*L'on voit que Jésus-Christ, achevant ce que Moïse avait commencé,
a voulu que la divinité fût l'objet, non seulement de notre
crainte et de notre vénération, mais encore de notre amour et de
notre tendresse.*

*Essai de
théodicée - Préface et abrégé*,
Gottfried
Wilhelm Leibnitz (1646-1716)

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!

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

- 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

- The outer product is distributive
- Law of oriented sines

- 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

- symmetric longitudinal
- antisymmetric longitudinal
- symmetric transverse
- antisymmetric transverse
- symmetric circular
- antisymmetric circular

- 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

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.