Algebras of rotations and their applications (FSI-9ARA)

Academic year 2022/2023
Supervisor: doc. Mgr. Petr Vašík, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech or English
Aims of the course unit:
Understanding the importance of advanced mathematical structures by their application in engineering.
Learning outcomes and competences:
The ability to apply groups of transformations in the task of rigid body motion. Implementation of simple motion algorithm in geometric algebra setting.
Prerequisites:
Foundations of linear algebra.
Course contents:
Survey on mathematical structures applied on rigid body motion, particularly various representations of Euclidean space and its transformations. We will focus on geometric algebras, i.e. Clifford algebras together with a conformal embedding of a Euclidean space.
Teaching methods and criteria:
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.
Controlled participation in lessons:
Lectures, attendance is non-compulsory.
Type of course unit:
    Lecture  10 × 2 hrs. optionally                  
Course curriculum:
    Lecture 1. Review of elementary notions of linear algebra: vector space, basis, change of basis matrix, transformation matrix.
2. Bilinear and quadratic forms, scalar product, outer product, exterior algebra.
3. Representations of a Euclidean space. quaternions, affine extension.
4. Clifford algebra.
5. Geometric algebra. conformal embedding of a Euclidean space.
6. Object representation, duality, inverse.
7. Euclidean transformations.
8. Foundations of geometric (Clifford) algebras, specifically the cases of G2, CRA (G3,1), CGA (G4,1) and PGA (G2,0,1).
9. Analytic geometry in CGA setting.
10. Algorithms for rigid body motion.
Literature - fundamental:
1. PERWASS, Christian. Geometric algebra with applications in engineering. Berlin: Springer, c2009. ISBN 354089067X.
2. MURRAY, Richard M., Zexiang LI a Shankar. SASTRY. A mathematical introduction to robotic manipulation. Boca Raton: CRC Press, c1994. ISBN 0849379814.
3. SELIG, J. M. Geometric fundamentals of robotics. 2nd ed. New York: Springer, 2005. ISBN 0387208747.
4. HILDENBRAND, Dietmar. Foundations of geometric algebra computing. Geometry and computing, 8. ISBN 3642317936.
5. HILDENBRAND, Dietmar. Introduction to geometric algebra computing. Boca Raton, 2018. ISBN 978-149-8748-384.
6. MOTL, Luboš a Miloš ZAHRADNÍK. Pěstujeme lineární algebru. 3. vyd. Praha: Karolinum, 2002. ISBN 80-246-0421-3.
7. 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.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
D-APM-K combined study --- -- DrEx 0 Recommended course 3 1 S
D-APM-P full-time study --- -- DrEx 0 Recommended course 3 1 S