Here’s the writeup for your last assignment, which plays the role of the final in the course. (Note that there is no final exam!)

This time, we’re taking off the “training wheels” and having you read a real paper, rather than course notes. Why? Because you’re ready for it! At this point you have all the fundamental knowledge you need to go out into the broader literature and start implementing all sorts of algorithms that are built on top of ideas from differential geometry. In fact, this particular algorithm is not much of a departure from things you’ve done already: solving simple equations involving the Laplacian on triangle meshes. As discussed in our lecture on the Laplacian, you’ll find many algorithms in digital geometry processing that have this flavor: compute some basic data (e.g., using a local formula at each vertex), solve a Laplace-like equation, compute some more basic data, and so on.

Your main references for this assignment will be:

- this video, which gives a brief (18-minute) overview of the algorithm, and
- this paper, which explains the algorithm in detail.

Written exercises for this assignment are found below.

Note that for handin you should use **DDG20A5**, not **DDG19A5** (as stated in the PDF).

Link for the paper in Q1 reading assignment is broken. I believe this is the correct link: http://ddg.math.uni-goettingen.de/pub/GeodesicsInHeat.pdf

Looks like the link on the webpage works; fixed the other one. Thanks!

This post doesn’t have the late day warning like the

extra assignment posts. Are we allowed to use late days on this assignment?

No, sorry! The issue is that we have to submit final grades to CMU by a hard deadline, and 5/14 is already cutting it close. Otherwise I’d love to give you all a little more slack.

For Section 4, where do we get the vector field $X$ from?

Sorry, what do you mean by Section 4? I see a vector field \(X\) in Exercises 5, and 6, but am not sure if this is what you mean.

Ah I meant for Exercises 7-9. Should we be using the expression for $\Delta\phi = \nabla\cdot X$ from Exercise 6?

Nope—for these problems you’re just considering the equation

\[ \Delta \phi = f \]

where the values of \(f\) are given below the diagram. (You could imagine that these values came from the divergence of a vector field if you like, but for the problem you can just treat them as some given data.)

Note that in Exercise 3 you may assume that \(\phi\) vanishes along the boundary. This assumption is sufficient for carrying out Exercises 4 and 5, which need only consider variations \(\psi\) that vanish along the boundary (since you’re considering Dirichlet boundary conditions, which fix the boundary values).