Assignment 4 (Written): Conformal Parameterization

The written part of your next assignment, on conformal surface flattening, is now available below. Conformal flattening is important for (among other things) making the connection between processing of 3D surfaces, and existing fast algorithms for 2D image processing. You’ll have the opportunity to implement one of these algorithms in the coding part of the assignment (to be announced soon).

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.

Grading

This assignment is worth 6.5% of your grade.

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…

A1Written

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