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
- For this assignment, you need to implement the following routines:
- in core/geometry.js
- in core/discrete-exterior-calculus.js
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).
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.
The written portion of assignment 1 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 a few things in mind:
- 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.
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 assignment, so that you can enjoy all the adventures yet to come…
This assignment is due on Tuesday, February 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 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 7, 2019 at 5:59:59pm Eastern (not at midnight!). Further hand-in instructions can be found on this page.
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).
- For this assignment, you will need to implement the following routines in projects/simplicial-complex-operators/simplicial-complex-operators.js:
- 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.
- 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.
Please rename your simplicial-complex-operators.js file to simplicial-complex-operators.txt and submit it in a zip file called solution.zip to Geometry.Collective@gmail.com.