More informations about this program and its mathematical motivations are available at the homepage http://www.math.sciences.univ-nantes.fr/~rollin/index.php?page=flow
The Discrete Moment Map Flow (DMMF) program provides an approximation of the flow defined for discrete surfaces in the four dimensional space computed and reprensented as a realtime movie. The user can interact with various parameters of the flow and explore its behavior via keyboard and mouse inputs.
The DMMF program implements a particular evolution equation for quadrangular meshes in the four dimensional Euclidean space, associated to a quadrangulation of the torus that came up in our paper Discrete Geometry and Isotropic Surfaces (F. Jauberteau, Y. Rollin, S. Tapie).
When running the program, the user may see the flow evolving in real time. The program uses a collection of parametrized tori of R^4 (like the Clifford torus, Chekanov torus, (p,q)-knotted tori, etc...) to define the inital mesh of the flow, at time 0. Since it is difficult to visualize a surface in dimension 4, we choose a radial projection on the 3-sphere followed by a stereographic projection onto the 3-dimensional Euclidean space. The faces of the quadrangular mesh come with a color gradient which depends on their symplectic density. In all experiments so far, we observe that the flow is converging toward a Lagrangian quadrangular mesh (with zero symplectic density).
Limits of the flow provide many examples of discrete immersed Lagrangian surfaces in the four dimensional Euclidean space, at least from an experimental perspective.
The program is coded using the Processing language. In order to execute the source code, first install Processing3, which may be downloaded from the website: https://www.processing.org
The source code (i.e. every *.pde files) must be placed in a directory named dmmf, located in your Processing sketchbook.
- The source code may be downloaded from github as a zip file at https://github.com/yannrollin/dmmf
- It is also possible to download the file using the git command line. The idea is to clone the repository:
- Open a terminal
- Change the current directory to your sketckbook location. This may change from one system to another. For instance if your directory is ~/Processing type
cd ~/Processing - Clone the repository using the command
git clone https://github.com/yannrollin/dmmf.git - This should create a dmmf directroy in your sketchbook. You can now open the dmmf scketch in the processing IDE and run the program.
- You may keep the dmmf program up to date. For this, just go to the sketchbook directory
cd ~/Processing/dmmf/and run the commandgit pull
- Once this is done, launch the Processing IDE, and go to the menu File->Sketchbook to open the dmmf sketch. Then run the program by pressing the "play" button of the processing IDE.
Yann Rollin
Contact: yann.rollin@univ-nantes.fr
GNU GPL v3