## Assignment 1 (Written): Exterior Calculus (Due 3/18)

The written portion of assignment 1 is now available, which covers some of the fundamental tools we’ll be using in our class. Initially this assignment may look a bit intimidating, but 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.

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 assignment, so that you can enjoy all the adventures yet to come…

This assignment is due on Thursday, March 18.

## Assignment 1 (Coding): Exterior Calculus (Due 3/18)

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 “$\star$” 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

• For this assignment, you need to implement the following routines:
1. in core/geometry.[js|cpp]
1. cotan
2. barycentricDualArea
2. in core/discrete-exterior-calculus.[js|cpp]
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 $i$.

EDIT: You can compute the ratio of dual edge lengths to primal edge lengths using the cotan formula, which can be found on Slide 28 of the Discrete Exterior Calculus lecture, or in exercise 36 of the notes (you don’t have to do the exercise for this homework).

Submission Instructions

Please submit your geometry.[js|cpp] and discrete-exterior-calculus.[js|cpp] files to Gradescope.  You should not submit any other source files (and therefore, should not edit any other source files to get your code working!).

## Assignment 0 (Coding): Combinatorial Surfaces — due 2/26

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

• Decide whether you want to use the web skeleton (in JavaScript) or the desktop skeleton (in C++). The web skeleton should “just work” for anyone with a web browser. Setting up the C++ skeleton is also fairly automatic (just a few git commands), but requires a little more familiarity with coding environments. (The benefit of the C++ version is that it’s built on a richer, real-world mesh processing/visualization library than the web version.)
• To use the JavaScript version, download or clone the files in the ddg-exercises-js repository.
• To use the C++ version, follow the instructions in the ddg-exercises repository. It is important that you follow the “Getting Started” instructions and do not simply git clone the repo. Otherwise, dependencies will not be installed correctly, and the code will not build. If you struggle to get the C++ version working on your platform, we recommend you switch to the JavaScript version.
• Either way, you should need to download just one code skeleton for the whole semester (though we may push periodic updates to fix bugs, etc.).
• For this assignment, you will need to implement the following routines in projects/simplicial-complex-operators/simplicial-complex-operators.[js|cpp]:
• assignElementIndices
• buildVertexVector
• buildEdgeVector
• buildFaceVector
• star
• closure
• isComplex
• isPureComplex
• boundary

Notes

• The JavaScript 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. Likewise, the C++ version comes with a viewer with similar functionality. (Viewers will be available for all assignments throughout the semester.)
• 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 web 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

The assignment is due on February 26, 2020 at 5:59:59pm Eastern (not at midnight!). Remember to turn in the whole coding assignment via a single ZIP file containing the modified source files. Further hand-in instructions can be found on this page.

## Assignment 0 (Written): Combinatorial Surfaces — due 2/26

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 all 15 exercises from the Written Exercises section of Chapter 2 of the course notes. If these exercises seem scary and unfamiliar, and you don’t know where to start, that’s ok! This is not typical computer science stuff, and you shouldn’t necessarily know how to do it right off the bat. If you find yourself stumbling, please reach out to the instructor/TAs and your classmates on Piazza, Discord, or during the office hours/Q&A sessions. We’ll have you up and running in no time. 🙂

The assignment is due on February 26, 2021 at 5:59:59pm Eastern (not at midnight!). Hand-in instructions can be found on this page.