algebra

Our next 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!)

These slides should also be helpful for those who have started on the homework. 🙂

Assignment 1 (Written): A First Look at Exterior Algebra and Exterior Calculus

The written portion of your first assignment is now available (below), which covers some of the fundamental tools we’ll be using in our class. Initially this assignment may look a bit intimidating but keep in mind a few things:

• The homework is not as long as it might seem: all the text in the big gray blocks contains supplementary, formal definitions that you do not need to know in order to complete the assignments.
• Moreover, note that you are required to complete only three problems from each section.
• If the $\wedge$ and $\star$ symbols look alien to you, don’t sweat: this is not something you should know already! We’ll be talking about these objects in our lectures next week. Until then, you can (and should) get a jump on the lectures by reading the first few sections of Chapter 3 in our course notes.

Finally, don’t be shy about asking us questions here in the comments, via email, or during office hours.  We want to help you succeed on this first assignment, so that you can enjoy all the adventures yet to come…

Here is a zip file with the $\LaTeX$ source.

Reading: Overview of DDG (Due 9/7)

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”.
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!

Linear Algebra and Vector Calculus Review

The prerequisites for 15-458/858B include courses or other past experience in linear algebra and vector calculus. However, we realize that for many of you it may have been a while since you studied this material. The worksheets below provide optional practice for those who wish to brush up on their skills. To emphasize again: this material is purely for your own review; it is not a required assignment, nor will it be graded. You may also wish to review this material from time to time during the semester (e.g., when we study exterior calculus).

[Linear Algebra Review]
[Vector Calculus Review]

Please email us if you’d like to see the solutions! (We keep these solutions private so they can be re-used for other classes in future years.)