Learning Geometric Algebra with CLUCalc

chapters, the software tool CLUCalc [133] is presented, which can help one to understand the algebra's algebraic and geometric properties. CLUCalc allows the user to enter geometric algebra expressions in much the same way as they are written in this text, and to investigate and visualize their outcome. In fact, a whole programming language, called CLUScript, has been developed by the author that combines structured programming with an intuitive representation of mathematical operations and simple methods to generate high-quality visualizations.

In particular, readers who are interested in the geometric representation of multivectors, as discussed in Chap. 4, will nd that CLUCalc facilitates their understanding considerably; this will lead to a quicker and more intuitive insight into the structure of geometric algebra. However, CLUCalc can also be used to implement complex algorithms, to generate illustrations for publications, and to give presentations including animated and userinteractive embedded 3D visualizations. All drawings shown in this text, for example, were generated using CLUCalc. Some additional interesting features of CLUCalc are:

• automatic analysis and visualization of multivectors with respect to their geometric content;

• user-interactive and animated visualizations;

• rendering of arbitrary LATEX text1 for the annotation of graphics and for texture mapping onto arbitrary surfaces;

• support for solving (multi)linear multivector equations;

• full support for the tensor representation of multivector equations, as discussed in Chap. 5;

• support for error propagation in algebraic operations;

• reading, writing, and texture mapping of images;

• control of external hardware through a serial port.

These features indicate that CLUCalc is more than a visualization tool for geometric algebra. In fact, it has been used in various applications. For example, the monocular pose estimation algorithm presented in Chap. 8 was implemented and tested with CLUCalc. CLUCalc was also used for the development and implementation of an inverse kinematic algorithm, which was used to control a robotic arm [95, 96]. CLUScripts that implement these algorithms can be found on [94]. Another application is the visualization of point groups, and wallpaper groups with CLUCalc [97, 142, 99, 98, 143, 100]. The visualizations scripts are freely available from [141]. The Space Group Visualizer, which is the rst tool to visualize all 230 space groups in 3D space, is also based on CLUCalc. There are many CLUScripts available that visualize physical and geometrical constellations. For example, there are scripts that simulate the gravitational three-body problem, visualize parts of the solar system, and demonstrate the constraints of the fundamental matrix. All of these can be found in the distribution of CLUCalc and on [133]. The current, freely available versions of CLUCalc for various operating systems, as well as a complete documentation of all its features, can be found at that site.