Algebras of rotations and their applications
FSI-9ARAAcad. year: 2020/2021
Survey on mathematical structures applied on rigid body motion, particularly various representations of Euclidean space and its transformations. In detail, we will study groups SO(2), SO(3) and their Lie algebras, groups Spin(2), Spin(3), quaternions, their construction, properties and applications. Introduction to geometric algebras.
Learning outcomes of the course unit
The ability to apply groups of transformations in the task of rigid body motion. Implementation of simple motion algorithm in geometric algebra setting.
Foundations of linear algebra.
Recommended optional programme components
Recommended or required reading
PERWASS, Christian. Geometric algebra with applications in engineering. Berlin: Springer, c2009. ISBN 354089067X. (EN)
MURRAY, Richard M., Zexiang LI a Shankar. SASTRY. A mathematical introduction to robotic manipulation. Boca Raton: CRC Press, c1994. ISBN 0849379814. (EN)
SELIG, J. M. Geometric fundamentals of robotics. 2nd ed. New York: Springer, 2005. ISBN 0387208747. (EN)
HILDENBRAND, Dietmar. Foundations of geometric algebra computing. Geometry and computing, 8. ISBN 3642317936. (EN)
HILDENBRAND, Dietmar. Introduction to geometric algebra computing. Boca Raton, 2018. ISBN 978-149-8748-384. (EN)
MOTL, Luboš a Miloš ZAHRADNÍK. Pěstujeme lineární algebru. 3. vyd. Praha: Karolinum, 2002. ISBN 80-246-0421-3. (CS)
GONZÁLEZ CALVET, Ramon. Treatise of plane geometry through geometric algebra. 1. Cerdanyola del Vallés: [nakladatel není známý], 2007. TIMSAC. ISBN 978-84-611-9149-9. (EN)
Planned learning activities and teaching methods
Lectures together with hosted consultations. Elementary notions nad their connections will be presented and explained.
Assesment methods and criteria linked to learning outcomes
Final exam is oral. It is necessary to know elementary notions, their definitions and basic properties. Implementation of a simple algorithm for rigid body motion is considered as a part of the exam.
Language of instruction
Understanding the importance of advanced mathematical structures by their application in engineering.
Specification of controlled education, way of implementation and compensation for absences
Lectures, attendance is non-compulsory.
Type of course unit
20 hours, optionally
Teacher / Lecturer
1. Review of elementary notions of linear algebra: vector space, basis, change of basis matrix, transformation matrix.
2. Groups SO(2), SO(3), definitions, properties, matrix representations.
3. Algebras so(2), so(3), definitions, properties, matrix representations.
4. Matrix exponential, Baker-Campbell-Hausdorff formula.
5. Moving frame method, piecewise constant input on so(3).
6. Groups Spin(2) and Spin(3) as a double-cover of groups SO(2) and SO(3), respectively. Their topological properties.
7. Algebra of quaternions and the identification of unit quaternions with the group Spin(3).
8. Analytic geometry in terms of quaternions and dual quaternions.
9. Foundations of geometric (Clifford) algebras, specifically the cases of G2, CRA (G3,1) and CGA (G4,1).
10. Analytic geometry in CGA setting.