• # Projects for Differential Geometry

Important Information for the MAA Edition! The second edition has been expanded by over 100 pages to try to take account of various useful suggestions made by users since the appearance of the first edition. The version of Maple that has been tested for all Maple work in the new edition is Maple 10.

# Enneper's Surface

The point of this book is to mix together differential geometry, the calculus of variations and some applications (e.g. soap film formation, constrained particle motion, Foucault's pendulum) to see how geometry fits into science and mathematics. The book includes many Maple procedures that allow students to view geometry and calculate things such as Euler-Lagrange equations.

In particular, Chapter 5 on geodesics contains a procedure to plot geodesics on surfaces and this procedure gives beautiful illustrations of the Clairaut relation for example. The same type of procedure also allows students to visualize the motion of a particle constrained to move in bowls (of various shapes) under gravity. These are the kinds of connections between geometry and applications which I like and which I think are important for students to see. Here is an example of a geodesic on the surface of revolution obtained by revolving the Witch of Agnesi about the x-axis. Notice how the geodesic is bounded between two parallels. This is the Clairaut relation in action. By the way, the following picture is only a first attempt at using Maple to create a JPEG file for the web --- better things will surely come later!

The picture above was created by a procedure called `plotgeo' which may be found in Chapter 5 of the book. Here are a few other examples of geodesics on surfaces constructed from this procedure.
Geodesic on a Torus Geodesic on a Cylinder