## Assignment 6 [Coding]: Vector Field Decomposition and Design (Due 5/21)

Note: For the final assignment, you can do either this assignment or A5. Overall, you just need to be sure you completed A0 and A1, as well as 3 of the assignments A2–A6 (your choice which ones. See this Piazza post for more details.) Also, you cannot use late days on this assignment since it’s the last one.

Please implement the following routines in:

1. projects/vector-field-decomposition/tree-cotree.[js|cpp]:
1. buildPrimalSpanningTree
2. inPrimalSpanningTree
3. buildDualSpanningCotree
4. inDualSpanningCotree
5. buildGenerators
2. projects/vector-field-decomposition/harmonic-bases.[js|cpp]:
1. buildClosedPrimalOneForm
2. compute

In addition, please implement either Hodge Decomposition, or Trivial Connections (on surfaces of genus 0):

Hodge Decomposition: Please implement the following routines in

1. projects/vector-field-decomposition/hodge-decomposition.js:
1. constructor
2. computeExactComponent
3. computeCoExactComponent
4. computeHarmonicComponent

Trivial Connections on Surfaces of Genus 0: Please implement computeConnections in projects/direction-field-design/trivial-connections.js. This file has function signatures for trivial connections on arbitrary surfaces, but for this assignment we are only requiring you to compute the connection form on simply-connected surfaces where you don’t have to worry about the period matrix. You are, of course, welcome to implement the general algorithm if you would like to!

Notes:

• Recall that homology generators are a set of loops which somehow describe all of the interesting loops on a surface. For example, here are the two homology generators for the torus.
• Your buildGenerators function should implement the Tree-Cotree algorithm, which is Algorithm 5 of section 8.2 of the notes.
• In tree-cotree.js the trees are stored using the dictionaries vertexParent and faceParent. You should store the primal spanning tree by storing the parent of vertex v in vertexParent[v]. The root should be its own parent. Similarly, you should store the dual spanning cotree by storing the parent of face f in faceParent[f].
• The Tree Cotree algorithm finds homology generators on the dual mesh. That is to say, you should take each dual edge which is not in either spanning tree, and form a loop by following its endpoint back up the dual cotree until they meet at the root of the dual cotree. Even though we’re thinking of these homology generators as loops on the dual mesh, we still store them as lists of ordinary halfedges, since edges in the dual mesh are in 1-1 correspondence with edges in the primal graph.
• buildClosedPrimalOneForm should use a homology generator to construct a closed primal 1-form as described in section 8.22 of the notes.
• The compute function in HarmonicBases should compute a harmonic basis using Algorithm 6 in section 8.2 of the notes. If you choose to implement Hodge decomposition for the assignment, you are welcome to use your Hodge decomposition code to solve for $d\alpha$. Otherwise, you can just ignore the hodgeDecomposition parameter and directly solve the linear system $\Delta \alpha = \delta \omega$ to find $\alpha$. (Recall that $\delta$ is the codifferential, which we talked a lot about in the Discrete Exterior Calculus assignment).

Notes (Hodge Decomposition):

• The procedure for Hodge Decomposition is given as Algorithm 3 in section 8.1 of the Notes.
• Note that the SparseMatrix class has an invertDiagonal function that you can use to invert diagonal matrices.
• For computeCoExactComponent, you should compute the coexact component $\delta \beta$ of a differential form $\omega$ by solving the equation $d\delta \beta = d\omega$. As stated in the notes, the most convenient way of doing this is to define a dual 0-form $\tilde \beta := *\beta$, and then to solve $d*d \tilde \beta = d \omega$. Then you can compute $\delta \beta$ using the fact that $\delta \beta = *d*\beta = *d\tilde\beta$. When doing these computations, you should keep tract of whether your forms are primal forms or dual forms, recalling that hodge stars take you from primal forms to dual forms, and hodge star inverses take you from dual forms to primal forms. (THis is discussed in detail in section 8.1.3 of the notes.
• Note that the 2-form Laplacian B is not necessarily positive definite, so you should use an LU solver instead of a Cholesky solver when solving systems involving B.

Notes (Trivial Connections):

• The trivial-connection-js file is structured for implementing the full trivial connections algorithm on arbitrary surfaces. Since that’s tricky, we’re only requiring that you compute connections on topological spheres. You should implement this in the computeConnections() function, and it should pass the tests labeled “computeConnections on a sphere”.
• Even though we’re working with spheres, it is still helpful to use the formulation of Trivial connections given in section 8.4.1 of the notes. In particular, you should solve the optimization problem given in Exercise 8.21.
• On a sphere, there are no harmonic 1-forms, so the $\gamma$ part of your Hodge decomposition will always be 0. Furthermore, the sphere has no homology generators, so the problem simplifies to $\min_{\delta \beta} \;\|\delta\beta\|^2\;\text{s.t.}\; d\delta\beta = u$
• Following the notes, we observe that the constraint $d\delta\beta = u$ determines $\beta$ up to a constant, and that constant is annihilated by $\delta$. So you can find the connection 1-form $\delta \beta$ with a single linear solve.

Submission Instructions
Please submit all source files to Gradescope by 5:59pm ET on May 21, per the usual submission instructions.

Warning: You cannot use late days on this assignment since it’s the last one.

## Assignment 6 [Written]: Vector Field Decomposition and Design (Due 5/21)

Note: For the final assignment, you can do either this assignment or A5. Overall, you just need to be sure you completed A0 and A1, as well as 3 of the assignments A2–A6 (your choice which ones. See this Piazza post for more details.) Also, you cannot use late days on this assignment since it’s the last one.

In this assignment, you will investigate tools for working with tangent vector fields on surfaces. Tangent vector fields are central to classical differential geometry, and have many interesting applications. For this homework, we’ll look at one algorithm for designing vector fields, and along the way we’ll cover a lot of deep facts about surfaces.

There’s no PDF this week since the exercises are all from the notes. The notes will also give you all the background you’ll need to complete this assignment. It builds on a lot of the stuff we’ve already done in the class, especially discrete exterior calculus and the Laplacian.

Submit your written responses for Exercise 8.1 – Exercise 8.21 in the notes to Gradescope (per the usual submission instructions), except for Exercise 8.13.

Warning: You cannot use late days on this assignment since it’s the last one.

## Supplemental Videos: Geodesic Distance and Beyond

In completing your assignment for finals next week, we thought you might find a couple videos helpful (though totally optional). The first was already linked to in the assignment writeup, which gives a reasonably short (18 minutes) motivation for and description of the algorithm you’ll be implementing. The second is a longer (50 minute) talk that covers the algorithm in more detail, and describes some fun extensions and applications of the same basic technique:

Enjoy!

## Assignment 5 [Coding]: Geodesic Distance (Due 5/21)

Note: For the final assignment, you can do either this assignment or A6. Overall, you just need to be sure you completed A0 and A1, as well as 3 of the assignments A2–A6 (your choice which ones. See this Piazza post for more details.) Also, you cannot use late days on this assignment since it’s the last one.

For the coding portion of your final assignment, you will implement the heat method, which is an algorithm for computing geodesic distance on curved surfaces. All of the details you need for implementation are described in Section 3 of the paper, up through and including Section 3.2. Note that you need only be concerned with the case of triangle meshes (not polygon meshes or point clouds); pay close attention to the paragraph labeled “Choice of Timestep.”

Please implement the following routines in:

1. projects/geodesic-distances/heat-method.[js|cpp]:
1. constructor
2. computeVectorField
3. computeDivergence
4. compute

Notes

• Refer to sections 3.2 of the paper for discretizations of Algorithm 1 (page 3).
• Recall that our Laplace matrix is positive semidefinite, which might differ from the sign convention the authors use.
• The tests for computeVectorField and computeDivergence depend on the A and F matrices you define in your constructor. So if you fail the tests but your functions look correct, check whether you have defined the flow and laplace matrices properly.
• Your solution should implement zero neumann boundary conditions (which are the “default behavior” of the cotan Laplacian) but feel free to tryout other Dirichlet and Neumann boundary conditions on your own.

Submission Instructions

Please submit your source files to Gradescope by 5:59pm ET on May 21, 2021.

## Assignment 5 [Written]: Geodesic Distance (Due 5/21)

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!)

Note: For the final assignment, you can do either this assignment or A6. Overall, you just need to be sure you completed A0 and A1, as well as 3 of the assignments A2–A6 (your choice which ones. See this Piazza post for more details.) Also, you cannot use late days on this assignment since it’s the last one.

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. Submit your responses to Gradescope by 5:59pm ET, May 21, 2021.

## Lectures 21 & 22 — Geodesics

Our final lectures for the term focus on geodesics, which generalize the notion of “straight line” to curved spaces; this material also connects to your final assignment, on computing geodesic distance. Once again we’ll play “The Game” of discrete differential geometry, and see how two natural characterizations from the smooth setting (straightest and locally shortest) lead to two distinct, and fascinating definitions in the discrete setting. Along the way we’ll also encounter many rich topics from (discrete) differential geometry including the cut locus, the medial axis, the exponential/log map, the covariant derivative, and the Lie bracket! We’ll also see how all this stuff connects with practical algorithms for things like surface reconstruction from points, and give two different algorithms for tracing out geodesics on curved surfaces.

Note that the lecture video comprises about two lectures (about two hours total). I’d recommend watching it in two logical chunks:

• Part I: Shortest Geodesics – 0:00—1:04
• Part II: Straightest Geodesics – 1:04–1:55