Assignment 5

Geometry Processing, Spring 2026

Michael Kazhdan

Assignment 5: The Wave Equation -- Implicit and Spectral Time-Stepping

Due on April 25 (Saturday) at 11:59 PM

Announcements

Overview

In this assignment you will implement implicit and spectral time-stepping of the wave equation PDE. Given initial displacements (with zero velocity), at each time-step you will compute the per-vertex displacements, which will be visualized by offsettting the geometry of the surface in the normal direction.

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 the WaveEquation/WaveEquation.cpp and SpectralWaveEquation/SpectralWaveEquation.cpp files.

After you download the files, the first thing to do is compile the WaveEquation and SpectralWaveEquation executables.

On a Windows Machine
Begin by double-clicking on GeometryProcessing.sln to open the workspace in Microsoft Visual Studios.
- Compile the WaveEquation and SpectralWaveEquation executables 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 executables WaveEquation and SpectralWaveEquation are 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 WaveEquation and SpectralWaveEquation executables. (You can specify WaveEquation and SpectralWaveEquation as targets so you don't rebuild everything.)
- The executables WaveEquation and SpectralWaveEquation are compiled in Release mode and will be placed in the directory Bin/Linux.

How to Invoke the Executable

WaveEquation

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 supporting the prescription of displacement impulses, and time-step the wave equation PDE to get the evolution of displacements over time.
To see the supported command line arguments, run the executable without an argyments:

% WaveEquation

To run the visualization, invoke the executable as:

% WaveEquation --in <input geometry> [--timeStep <integration time-step>] [--rain <rain frequency>]

With:

<input geometry> the geometry to be processed.
<integration time-step> the time-step used for time-stepping the PDE using implicit integration. (The default value is 0.001.)
<rain frequency> the number of frames between rain drops. (The default value is 0, corresponding to no rain.)

SpectralWaveEquation

% SpectralWaveEquation

To run the visualization, invoke the executable as:

% SpectralWaveEquation --in <input geometry> [--timeStep <integration time-step>] [--sDimension <spectral dimension>] [--rain <rain frequency>]

With:

<input geometry> the geometry to be processed.
<integration time-step> the time-step used for time-stepping the PDE using implicit integration. (The default value is 0.001.)
<rain frequency> the number of frames between rain drops. (The default value is 0, corresponding to no rain.)
<spectral dimension> the numer of (lower) eigen-functions to use solving the PDE. (The default value is 400.)

How to Interact with the Executable

WaveEquation and SpectralWaveEquation

Upon invocation, the executable will open an OpenGL window where local impulses can be prescribed.
The basic supported interfaces are:

[space] pauses/resumes the animation
[left mouse] rotates
[right mouse] zooms
[left mouse]+[ctrl] pans
'[' decreases the time-step
']' increases the time-step
[left mouse click] adds a negative impulse to the geometry under the moues
[right mouse click] adds a positive impulse to the geometry under the moues
'r' prompts for rain frequency

What You Have to Do

WaveEquation/WaveEquation.cpp:

You will need to modify this file to support the implicit time-stepping of the wave equation PDE in three places:

In the body of the WaveEquation constructor, you will need to initially set up the solver, WaveEquationViewer:_solver.
You will need to fill in the body of the WaveEquationViewer::_updateNumericalFactorization member function to update the solver when the system matrix has changed. (Recall that, since the connectivity of the mesh has not changed, you only need to update the numerical factorization, not the symbolic one.)
This will be invoked if you modify the size of the time-step (with the '[' and ']' keys).
You will perform the update of the displacements, stored in the WaveEquationViewer::_mesh.values data member, in the body of the WaveEquationViewer::animate member function, using implicit time-stepping.
[NOTE] Since the PDE is second order, updating to get the values at the current time-step requires tracking more than just the values in the previous time-step. Feel free to add additional data members to the WaveEquationViewer as needed.

SpectralWaveEquation/SpectralWaveEquation.cpp:

You will need to modify this file to support the spectral time-stepping of the wave equation PDE in five places:

You will need to implement the body of ForwardFourier to compute the Fourier coeffiicents from a signal (given the mass matrix and the pre-computed solution to the generalized stiffness eigen-problem).
You will need to implement the body of InverseFourier to compute the signal from its Fourier coeffiicents (the pre-computed solution to the generalized stiffness eigen-problem).
You will need to implement the body of GetWaveValueCoefficients to compute the Fourier coefficients of the displacement's value given the initial conditions and the current time.
You will need to implement the body of GetWaveVelocityCoefficients to compute the Fourier coefficients of the displacement's value given the initial conditions and the current time.
You wil need to implement the body of ShiftWaveCoefficients to temporally shift the initial conditions to prescribed time.

Please see the assignment description for additional details.

Question

For both WaveEquation and SpectralWaveEquation, what happens to the displacement over time?
For SpectralWaveEquation, how does performance and quality behave with increased spectral dimension?

What to Submit

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

Your modified Include/RiemannianMesh.[h/inl], GradientDomainFiltering/GradientDomainFiltering.cpp, and GeodesicsInHeat/GeodesicsInHeat.cpp files.
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.)