Parametric Equations of Circles and Ellipses

This applet demonstrates how to describe and graph circles and ellipses using parametric techniques, using two equations rather than one. This approach also leads to the wonderful Lissajous Figures, named after Jules Antoine Lissajous 1822-1880.

Author and programmer: Ron Barrow

Instructions below See also:   Cartesian Equations of Circles and Ellipses    Other Parametric Equations

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How to Use this Applet

Instead of using a single equation to generate a curve, two parametric equations are used. Here the letter t denotes the parameter, and the x- and y-coordinates are calculated from the formulas.
You can vary the values of A, B, C and D using the buttons at the bottom. By varying A and C you can create a series of circles and ellipses. Varying B and D complicates the situation by drawing the strikingly beautiful Lissajous Figures.
You can hide the axes by unchecking the box marked "Show Axes". You can also generate a random figure by clicking the "Random" button with your mouse. Play around - enjoy yourself!