Geometrical Algorithms (FSI-0AV)

Academic year 2022/2023
Supervisor: doc. Mgr. Petr Vašík, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:
Introduction of advanced mathematical structures and their applications in engineering.
Learning outcomes and competences:
Enhancement of skills that are necessary for applying advanced mathematical structures.
Prerequisites:
Elementary notions of algebra and linear algebra.
Course contents:
A survey on advanced structures om multi-linear algebra and, consequently, their application in Euclidean space transformation. Introduction to the theory of geometric algebras and algorithms for elementary tasks of analytic geometry. Simple geometric algorithms for the rigid body motion using Euclidean transformations.
Teaching methods and criteria:
The course is taught in lectures explaining the basic principles and theory of the discipline. Calculations in an appropriate software will be presented.
Assesment methods and criteria linked to learning outcomes:
Graded assessment: semester project, oral exm.
Controlled participation in lessons:
Lectures, non-compulsory attendance.
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
Course curriculum:
    Lecture 1. Review: vector space, basis, dimension, scalar product, bilinear and quadratic forms.
2. Euclidean transformations of two and three dimensional space.
3. Inner and outer product, exterior algebra.
4. Clifford algebra.
5.-6. Introduction to geometric algebras, special cases of CRA (G3,1) and CGA (G4,1).
7.-8. Computation in geometric algebras.
9. Fundamental tasks of analytic geometry in geometric algebras.
10. Software for symbolic calculations and visualisation in geometric algebras (Python, CLUCalc).
11.-12. Euclidean transformations in geometric algebra, rigid body motion.
13. Consultations to semester project.
Literature - fundamental:
1. DORST, Leendert, D.H.F FONTIJNE a Stephen MANN. Geometric algebra for computer science: an object-oriented approach to geometry. Rev. ed. Burlington, Mass.: Morgan Kaufmann Publishers, c2007. Morgan Kaufmann series in computer graphics. ISBN 978-0-12-374942-0.
2. HILDENBRAND, Dietmar. Foundations of geometric algebra computing. Geometry and computing, 8. ISBN 3642317936.
3. HILDENBRAND, Dietmar. Introduction to geometric algebra computing. Boca Raton, 2018. ISBN 978-149-8748-384.
5. PERWASS, Christian. Geometric algebra with applications in engineering. Berlin: Springer, c2009. ISBN 354089067X.
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.
Literature - recommended:
1. HILDENBRAND, Dietmar. Introduction to geometric algebra computing. Boca Raton, 2018. ISBN 978-149-8748-384.
2. MOTL, Luboš a Miloš ZAHRADNÍK. Pěstujeme lineární algebru. 3. vyd. Praha: Karolinum, 2002. ISBN 80-246-0421-3.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
CŽV full-time study CZV Bases of Mechanical Engineering -- GCr 3 Compulsory 1 1 S
B-MAI-P full-time study --- no specialisation -- GCr 3 Compulsory 1 2 S