**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.

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

]]>- 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.

]]>**(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.**(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.**(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 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.)