I originally learned about Cinderella from Leo Dorst (Amsterdam). He also used it to do some Geometric Algebra illustrations. Here I have elaborated his approach, producing a variety of online JAVA applets, part of them animated and part of them interactive. Especially the interactive applets invite you to explore the full meaning of geometric relationships in a visual way. I have freely drawn on the material of David Hestenes, New Foundations for Classical Mechanics, Kluwer 1999, 2nd ed. You can tour the applets without using the book. But if you have it, it might increase the fun of reading it. Another important source is Dorst, Mann and Bouma's geometric algebra MATLAB tutorial GABLE.

Particularly in the section on conics I have compiled an instructive variety of ways to obtain conics (points, pairs of intersection lines, circles, ellipses, hyperbolas and parabolas).

Finally, W.K. Clifford's circle chain theorem in the ordinary Euclidean plane refers to a "chain of theorems" of increasing complexity. Every one of this infinite sequence of theorems must be true for the whole to be true. You will find the illustrations for n=2 to n=8 circles through one point O.

The applets work with Netscape 6.2, Explorers 5 and 6, but not with Netscape 4.7. In each group the first applet may take some time, because your browser has to load the 412k cindyrun.jar file. Later on some of the more involved applets may also take a few minutes to appear on your screen. If the applets do not display properly, please try the page with support for Java 1.4.2 and newer.

Latest additions: 1) Luca Redaelli (Milan) illustrates how to visualize the structural mechanics of a simply supported beam in terms of Geometric Algebra. 2) Point groups in two dimensions (E. Hitzer)

Please send any suggestions+corrections+improvements to Eckhard Hitzer. Especially if you detect any geometric errors in the constructions!

Vectors

• Congruence
• Dilation
• Definition of addition
• Inverse (addition)
• Commutativity of addition
• Associativity of addition
• Addition and subtraction
• Vector division and vector - vector projection and rejection

Bivectors

• Duality
• Inner product
• Vector projection and rejection
• Meet of planes

Outer product

• Grassmann's definition
• Outer product a^b (interactive, changing b)
• Straight line by outer product (interactive, animated)
• Trivector (a^b)^c = a^b^c
• Trivector (b^c)^a = a^b^c
• Trivector (c^a)^b = a^b^c
• Displacement direction & orientation

Triangle

• The outer product is distributive
• Law of oriented sines

Rotations

• by unit bivector (2 dim.)
• by bivector exponential (2 dim.)
• by reflection (2 dim.)
• by reflection (3 dim.)
• by rotor
• by Euler angles
• interpolation
• Compare also Leo Dorst's Combination of rotations

Oscillations

• symmetric longitudinal
• antisymmetric longitudinal
• symmetric transverse
• antisymmetric transverse
• symmetric circular
• antisymmetric circular

Circular polarized waves

• left circular polarized wave (interactive)
• left circular polarized traveling wave
• right circular polarized wave (interactive)
• right circular polarized traveling wave
• right and left circular polarized wave (interactive)
• right and left circular polarized traveling wave

Conics

Clifford's circle chain theorem

Simple structural mechanics computations by Luca Redaelli (Milan)

Point groups in two dimensions

2D lattice symmetries

Soli Deo Gloria. Created with Cinderella by Eckhard Hitzer (Fukui).
Last Modified 11 Jun 2004
EMS Hitzer is not responsible for the content of external internet sites.