M4522 Geometry 3

Faculty of Science
Spring 2020
Extent and Intensity
2/2/0. 3 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
Teacher(s)
prof. RNDr. Josef Janyška, DSc. (lecturer)
Mgr. Kristýna Bisová (seminar tutor)
Bc. Karolína Klempířová (seminar tutor)
Guaranteed by
prof. RNDr. Josef Janyška, DSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Thu 14:00–15:50 M1,01017
  • Timetable of Seminar Groups:
M4522/01: Mon 18:00–19:50 M6,01011, K. Bisová
M4522/02: Tue 18:00–19:50 M5,01013, K. Klempířová
Prerequisites
Knowledge of M1500 Algebra 1, M2500 Algebra 2 and M3521 Geometry 2.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The goal of the course is:
- analytical theory of affine mappings of affine spaces, especially in plane and three-dimensional space;
- analytical theory of isometric and similar mappings of Euclidean point spaces, especially in plane and three-dimensional space;
- theory of circle inversion in a plane;
- mastering relevant computing techniques;
- supporting students' spatial imagination.
Learning outcomes
Student will be able to:
- solving problemses with affine mappings;
- solving problems using isometric and similar mappings;
- solving problemss using circle inversion.
Syllabus
  • Invariant subspaces of linear transformations of the vector space.
  • Invariant subspaces of orthogonal transformations of a vector space with a scalar product.
  • Afine mappings:
  • - associated linear mappings;
  • - coordinate expression of affine mappings;
  • - affine transformations of an affine space, fix points and eigenvectors;
  • - homotheties;
  • - basic affine mappings, decomposition of an affine mapping into basic affine mappings.
  • Isometric mappings:
  • - coordinate expression of isometric mappings;
  • - group of isometric transformations, symmetries with respect to subspaces;
  • - decomposition of isometries by reflections;
  • - classification of isometries in plane and space.
  • Similar mappings.
  • - coordinate representation of similar mappings;
  • - a group of similarities;
  • - decomposition of similar mappings to homothetic transformations and isometriess.
  • Circle inversion and its using to solve planimetric problems.
Literature
    recommended literature
  • JANYŠKA, Josef. Geometrická zobrazení, Učební text, jarní semestr 2017
    not specified
  • SEKANINA, Milan. Geometrie. 1. vyd. Praha: Státní pedagogické nakladatelství, 1988, 307 s. info
  • HORÁK, Pavel and Josef JANYŠKA. Analytická geometrie. Brno: Masarykova univerzita v Brně, 1997, 151 s. ISBN 80-210-1623-X. info
  • SEKANINA, Milan. Geometrie. 1. vyd. Praha: Státní pedagogické nakladatelství, 1986, 197 s. URL info
  • KADLEČEK, Jiří and Jan TROJÁK. Geometrie. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1984, 249 s. info
  • BOČEK, Leo and Jaroslav ŠEDIVÝ. Grupy geometrických zobrazení. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1979, 213 s. info
  • ŠMARDA, Bohumil. Analytická geometrie. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1978, 157 s. info
Teaching methods
Lectures: theoretical explanations with examples of practical applications.
Exercises: solving problems focused on basic concepts and theorems, individual problem solving by students.
Assessment methods
Examination consists of two parts: written and oral.
Current requirements: Written tests in exercises.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
The course is also listed under the following terms Spring 2008 - for the purpose of the accreditation, Spring 2011 - only for the accreditation, Spring 2000, Spring 2001, Spring 2002, Spring 2003, Spring 2004, Spring 2005, Spring 2006, Spring 2007, Spring 2008, Spring 2009, Spring 2010, Spring 2011, Spring 2012, spring 2012 - acreditation, Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, spring 2018, Spring 2019.
  • Enrolment Statistics (recent)
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