Course detail

Algebras of rotations and their applications

FSI-9ARA Acad. year: 2025/2026 Summer semester

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

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.

The study programmes with the given course

Programme D-APM-P: Applied Mathematics, Doctoral, recommended course

Type of course unit

 

Lecture

20 hours, optionally

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.