Classical curves known as elastica were found by Leonhard Euler in 1744. An elastic curve models a bent rod minimizing bending energy. This theory has been generalized to finite type curves, which form an infinite hierarchy starting at the elastic curves.
SpaceCurve is a Java Web Start designed to experiment with finite type curves.
It requires the Java Runtime Environment which can be downloaded from here. The source code of the application can be downloaded from this link.
The curves are computed from the differential equations
where γ is the curve, F its frame, and X an associated polynomial Killing field. The initial conditions for the above equations can be changed interactively by manipulating the red points on the graph of the Web Start.
For more information on polynomial Killing fields see Minimal tori in S4, D. Ferus, F. Pedit, U. Pinkall, I. Sterling.
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To add a branch point (red), double-click in empty space on the graph. To remove a branch point, double-click on any branchpoint. To change the sym point (green), slide it along the real axis. |
 
 | Toggle the isospectral deformation of the curve. |
 
| Toggle the branchpoints to be symmetric to the imaginary axis. |
 | Reset the curve to the current window size. |
 | Change the appearance of the curve and background. |
 | Iterate through the list of curve presets. |
This project was created by Ulf Wagner. It is based on the courses Experimental Mathematics I and II, taught by Dr Nick Schmitt in the Summer Semester 2011 and Winter Semester 2011/2012 at the Mathematics Institute of the University of Tübingen, and is part of my diploma thesis. I especially thank Dr Nick Schmitt for his time, patience and programming expertise. I also thank Dhia Mansour for assisting me throughout the courses and Prof Dr Franz Pedit for suggesting the idea.
The project was supported by SFB TR 71 B5.
© 2016 Ulf Wagner