• Drag the numbers to get new curves. Add or remove numbers with the plus and minus buttons, and reset them with the refresh button.
• Clicking a number changes the color. Clicking the three dots button changes the style.
• The graph in the background is the curvature. Toggle it on and off with the “wave” button.
• Print the curve with the print button.

FourierCurve

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 c_k 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.

Download FourierCurve.jar.

The project was supported by SFB TR 71 B5.

© 2011 Dhia Mansour