Course detail

Algebras of rotations and their applications

FSI-9ARAAcad. year: 2024/2025

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.

Language of instruction

Czech

Mode of study

Not applicable.

Guarantor

doc. Mgr. Petr Vašík, Ph.D.

Department

Institute of Mathematics (ÚM)

Entry knowledge

Foundations of linear algebra.

Rules for evaluation and completion of the course

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.
Lectures, attendance is non-compulsory.

Aims

Understanding the importance of advanced mathematical structures by their application in engineering.
The ability to apply groups of transformations in the task of rigid body motion. Implementation of simple motion algorithm in geometric algebra setting.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

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)
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)
MURRAY, Richard M., Zexiang LI a Shankar. SASTRY. A mathematical introduction to robotic manipulation. Boca Raton: CRC Press, c1994. ISBN 0849379814. (EN)
PERWASS, Christian. Geometric algebra with applications in engineering. Berlin: Springer, c2009. ISBN 354089067X. (EN)
SELIG, J. M. Geometric fundamentals of robotics. 2nd ed. New York: Springer, 2005. ISBN 0387208747. (EN)

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme D-APM-P Doctoral 1 year of study, summer semester, recommended course
  • Programme D-APM-K Doctoral 1 year of study, summer semester, recommended course

Type of course unit

 

Lecture

20 hod., optionally

Teacher / Lecturer

doc. Mgr. Petr Vašík, Ph.D.

Syllabus

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.