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Course, academic year 2023/2024
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Geometry II - NMUM204
Title: Geometrie II
Guaranteed by: Department of Mathematics Education (32-KDM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2023
Semester: summer
E-Credits: 5
Hours per week, examination: summer s.:2/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: cancelled
Language: Czech
Teaching methods: full-time
Teaching methods: full-time
Guarantor: Mgr. Zdeněk Halas, DiS., Ph.D.
doc. RNDr. Jarmila Robová, CSc.
Class: M Bc. MZV
M Bc. MZV > Povinné
M Bc. MZV > 2. ročník
Classification: Mathematics > Mathematics, Algebra, Differential Equations, Potential Theory, Didactics of Mathematics, Discrete Mathematics, Math. Econ. and Econometrics, External Subjects, Financial and Insurance Math., Functional Analysis, Geometry, General Subjects, , Real and Complex Analysis, Mathematics General, Mathematical Modeling in Physics, Numerical Analysis, Optimization, Probability and Statistics, Topology and Category
Incompatibility : NMTM204, NMUM812, NUMP011
Interchangeability : NMTM204, NMUM812, NUMP011
Is incompatible with: NMTM204, NMUM812, NUMP011
Is interchangeable with: NMTM204, NMUM812, NUMP011
Annotation -
Last update: Mgr. Zdeněk Halas, DiS., Ph.D. (08.09.2013)
Continuation of Geometry I. Study of geometrical transformations in affine and Euclidean space, their basic properties, equations, invariant points and directions. The theory is based on linear algebra.
Course completion requirements -
Last update: Mgr. Zdeněk Halas, DiS., Ph.D. (29.10.2019)
Credit
Attendance at seminars is compulsory for full-time students, maximum 3 absences are allowed.

Possible absences above the limit will be solved by additional homework.

There will be 2 tests, one in the middle of the semester, one at the end of the semester, 2 correction terms are allowed.

Both tests will have the same score, from each test individually the student must earn at least 50% of the points, for both tests together they must obtain at least 2/3 of the total of points.

Exam
The requirements of the exam correspond to the syllabus of the subject to the extent that was presented at the lecture, including everything that was ordered for individual study.

The exam can be taken after obtaining the credit.

The examination consists of a written and an oral part, which are consecutive (they cannot be divided into two terms).

Successful completion of the written part is a prerequisite for admission to the oral part.

Literature -
Last update: Mgr. Zdeněk Halas, DiS., Ph.D. (29.10.2019)
Conic sections:
  • Pech, P.: Kuželosečky. České Budějovice, 2004. — základní literatura ke kuželosečkám (konstrukce, vlastnosti)
  • Janyška, J., Sekaninová, A.: Analytická teorie kuželoseček a kvadrik. Brno, 2013. — doplňující literatura ke kuželosečkám (metoda invariantů), z celého skripta je třeba jen několik stran

Mappings:

  • Sekanina, M. a kol.: Geometrie II. SPN, Praha, 1988. — základní literatura k tomuto předmětu (pouze kapitoly 1 a 2, tj. po stranu 100)

Additional literature:

  • Kubát, V., Trkovská, D.: Analytická geometrie v afinních a eukleidovských prostorech. Matfyzpress, Praha, 2011.
  • Lávička, M. Geometrie II. Pomocný učební text. Plzeň, 2006. Dostupné z < http://home.zcu.cz/~lavicka/subjects/G2/texty/G2_text.pdf >.
  • Jennings, G. A. Modern Geometry with Applications. Springer, 1996.

Syllabus -
Last update: Mgr. Zdeněk Halas, DiS., Ph.D. (08.09.2013)
Affine maps
  • Ratios of vectors along a line and the parameter in the parametric equation of a line; properties.
  • Affine map, basic properties, representation. Associated homomorphism.
  • Affine transformations, invariant points and vectors. Affine group.
  • Basic affine maps. Module of affine map, equiaffinity.
  • Translation group, homothety group.

Maps in Euclidean space

  • Linear isometry, basic properties, representation.
  • Classification of linear isometries in plane, reflections. Isometry group.
  • Similarity map, basic properties, representation. Similarity as composition of a homothety and an orthogonal transformation. Similarity group.
  • Circle inversion, basic properties, representation.
  • Groups of geometrical transformations.
 
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