Course detail
Algebras of rotations and their applications
FSI-9ARA Acad. year: 2020/2021 Summer semester
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.
Language of instruction
Czech
Supervisor
Department
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.
Prerequisites
Foundations of linear algebra.
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.
Aims
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.
The study programmes with the given course
Programme D-APM-K: Applied Mathematics, Doctoral, recommended course
Programme D4P-P: Applied Natural Sciences, Doctoral
branch D-APM: Applied Mathematics, recommended course
Type of course unit
Lecture
20 hours, optionally
Teacher / Lecturer
Syllabus
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.