FourierCurve is a Java applet designed to experiment with periodic curves. A plane curve is uniquely determined by its curvature κ by the formula
If κ is periodic then so is the curve; this can be achieved by choosing the curvature as the a real Fourier series
The numbered buttons in the applet correspond to the Fourier coefficients ck in the curvature κ. The zero button, controlling the constant term c0 in the Fourier series, moves discretely through rational numbers and usually produces a closed curve. When c0 is an integer, then the curve generically does not close. The other coefficients (buttons) change the shape of the curve, and for noninteger c0, do not affect the periodicity.
More information on the question, when a plane curve is closed, can be found in When Is a Periodic Function the Curvature of a Closed Plane Curve?, J. Arroyo, O. J. Garay, and J. J. Mencía.
This project was created by Dhia Mansour in the course Experimental Mathematics I, taught by Dr Schmitt in the Winter Semester 2010/11 at the Mathematics Institute of the University of Tübingen. I especially thank Dr Nick Schmitt for his time and programming expertise. I also thank Prof Dr Franz Pedit for suggesting the idea.
The project was supported by SFB TR 71 B5.
© 2011 Dhia Mansour