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Lecturer(s)
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Káňa Michal, doc. RNDr. Ph.D.
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Breitfelder Ondřej, Mgr.
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Pařez Klaus, doc. RNDr. Ph.D.
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Course content
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Affine space, affine mappings and affine coordinate system. Affine subspaces and their parametric description. Equations of affine subspaces. Relative position of affine subspaces. Pencils of hyperplanes. Affine combination of points, barycentric coordinates. Vector spaces with scalar product. Euclidean space and Cartesian coordinate system. Orthogonality of Euclidean subspaces. Distances of Euclidean subspaces. Angles of Euclidean subspaces. Volumes of parallelotopes. Möbius extension of Euclidean space. Conics - ellipse, hyperbola, parabola. Intersections of the surface of a cone with a plane. Conic sections - quadratic curves in plane. Quadrics - quadratic surfaces in space. Selkected types of quadrics and their prroperties.
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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, Textual studies, Lecture, Practicum
- Contact hours
- 39 hours per semester
- Preparation for an examination (30-60)
- 50 hours per semester
- Preparation for formative assessments (2-20)
- 10 hours per semester
- Preparation for comprehensive test (10-40)
- 10 hours per semester
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| prerequisite |
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| Knowledge |
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| explain the basic lessons of plane, possibly spatial analytic geometry at the high school level |
| describe and explain the basic procedures for solving geometric problems |
| describe and explain the basic principles of linear algebra and vector calculus |
| describe and explain the basic principles of calculus |
| Skills |
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| apply the acquired procedures to elementary geometric problems at the secondary school level |
| calculate vectors, matrices and determinants and solve systems of linear and quadratic equations |
| use the calculus apparatus for basic and intermediate problems |
| Competences |
|---|
| N/A |
| N/A |
| N/A |
| N/A |
| learning outcomes |
|---|
| Knowledge |
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| define an affine space and introduce a suitable system of coordinates, understand the problem of affine subspaces, derive their equations and determine their relative positions |
| define Euclidean space, introduce the Cartesian coordinate system as a specialization of the general affine coordinate system, construct equations of orthogonal subspaces, determine distances and deviations of Euclidean subspaces |
| define and classify conics in the Euclidean plane, convert their expressions into canonical forms and recognize them |
| define and classify quadrics in three-dimensional Euclidean space, convert their expressions into canonical forms, and recognize and actively use them |
| Skills |
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| actively use the analytical method in solving mathematical and application problems |
| solve geometric problems with linear and quadratic objects |
| actively use the analytical method to visualise a variety of mathematical concepts |
| find and use application possibilities not only in geometry and other mathematical disciplines, but also in natural sciences, computer graphics, etc. |
| solve geometric problems using the synthetic method |
| Competences |
|---|
| N/A |
| N/A |
| teaching methods |
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| Knowledge |
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| Lecture |
| Lecture supplemented with a discussion |
| Practicum |
| Task-based study method |
| Self-study of literature |
| Skills |
|---|
| Lecture |
| Lecture supplemented with a discussion |
| Practicum |
| Task-based study method |
| Self-study of literature |
| Competences |
|---|
| Lecture |
| Lecture supplemented with a discussion |
| Practicum |
| Task-based study method |
| Self-study of literature |
| assessment methods |
|---|
| Knowledge |
|---|
| Combined exam |
| Test |
| Seminar work |
| Skills |
|---|
| 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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Bican, Ladislav. Lineární algebra a geometrie. Vyd. 1. Praha : Academia, 2000. ISBN 80-200-0843-8.
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Gallier, Jean. Geometric methods and applications : for computer science and engineering. New York : Springer, 2001. ISBN 0-387-95044-3.
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Horák, P. a Janyška, J. Analytická geometrie. Brno, 1997.
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Lávička, M. a Vršek J. Geometrie 1: Úvod do afinních a eukleidovských prostorů. Plzeň, 2020.
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Mahel a kol. Sbírka úloh z lineární algebry a analytické geometrie. ČVUT, 1980.
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Sekanina, M. a kol. Geometrie I., Státní pedagogické nakladatelství. Praha, 1986.
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Tarrida, R. Affine Mapsa, Euclidean Motions and Quadrics. London, 2011.
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