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Non-Euclidean Geometry I - NMUG401

Title:	Neeukleidovská geometrie I
Guaranteed by:	Department of Mathematics Education (32-KDM)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2021
Semester:	winter
E-Credits:	5
Hours per week, examination:	winter s.:2/2, C+Ex [HT]
Capacity:	unlimited
Min. number of students:	unlimited
4EU+:	no
Virtual mobility / capacity:	no
State of the course:	taught
Language:	Czech
Teaching methods:	full-time
Teaching methods:	full-time
Additional information:	http://www.karlin.mff.cuni.cz/~krump/neeuklid_1718.htm

Guarantor:	Mgr. Lukáš Krump, Ph.D.
Class:	M Bc. DGZV M Bc. DGZV > Povinné M Bc. DGZV > 2. ročník
Classification:	Mathematics > Geometry
Incompatibility :	NDGE020, NMTD401
Interchangeability :	NDGE020, NMTD401
Is incompatible with:	NMTD401, NDGE020
Is interchangeable with:	NMTD401, NDGE020

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Annotation -

Last update: T_KDM (04.05.2015)

Axiomatic of geometry, non-Euclidean geometries, models of non-Euclidean geometries (Beltrami-Klein, Poincare), groups of transformations.

Aim of the course -

Last update: T_KDM (04.05.2015)

This course helps to obtain theoretical background for teaching mathematics at high school.

Course completion requirements -

Last update: Mgr. Lukáš Krump, Ph.D. (29.10.2019)

The course credit (="zápočet") is obtained for an oral essay; in well-reasoned cases (longer justified absence), the essay can be in a written form.

The nature of this study control excludes repeating.

The course credit is a necessary condition for admission to the exam.

Literature -

Last update: T_KDM (04.05.2015)

1. Kutuzov, B.V.: Lobačevského geometrie a elementy základů geometrie, ČSAV, Praha, 1953

2. Trajnin, J.L.: Osnovanija geometrii, Moskva, 1961

3. Hlavatý, V.: Úvod do neeuklidovské geometrie, JČMF, Praha, 1949

4. Čech, E.: Základy analytické geometrie II., Praha, 1952

5. Boček, L. & Šedivý J.: Grupy geometrických zobrazení, SPN, Praha

6. Weblen, O. & Young, J.W.: Projective geometry I.II., Blaisdell P. C., New York, 1938

7. Gans, D.: An Introduction to Non-Euclidean Geometry, Academic Press, New York, 1973

8. Tuller, A.: Introduction to Geometries,

9. Springer, C.E.: Geometry and Analysis of Projective Spaces,

10. Wolfe, H.E.: Introduction to Non-Euclidean Geometry, Holt, Rinehart & Winston, Inc., New York, 1966

Teaching methods -

Last update: T_KDM (04.05.2015)

Lectures and exercises.

Requirements to the exam -

Last update: Mgr. Lukáš Krump, Ph.D. (29.10.2019)

The exam is oral, its contents corresponds to the syllabus in the extent taught.

Syllabus -

Last update: T_KDM (04.05.2015)

Spherical geometry, excess of angles in spherical triangle, solution of spherical triangles.

Stereographic projection and circular inversion. Solutions of problems of Apollonios.

Axiomatisation of geometry, absolute geometry, the 5th postulate, mutual position of two lines in non-Euclidean geometry, defect of angles and area of triangle. Sheaves of lines and sets of corresponding points.

Models of non-Euclidean geometry. Distances and angles in the Poincare and Beltrami- Klein models. Riemannian metric and groups of transformations of models.