A lissajous curve is one that is described by a parametric equation, and a popular example is the logo for the ABC. However instead of creating them with an Arduino and mathematics, YouTube user 'mkvenkit' has used simple physics to do so. By mounting a mirror each on the end of the shafts of two small DC motors, and reflecting a laser pointer beam on the mirrors - the beam can be reflected at high speed to create some interesting displays, for example:
If this type of project is of interest, you will enjoy the book "Practical Arduino" by Jonathan Oxer and Hugh Blemings:
Using this book you can create your own Arduino-based designs, gain an in-depth knowledge of the architecture of Arduino, and learn the easy-to-use Arduino language all in the context of practical projects that you can build yourself at home. Get hands-on experience using a variety of projects and recipes for everything from home automation to test equipment. For more information and to order, click here.