Rotor Polynomials: Algebra and Geometry of Conformal Motions

A typical topic of mathematics in high school is the computation of zeros of a function and in particular of a polyno- mial function. High school students also learn about the close relation between these zeros and factorizations of the polynomial. By using polynomial division, the polynomial degree can be reduced whence the computation of further zeros is simplified. Large parts of this theory remain true if real (or complex) numbers from high school are replaced by other types of numbers. In recent years it has been unveiled that some of these seemingly weird number systems are well suited for describing rigid body kinematics the theoretical foundation of many mechanical and robotic systems. Polyno- mial factorization in this context can be seen as the decomposition of complicated motions into simpler ones sim- ple enough to be amenable to mechanical creation by means of revolute joints. An important difference to high school polynomials is the non-uniqueness of factorizations. In the context of rigid body motions this is actually an advantage as it gives rise to different mechanical systems that can describe the same motion and may be combined advantageously. The project Rotor Polynomials: Algebra and Geometry of Conformal Motions aims to extend the factorization theory to polynomials that not only describe rigid body motions but conformal motions. While rigid body motions preserve lengths and dependent geometric entities such as angle, surface area, or volume, conformal motions just preserve angles. Being less relevant (but not irrelevant either!) for robotics, they still have important applications in disciplines where comparing shape is important: Computer graphics, archaeology, biology, or surveying to name but a few. Having algorithms for polynomial interpolation, factorization, and division paves the way to the applica- tion of advanced algebraic techniques in these areas. A deep understanding of polynomial factorization not only requires algebra but also geometry. It comes into play when viewing the numbers of conformal algebra as points and its polynomials as curves in a space of no less than 15 dimensions. While not being easily accessible to human imagination, it still provides a fruitful environment for ab- stract geometric and algebraic reasoning that has direct implications to the world that surrounds us.

As part of the project "Rotor Polynomials: Algebra and Geometry of Conformal Movements", motions of a rigid body and also of objects that can be deformed in the sense of conformal kinematics were investigated. Particular attention is paid to the decomposition (factorization) of these motions into simple elementary motions, for example rotations or translations. These decompositions are important for the mechanical realization of the motion with the help the simplest possible mechanisms. The elementary movements are generated by mechanical joints (revolute joint, translational joint). Decompositions into elementary motions are generically possible, but not unique. There are also very special motions with no or even an infinite number of decompositions. The conformal kinematics approach allows a deeper understanding of the causes of this initially surprising behavior. An important result of the project is a new geometric algorithm for the factorization of rational motions. In contrast to known factorization algorithms, it can be generalized to the very important class of "algebraic" motions. While rational movements can only be generated by very special mechanisms, algebraic movements occur with very general mechanisms. However, it was also shown that factorizations are only possible in special cases. Further results deal with examples of motions with an infinite number of factorizations. For these, factorization algorithms show an abnormal behavior that can be described by algebraic equations. Somewhat surprisingly, the class of these specially factorizable motions also includes well-known motions from conformal kinematics that have already appeared in other contexts. There, however, they were developed to describe certain physical phenomena and without any connection to factorization theory. Finally, the synthesis of simple rational motions from given elements, in this specific case the given path of a moving plane or straight line, was also investigated. It turns out that these can be described and calculated very easily. The development of mechanisms for generating these motions is reserved for future research.