Each animation is embedded in a web page that elaborates on what you will see, but some web-browsers may not like those embeddings; for that reason I've provided direct links to each of those embedded animations on the web page in which it's supposed to appear. If clicking on a link from this page opens a page that fails to display a movie, click on the link to that movie.

You can save these animations to your hard drive. Performance may improve when you run them from copies stored on your own hard drive. Select "Save As" from your browser's "File" menu. Set the format to "Source" and then save to your hard drive. For machines that don't have QuickTime installed, control-click (Mac) or right-click (Windoze) on the direct link to the .mov file you want. Use the resulting menu to download the movie; you can transfer it to another machine using a Zip disk, a flash drive, or a network connection.

Bill Emerson, Brad Kline, and I gave an MAA Minicourse in creating and exporting animations like these to the Web at the Joint Mathematics Meetings in New Orleans during January of 2001 and in San Diego during January of 2002. Bill and I also gave a workshop at the Spring 2002 meeting of the Rocky Mountain Section of the MAA in Laramie, WY, in April of 2002. We're always happy to discuss these techniques. Write.

Owing to the improved graphics capabilities that Wolfram Research, Inc., has built into their recent release of version 6 of Mathematica, I'm currently reworking many of the animations accessible from this page. Thus, some of what you can find here has changed recently, and more may change before long.

If you found this page through a web search engine, you may be looking at a copy that your search service cached. In that case, you may not be seeing the page as it currently exists. So you may want to visit Mathematics Animated.

Every once in a while, I get a report of one or more non-functioning animations. If you encounter one, please let me know.

Permission is granted for non-commercial educational use; all other rights reserved.

The pictures on this page do not move; click on a picture or on one of its associated links in order to access the movies.

A Visual Proof of the Pythagorean Theorem: This is an animated, visual proof of the Pythagorean Theorem.

A Visual Proof of Pappus' Generalization of the Pythagorean Theorem: This is an animated, visual proof of Pappus' theorem.

The Sine Curve: This animation shows how a point moving around the unit circle generates the sine function.

The Tangent Curve: This animation shows how a point moving around the unit circle generates the tangent function.

The Conic Sections: This shows how a plane intersects a cone to form the curves traditionally known as the conic sections. It's based on an idea I found on Preston Nichols' web site at Wittenberg University. (06/26/07)

Tracing Out An Ellipse: Using the definition to trace out an ellipse. (07/14/07)

A Proof of Quetelet & Dandelin: The Quetelet/Dandelin proof that a plane parallel to a generator of a cone intersects that cone in a curve that satisfies the focus/directrix definition of a parabola. (06/24/07)

Another Proof of Quetelet & Dandelin: The Quetelet/Dandelin proof that a plane whose angle from the vertical is greater than the vertex angle of a cone meets that cone in an ellipse. (06/24/07)

An Elliptic Reflector: A light pulse is emitted from one focus of an elliptical reflector and is reflected to the other focus. (06/28/07)

A Parabolic Reflector: A planar wave-front enters a parabolic reflector and is reflected to the focus of the parabola. (06/29/07)

A Hyperbolic Reflector: The right half of the hyperbola x² – y² = 1 is a double-sided reflector here. The left focus emits a pulse of red light. A short time later, the right focus emits a pulse of green light, timed so that the spherical wavefronts resulting from the two pulses both reach the vertex of the reflector simultaneously. Can you predict what happens then? (06/29/07)

The Moving Secant Line: Approximating a tangent line by secant lines.

Measuring Slope and Plotting the Derivative: Shows how to use slopes of tangent lines to plot the derivative.

The Cycloid: The curve traced out by a point attached to a circle that rolls along a straight line without slipping.

An Epicycloid: The curve traced out by a point attached to a small circle that rolls without slipping around the outside of another circle.

A Hypocycloid: The curve traced out by a point attached to a small circle that rolls without slipping around the inside of a larger circle.

Visualizing the Fundamental Theorem of Calculus: This movie helps students visualize the area function that lies at the center of the Fundamental Theorem of Calculus.

A Simple Polar Area Problem: Find the area inside one loop of a curve given in polar coordinates.

A Standard Polar Area Problem: Find the area inside one curve but outside another.

A Harder Polar Area Problem: Find the area inside one curve but outside another.

A Volume of Revolution About The X-Axis: The volume of revolution generated by the region between the x-axis and the curve y = Sqrt[x], with x between 1 and 4. (11/18/07)

Another Volume Of Revolution About The X-Axis: The volume of revolution that results when another plane region is revolved about the x-axis. (11/18/07)

A Detailed Look at a Volume of Revolution About the X-Axis: Generating a surface by revolution and then filling it. (06/26/07)

A Volume of Revolution About the Y-Axis: A volume of revolution generated by revolving a plane region about the y-axis. (11/18/07)

Integrating Over a Region in the Plane: y-axis first: How should we set up an integral over a plane region?

Integrating Over a Region in the Plane: x-axis first: How should we set up an integral over a plane region?

The Moving Triplet: The moving triplet is made up of the unit tangent vector, the unit normal vector, and the unit binormal vector. (11/25/07)

How to Make a Contour Map: Shows how a contour map reflects the surface from which it comes. (07/06/07)

A surface pictured with a tangent plane: Shows how a tangent plane makes contact with a surface.

A Singular Surface in Three Dimensions: A standard example from multivariable calculus. (06/24/07)

A Non-Differentiable Surface: Another standard example from multivariable calculus. (07/10/07)

Another Non-Differentiable Surface: A less familiar example from multivariable calculus.

A Singular Surface in Three Dimensions: This surface has a very interesting singularity at x = 0, y = 0. (06/24/07)

A vibrating drumhead: The vibrations of a drumhead after a single, centrally placed hit with a drumstick.

Another vibrating drumhead: The vibrations of a drumhead after a single hit with a drumstick at a point halfway from the center to the rim.

Inversion In The Complex Plane: Illustrates the action of the reciprocal function, ƒ(z) = 1 ⁄ z, in the complex plane.

More Inversion In The Complex Plane: Another look at the action of the reciprocal function in the complex plane.

Still more Inversion In The Complex Plane: A third look at the action of the reciprocal function in the complex plane.

Squaring In The Complex Plane: Illustrates the action of the squaring function in the complex plane.