Assignment 2

Geometry Processing, Spring 2026

Michael Kazhdan

Assignment 2: Combinatorial Smoothing

Due on February 21 (Saturday) at 11:59 PM

Overview

In this assignment you will impliment combinatorial smoothing, comparing the performance of explicit and implicit time-stepping.

The code skeleton can be downloaded here.
Example models you can use for testing can be found here.
A detailed description of the assignment can be found here.

Getting Started

You should use the code (GeometryProcessing.zip) as a starting point for your assignment. We provide you with numerous files, but you should only have to change CombinatorialSmoothing/CombinatorialSmoothing.cpp.

After you download the files, the first thing to do is compile the CombinatorialSmoothing executable.

On a Windows Machine
Begin by double-clicking on GeometryProcessing.sln to open the workspace in Microsoft Visual Studios.
- Compile the CombinatorialSmoothing executable by clicking on "Build" and then selecting "Build Solution". Please be sure that you are compiling for Release under x64 mode, as only those project settings were set. (If you need to compile under Debug, you will need to copy over the settings.)
- The executable CombinatorialSmoothing.exe is compiled in Release mode for the 64-bit architecture and will be placed in the directory Bin/x64.
On a Unix/Mac Machine
- Type make to compile the CombinatorialSmoothing executable.
- The executable CombinatorialSmoothing is compiled in Release mode and will be placed in the directory Bin/Linux.

How to Invoke the Executable

The executable runs on the command line. It takes a geometry file as input, either in ply or in obj file format (determined by the file extension) and opens up an OpenGL-based viewer visualizing progressive smoothing the vertex positions.
To see the supported command line arguments, run the executable without an argyments:

% CombinatorialSmoothing

To smooth the geometry, invoke the executable as:

% CombinatorialSmoothing --in <input geometry> --stepSize <gradient-descent step-size>

With:

<input geometry> the geometry to be processed.
<gradient-descent step-size> the step-size to be used for time-stepping the gradient-descent PDE. (Default value is 10^-4.)

The executable also supports an optional --implicit flag. If specified, the smoothing should be performed using implicit time-stepping.

How to Interact with the Executable

Upon invocation, the executable will open an OpenGL window where the smoothing will be animated.
The basic supported interfaces are:

[space] pauses/resumes the animation
[left mouse] rotates
[right mouse] zooms
[left mouse]+[ctrl] pans

What You Have to Do

You will need to mofiy the CombinatorialSmoothing/CombinatorialSmoothing.cpp file to support explicit and implicit time-stepping of the gradient-descent PDE.
To achieve this, you will have to make changes in two places:

In the CombinatorialSmoothingViewer::CombinatorialSmoothingViewer constructor you will need to set the combinatorial Laplacian, CombinatorialSmoothingViewer::_L, and initialize the sparse Cholesky solver, CombinatorialSmoothingViewer::_solver. (Note that the identity matrix, CombinatorialSmoothingViewer::_Id, is already set for you.)
In the CombinatorialSmoothingViewer::animate member function you will need to update the the vertex positions, Mesh::vertices, to represent the values at the next time-step.
- If CombinatorialSmoothingViewer::_implicit is set to true, you should be using implicit time-stepping.
- Otherwise, you should be using explicit time-stepping.
The size of the time-step is stored in the CombinatorialSmoothingViewer::_stepSize member data.

Please see the assignment description for additional details.

Questions

How do the explicit and implicit solvers compare when using a small time-step?
How do the explicit and implicit solvers compare when using a large time-step?

What to Submit

Please submit your implementation on canvas.
For this assignment, you should submit:

Your modified CombinatorialSmoothing/CombinatorialSmoothing.cpp file.
A write up describing what you have implemented. (Particularly, if things did not work as expected, please provide guidance that would allow us to give you partial credit.)