About This Project

This personal project started when I was learning calculus in 1993. I found a book Curves and their Properties by Robert C Yates at school library. The book tells various interesting properties that relates different curves. For example, if you roll a parabola along a line, the trace of its focus is a catenary — a shape assumed by a hanging chain. If the parabola rolls on a congruent parabola, the trace of its vertex is a cissoid — a curve used by the Greeks to double the cube.

I wrote some graphical programs in Mathematica to see these properties, partly as a programming exercise. These programs matured into several Mathematica Packages over the years. In summer 1995 I worked as a intern at Wolfram Research Inc., the company that makes Mathematica. There I studied and wrote more Mathematica programs on plane curves as a project.

The goal of the project is to produce materials that demonstrate interesting properties of plane curves visually. Many concepts or properties of plane curves such as cusp, tangent, evolute, involute, envelope, are more readily explained by a illustration or animation. The mathematics in these pages are beginning college level. Overall, this project is designed to educate and entertain.

At Wolfram Research, I got acquainted with the World Wide Web. It seems to be the perfect medium for making my notes public. I started to convert my notes to html in February 1996.

Note that the historical notes about these curves are gathered from second-hand resources. (That is, books and articles written by math historians.)

As of 2004-12, this site contains 80 animations, 110 Geometer's Sketchpad files, 450 illustrations, 90 Mathematica notebooks, and 90 HTML files. I hope you enjoy them.


This project is based on materials from several books (See: bibliography). In the beginning, the main reference and motivation was: Curves and Their Properties by Robert C Yates, 1952.

During my intern at Wolfram, many people have helped me in Mathematica programing. Especially Matthew Cook, Michael Trott, and Robby Villegas. Thanks to Terry Rob who gave me many good advice at a early stage of the project.

There are few companies I like to thank. Wolfram Research, Inc., for letting me work on this project during my intern in 1995. Also thanks to the following companies who donated their software.

Copyright and Usage

Every page on this site is copyrighted by Xah Lee unless otherwise indicated. These works are made in 1995 to 2004. If you need to include a image or two in your academic work, please feel free to do so. Please give proper credit such as “Image by Xah Lee (, used with permission”. I would appreciate it to hear from you.

If you need more than 3 images, or for other usage outside of education including commercial publications, please contact me via email at with subject Plane Curves exactly. (the subject line is to make sure your email passes thru spam filters.) High-resolution images are available for $10 each.