`*`” 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:
- in
`core/geometry.js``cotan``barycentricDualArea`

- in
`core/discrete-exterior-calculus.js``buildHodgeStar0Form``buildHodgeStar1Form``buildHodgeStar2Form``buildExteriorDerivative0Form``buildExteriorDerivative1Form`

- in

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 submit your `geometry.js` and `discrete-exterior-calculus.js` files in a **single zip file** called `solution.zip` by email to Geometry.Collective@gmail.com.

**Grading**

This assignment is worth 6.5% of your grade.