Assignment 3 [Coding]: The Laplacian (Due 4/2)

For the coding portion of this assignment, you will build the so-called “cotan-Laplace” matrix and start to see how it can be used for a broad range of surface processing tasks, including the Poisson equation and two kinds of curvature flow.

Getting Started

Please implement the following routines in

  1. core/geometry.js:
    • laplaceMatrix
    • massMatrix
  2. projects/poisson-problem/scalar-poisson-problem.js:
    • constructor
    • solve
  3. projects/geometric-flow/mean-curvature-flow.js:
    • buildFlowOperator
    • integrate
  4. projects/geometric-flow/modified-mean-curvature-flow.js:
    • constructor
    • buildFlowOperator


  • Sections 6.4-6 of the course notes describe how to build the cotan-Laplace matrix and mass matrices, and outline how they can be used to solve equations on a mesh. In these applications you will be required to setup and solve a linear system of equations $Ax=b$ where $A$ is the Laplace matrix, or some slight modification thereof. Highly efficient numerical methods such as Cholesky Factorization can be used to solve such systems, but require A to be symmetric positive definite. Notice that the cotan-Laplace matrix described in the notes is negative semi-definite. To make it compatible for use with numerical methods like the Cholesky Factorization, your implementation of laplaceMatrix should instead produce a positive definite matrix, i.e., it should represent the expression
    $$(\Delta u)_i=\frac12 \sum_{ij}(\cot \alpha_{ij}+\cot \beta_{ij})(u_i–u_j).$$(Note that $u_i−u_j$ is reversed relative to the course notes.) To make this matrix strictly positive definite (rather than semidefinite), you should also add a small offset such as $10^{-8}$ to the diagonal of the matrix (which can be expressed in code as a floating point value 1e-8). This technique is known as Tikhonov regularization.
  • The mass matrix is a diagonal matrix containing the barycentric dual area of each vertex.
  • In the scalar Poisson problem, your goal is to discretize and solve the equation $\Delta \phi = \rho$ where $rho$ is a scalar density on vertices and $\Delta$ is the Laplace operator. Be careful about appropriately incorporating dual areas into the discretization of this equation (i.e., think about where and how the mass matrix should appear); also remember that the right-hand side cannot have a constant component (since then there is no solution).
  • In your implementation of the implicit mean curvature flow algorithm, you can encode the surface $f:M \to \mathbb R^3$ as a single DenseMatrix of size $|V| \times 3$, and solve with the same (scalar) cotan-Laplace matrix used for the previous part. When constructing the flow operator, you should follow section 6.6 of the notes. But be careful – when we discretize equations of the form $\Delta f = \rho$, we create systems of the form $A f = M \rho$. So you’ll need to add in the mass matrix somewhere. Also, recall that our discrete Laplace matrix is the negative of the actual Laplacian.
  • The modified mean curvature flow is nearly identical to standard mean curvature flow. The one and only difference is that you should not update the cotan-Laplace matrix each time step, i.e., you should always be using the cotans from the original input mesh. The mass matrix, however, must change on each iteration.

Submission Instructions

Please rename your geometry.js, scalar-poisson-problem.js, mean-curvature-flow.js and modified-mean-curvature-flow.js files to geometry.txt, scalar-poisson-problem.txt, mean-curvature-flow.txt and modified-mean-curvature-flow.txt (respectively) and put them in a single zip file called These files and your solution to the written exercises should be submitted together in a single email to with the subject line DDG19A3.

Assignment 3 [Written]: The Laplacian (Due 4/2)

These exercises will lead you through two different derivations for the cotan-Laplace operator. As we’ll discuss in class, this operator is basically the “Swiss army knife” of discrete differential geometry and digital geometry processing, opening the door to a huge number of interesting algorithms and applications. Note that this time the exercises all come from the course notes—you will need to read the accompanying notes in order to familiarize yourself with the necessary material (though actually we’ve covered much of this stuff in class already!)


Assignment 2 (Written): Investigating Curvature (Due 3/19)

The written portion of Assignment 2 can be found below. It takes a look at the curvature of smooth and discrete surfaces, which we have been talking about in lecture. Note that you are required to complete only two problems from each section!

Warning: We renumbered the Exercises in the course notes to make more sense, so you make sure you refer to the updated notes when doing these exercises.

Assignment 2 (Coding): Investigating Curvature (Due 3/19)

For the coding portion of this assignment, you will implement various expressions for discrete curvatures and surfaces normals that you will derive in the written assignment. (However, the final expressions are given below in case you want to do the coding first.) Once implemented, you will be able to visualize these geometric quantities on a mesh. For simplicity, you may assume that the mesh has no boundary.

Getting Started
Please implement the following routines in core/geometry.js:

  1. angle
  2. dihedralAngle
  3. vertexNormalAngleWeighted
  4. vertexNormalSphereInscribed
  5. vertexNormalAreaWeighted
  6. vertexNormalGaussianCurvature
  7. vertexNormalMeanCurvature
  8. angleDefect
  9. totalAngleDefect
  10. scalarMeanCurvature
  11. circumcentricDualArea
  12. principalCurvatures


1. The dihedral angle between the normals $N_{ijk}$ and $N_{ijl}$ of two adjacent faces $ijk$ and $ijl$ (respectively) is given by
$$ \theta_{ij} := \text{atan2}\left(\frac{e_{ij}}{\|e_{ij}\|} \cdot \left(N_{ijk} \times N_{jil}\right), N_{ijk} \cdot N_{jil}\right)$$

where $e_{ij}$ is the vector from vertex $i$ to vertex $j$.

2. The formulas for the angle weighted normal, sphere inscribed normal, area weighted normal, discrete Gaussian curvature normal and discrete mean curvature normal at vertex $i$ are
N_i^\phi &:= \sum_{ijk \in F} \phi_i^{jk}N_{ijk}\\
N_i^S &:= \sum_{ijk \in F} \frac{e_{ij} \times e_{ik}} {\|e_{ij}\|^2\|e_{ik}\|^2}\\
N_i^A &:= \sum_{ijk \in F} A_{ijk}N_{ijk}\\
KN_i &= \frac 12 \sum_{ij \in E} \frac{\theta_{ij}}{\|e_{ij}\|}e_{ij}\\
HN_i &= \frac 12 \sum_{ij \in E}\left(\cot\left(\alpha_k^{ij}\right) + \cot\left(\beta_l^{ij}\right)\right)e_{ij}

where $\phi_i^{jk}$ is the interior angle between edges $e_{ij}$ and $e_{ik}$, and $A_{ijk}$ is the area of face $ijk$. Note that sums are taken only over elements (edges or faces) containing vertex $i$. Normalize the final value of all your normal vectors before returning them.

3. The circumcentric dual area at vertex $i$ is given by
\[A_i := \frac 18 \sum_{ijk \in F} \|e_{ik}\|^2\cot\left(\alpha_j^{ki}\right) + \|e_{ij}\|^2\cot\left(\beta_k^{ij}\right)\]

4. The discrete scalar Gaussian curvature (also known as angle defect) and discrete scalar mean curvature at vertex $i$ are given by
K_i &:= 2\pi – \sum_{ijk \in F} \phi_i^{jk}\\
H_i &:= \frac 12 \sum_{ij \in E} \theta_{ij}\|e_{ij}\|

Note that these quantities are discrete 2-forms, i.e., they represent the total Gaussian and mean curvature integrated over a neighborhood of a vertex. They can be converted to pointwise quantities (i.e., discrete 0-forms at vertices) by dividing them by the  circumcentric dual area of the vertex (i.e., by applying the discrete Hodge star).

5. You are required to derive expressions for the principal curvatures $\kappa_1$ and $\kappa_2$ in exercise 4 of the written assignment. Your implementation of principalCurvatures should return the (pointwise) minimum and maximum principal curvature values at a vertex (in that order).

Submission Instructions

Please rename your geometry.js file to geometry.txt and put it in a single zip file called This file and your solution to the written exercises should be submitted together in a single email to with the subject line DDG19A2.

Assignment 1 (Coding): Exterior Calculus (Due 2/26)

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 “$\star$” 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

  • 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 $i$.

EDIT: You can compute the ratio of dual edge lengths to primal edge lengths using the cotan formula, which can be found on Slide 28 of the Discrete Exterior Calculus lecture, or in exercise 36 of the notes (you don’t have to do the exercise for this homework).

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 by email to

Assignment 1 (Written): Exterior Calculus (Due 2/26)

The written portion of assignment 1 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 a few things in mind:

  • 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.

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 assignment, so that you can enjoy all the adventures yet to come…

This assignment is due on Tuesday, February 26.

Assignment 0 (Written): Combinatorial Surfaces — due 2/7

For the written part of your first homework you will do some exercises that will help familiarize you with basic descriptions and representations of combinatorial surfaces (simplicial surfaces, adjacency matrices, halfedge meshes), which will help prepare you to work with such surfaces as we continue through the course. (If any of this stuff seems abstract right now, don’t worry: we’ll use it over and over again to implement “real” algorithms starting in just a couple weeks!)

You must complete 8 out of 15 exercises in the Written Exercises section of Chapter 2 of the course notes. You may choose any set of 8 exercises you like, but if you do more than 8, please mark clearly on your submission which ones you would like us to grade.

The assignment is due on February 7, 2019 at 5:59:59pm Eastern (not at midnight!). Further hand-in instructions can be found on this page.

Assignment 0 (Coding): Combinatorial Surfaces — due 2/7

For the coding portion of your first assignment, you will implement some operations on simplicial complexes which were discussed in class and in Chapter 2 of the course notes. Once implemented, you will be able to select simplices and apply these operations to them by clicking the appropriate buttons in the viewer (shown above).

Getting Started

  • Please download or clone the files in 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.
  • For this assignment, you will need to implement the following routines in projects/simplicial-complex-operators/simplicial-complex-operators.js:
    • assignElementIndices
    • buildVertexEdgeAdjacencymatrix
    • buildEdgeFaceAdjacencymatrix
    • buildVertexVector
    • buildEdgeVector
    • buildFaceVector
    • star
    • closure
    • link
    • isComplex
    • isPureComplex
    • boundary


  • This assignment comes with a viewer projects/simplicial-complex-operators/index.html which lets you apply your operators to simplices of meshes and visualize the results.
  • Selecting simplices will not work until you fill in the assignElementIndices function.
  • The assignment also comes with a test script tests/simplicial-complex-operators/text.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).
  • All browsers come with tools for debugging (for instance the JavaScript Console in Chrome).

Submission Instructions

Please rename your simplicial-complex-operators.js file to simplicial-complex-operators.txt and submit it in a zip file called to