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
Supervisor
Department
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.