CLUViz Geometric Algebra Animation Software

Introduction

As a part of our project, we've developed a series of images and animations using CLUViz to visually demonstrate how an encryption system based on rotations in 3-dimensions works. CLUViz is a freely available software tool for 3D visualizations and scientific calculations that was conceived and written by Dr. Christian Perwass. CLUCalc interprets a script language called ‘CLUScript’, which has been designed to make mathematical calculations and visualisations very intuitive.

Follow the link http://www.clucalc.info/ for more information regarding CLUCalc and http://www.raytrix.de/tl_files/downloads/CLUVizSetup.exe to download the software.

3D Rotations Process

The animation shows a 3-dimensional sphere in 3D Euclidean Space, with the associated based vectors e1, e2 and e3 outlined in white. As demonstrated by the analytical proof of commuting rotations in 3-dimensions in the Semester A Progress Report (Provide Link...), the initial and final message vectors m and m' will only be the same if Alice's Rotation Axis and Bob's Rotation Axis are parallel or either of their Rotation Angle's are equal to 0 degrees. As such, the animation displays the initial and final message vectors and all intermediate message vectors as a result of the rotations by both Alice and Bob.

Sliders at the bottom of the visualisation allow the user to interact with the module and change various elements to gain a better understanding of how it works.

• Two sliders are used to alter the e1 and e2 elements of the initial message vector m, with the e3 element being automatically set to the highest value that ensures that m isn't greater than a unit vector in length.
• Two sliders are used to alter the rotation angle of Alice and Bob between 0 and 360 degrees. Naturally the initial rotation of each is considered to be a clockwise rotation, while the second is counter-clockwise.
• The final slider is used to alter the rotation speed of Alice's Rotation Axis to 1 of 8 different speeds. An option is also provided to halt the animation at any stage, allowing the user to take a closer look.

As mentioned, the animation is conducted through the movement of a single element- that being Alice's Rotation Axis. This movement, allows the user to accurately determine how the difference between the initial and final message vectors alters with the difference between Alice's and Bob's Rotation Axis. As you can see, at the exact point where Alice's Rotation Axis crosses over with Bob's Rotation Axis, the initial message vector crosses over with the final message vector, hence indicating that the four rotations commute.