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Course info
KMA / AXG
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Course description
Department/Unit / Abbreviation
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KMA
/
AXG
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Academic Year
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2023/2024
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Academic Year
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2023/2024
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Title
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Axioms of Geometry
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Form of course completion
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Exam
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Form of course completion
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Exam
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Accredited / Credits
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Yes,
3
Cred.
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Type of completion
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Combined
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Type of completion
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Combined
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Time requirements
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Lecture
2
[Hours/Week]
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Course credit prior to examination
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Yes
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Course credit prior to examination
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Yes
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Automatic acceptance of credit before examination
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No
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Included in study average
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YES
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Language of instruction
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Czech
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Occ/max
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Automatic acceptance of credit before examination
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No
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Summer semester
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3 / -
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0 / -
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1 / -
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Included in study average
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YES
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Winter semester
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0 / -
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0 / -
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0 / -
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Repeated registration
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NO
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Repeated registration
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NO
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Timetable
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Yes
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Semester taught
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Summer semester
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Semester taught
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Summer semester
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Minimum (B + C) students
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1
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Optional course |
Yes
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Optional course
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Yes
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Language of instruction
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Czech
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Internship duration
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0
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No. of hours of on-premise lessons |
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Evaluation scale |
1|2|3|4 |
Periodicity |
každý rok
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Evaluation scale for credit before examination |
S|N |
Periodicita upřesnění |
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Fundamental theoretical course |
No
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Fundamental course |
No
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Fundamental theoretical course |
No
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Evaluation scale |
1|2|3|4 |
Evaluation scale for credit before examination |
S|N |
Substituted course
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None
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Preclusive courses
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N/A
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Prerequisite courses
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N/A
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Informally recommended courses
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N/A
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Courses depending on this Course
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N/A
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Histogram of students' grades over the years:
Graphic PNG
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XLS
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Course objectives:
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The aim of this course is to introduce students with the theory of Euclidean and non-Euclidean geometries based on the axiomatic approach and active understanding of the fundamental concepts such as:
- Euclid's Elements, Hilbert's axiomatic system,
- non-Euclidean geometries and their models, hyperbolic and elliptic geometries,
- proving in Euclidean and non-Euclidean geometries,
- geometric transformations in hyperbolic and elliptic geometries.
The course also aims at enabling the student to apply the methods in other areas of science, e.g. in physics.
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Requirements on student
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The final examination is in the form of a written exam which is supplemented by an oral examination. All assessment tasks will assess the learning outcomes, especially, the ability to provide logical and coherent proofs of chosen theoretical results and to use the methods from the course on solving given problems.
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Content
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This course is a survey of main concepts of Euclidean geometry with the emphasis on the axiomatic approach, constructions and logic of proof including historical aspects. Consequences of the Euclidean parallel postulate. A study of axioms of Euclidean geometry, inference rule, some basic theorems of Euclidean geometry and rigorous proofs will be offered. Non-Euclidean geometry is introduced. The similarities and differences between Euclidean and non-Euclidean geometries will be discussed. Transformations in non-Euclidean geometries. Models of hyperbolic and elliptic geometries.
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Activities
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Fields of study
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Guarantors and lecturers
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Literature
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Basic:
Anderson, J. W. Hyperbolic geometry. London : Springer, 1999. ISBN 1-85233-156-9.
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Extending:
Hilbert, D., Hallett, M., Majer, U. David Hilbert´s Lectures on the foundations of geometry. 1891-1902. Berlin : Springer, 2004. ISBN 3-540-64373-7.
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Extending:
Stillwell, John. Mathematics and its history. 2nd ed. New York : Springer, 2002. ISBN 0-387-95336-1.
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Recommended:
Lávička, M. Geometrie 1 : Základy geometrie v rovině. 1. vyd. Plzeň : Západočeská univerzita, 2002. ISBN 80-7082-861-7.
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Recommended:
Hartshorne, Robin. Geometry: Euclid and beyond. New York : Springer, 2000. ISBN 0-387-98650-2.
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Recommended:
Ramírez Galarza, Ana Irene; Seade, José. Introduction to classical geometries. 2007. ISBN 978-3-7643-7517-1.
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Recommended:
Coxeter, H. S. M. Non-Euclidean geometry. 6th ed. Washington : Mathematical Association of America, 1998. ISBN 0-88385-522-4.
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On-line library catalogues
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Time requirements
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All forms of study
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Activities
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Time requirements for activity [h]
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Presentation preparation (report) (1-10)
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5
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Undergraduate study programme term essay (20-40)
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20
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Preparation for an examination (30-60)
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40
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Contact hours
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26
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Total
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91
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Prerequisites
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Knowledge - students are expected to possess the following knowledge before the course commences to finish it successfully: |
popsat a vysvětlit pokročilé principy z lineární algebry a vektorového počtu |
rozumět základním pojmům z teorie algebraických struktur |
popsat a vysvětlit vybrané postupy pro řešení úloh afinní a euklidovské, případně projektivní geometrie |
rozumět vlastnostem a použití Möbiových transformací |
Skills - students are expected to possess the following skills before the course commences to finish it successfully: |
aplikovat osvojené postupy na vybrané geometrické úlohy v n-rozměrných afinních a euklidovských, resp. projektivních prostorech |
vhodně používat aparát lineární algebry |
aplikovat geometrické transformace, využívat jejich vlastností |
používat základní nástroje komplexní analýzy |
Competences - students are expected to possess the following competences before the course commences to finish it successfully: |
N/A |
N/A |
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Learning outcomes
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Knowledge - knowledge resulting from the course: |
chápat vývoj axiomatických systémů a na jejich příkladě demonstrovat rostoucí úroveň abstraktního geometrického myšlení |
rozumět důsledkům pátého Eukleidova postulátu na vývoj eukleidovské a neeukleidovských geometrií |
rozlišovat mezi různými typy neeukleidovských geometrií a umět popsat jejich modely |
demonstrovat vhodnou kombinací příkladů a protipříkladů základní tvrzení abstraktní teorie, vyhledávat analogie a provádět zobecnění |
Skills - skills resulting from the course: |
demonstrovat souvislosti s eukleidovskou geometrií, především na příkladu hyperbolické a eliptické geometrie |
samostatně řešit problémy a dokazovat věty neeukleidovských geometrií |
provádět důkazy vět v axiomaticky budované teorii |
Competences - competences resulting from the course: |
N/A |
N/A |
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Assessment methods
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Knowledge - knowledge achieved by taking this course are verified by the following means: |
Combined exam |
Skills - skills achieved by taking this course are verified by the following means: |
Combined exam |
Competences - competence achieved by taking this course are verified by the following means: |
Combined exam |
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Teaching methods
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Knowledge - the following training methods are used to achieve the required knowledge: |
Lecture |
Lecture supplemented with a discussion |
Interactive lecture |
Task-based study method |
Textual studies |
Self-study of literature |
Skills - the following training methods are used to achieve the required skills: |
Lecture |
Lecture supplemented with a discussion |
Interactive lecture |
Task-based study method |
Textual studies |
Self-study of literature |
Competences - the following training methods are used to achieve the required competences: |
Lecture supplemented with a discussion |
Interactive lecture |
Lecture |
Task-based study method |
Self-study of literature |
Textual studies |
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