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Lecturer(s)
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Káňa Michal, doc. RNDr. Ph.D.
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Vincík Jáchym, prof. RNDr. Ph.D.
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Course content
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Affine mappings in affine spaces and their classification. Isometries and similarities in Euclidean spaces. Composition of transformations. Projective embedding, projective space and its subspaces, Principle of duality. Projective mappings and transformations. Quadrics (especially conic sections in the plane and quadrics in the 3-dimensional space), their projective, affine and metric classification. Geometry and group theory, classification of geometries. Klein-Cayley's geometries.
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Learning activities and teaching methods
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Interactive lecture, Lecture supplemented with a discussion, E-learning, Cooperative instruction, Discussion, Task-based study method, Students' self-study, Lecture, Practicum
- Contact hours
- 39 hours per semester
- Preparation for an examination (30-60)
- 50 hours per semester
- Preparation for formative assessments (2-20)
- 20 hours per semester
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| prerequisite |
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| Knowledge |
|---|
| to describe and explain advanced principles from linear algebra and vector calculus |
| to describe and explain selected procedures for solving problems of affine and Euclidean geometry |
| to understand basic concepts of group theory |
| Skills |
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| to apply adopted methods to selected geometric problems in n-dimensional affine and Euclidean spaces |
| to use the apparatus of linear algebra for mid-level problems |
| Competences |
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| N/A |
| N/A |
| learning outcomes |
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| Knowledge |
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| to understand affine, isometric and similar mappings, derive their equations, analyse their properties and applicability and decide whether a given mapping is affine, isometric or similar |
| to define projective space and its subspaces, understand their mutual relations and work with them using basic methods of projective geometry (especially using the Principle of Duality) |
| to define and classify quadrics in n-dimensional projective, affine and Euclidean space, convert their expressions into canonical forms, recognize them and use them actively |
| to classify projective mappings and understand the structure of the projective group |
| Skills |
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| to provide logical proofs of selected important theorems of the studied theory |
| to compare and relate different types of geometry (e.g. projective, affine, Euclidean, hyperbolic, elliptic, Möbius) |
| to analyze the basic characteristics of quadrics and use their properties to solve selected problems based on specific situations in real life and engineering practice |
| to demonstrate the fundamental propositions of an abstract theory using an appropriate combination of examples and counterexamples, look for analogies and make generalizations |
| Competences |
|---|
| N/A |
| N/A |
| teaching methods |
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| Knowledge |
|---|
| Lecture |
| Lecture supplemented with a discussion |
| Interactive lecture |
| Practicum |
| Task-based study method |
| Self-study of literature |
| Discussion |
| Textual studies |
| Skills |
|---|
| Lecture |
| Lecture with visual aids |
| Interactive lecture |
| Task-based study method |
| Textual studies |
| Discussion |
| Self-study of literature |
| Competences |
|---|
| Lecture |
| Lecture with visual aids |
| Interactive lecture |
| Practicum |
| Task-based study method |
| Textual studies |
| Self-study of literature |
| Discussion |
| assessment methods |
|---|
| Knowledge |
|---|
| Combined exam |
| Test |
| Seminar work |
| Skills |
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| Combined exam |
| Test |
| Seminar work |
| Competences |
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| Combined exam |
| Test |
| Seminar work |
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Recommended literature
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Audin, Michéle. Geometry. Berlin : Springer, 2003. ISBN 3-540-43498-4.
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Casas-Alvero, E. Analytic Projective Geometry. Zürich, 2014. ISBN 978-3-03719-138-5.
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Coxeter, Harold Scott MacDonald. The real projective plane : with an appendix for mathematica by George Beck : Macintosh version. 3rd ed. New York : Springer, 1993. ISBN 0-387-97889-5.
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Čižmár, J. Grupy geometrických transformácií. 1. vyd. Bratislava : Alfa, 1984.
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Lávička, M. Geometrie 2: Projektivní prostory, geometrická zobrazení a kvadriky. Plzeň, 2021.
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Reid., M., Szebdroi, B. Geometry and topology. Cambridge University Press, 2005. ISBN 978-0-521-61325-5.
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Sekanina, M. a kol. Geometrie. 1. díl..
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Sekanina, M. a kol. Geometrie. 2. díl.. 1. vyd. Praha : Státní pedagogické nakladatelství, 1988.
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Sekaninová, A. a Janyška, J. Analytická teorie kuželoseček a kvadrik. MUNI, Brno, 2001. ISBN 80-210-2604-9.
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