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Main menu for Browse IS/STAG
Course info
KMA / G2
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Course description
Department/Unit / Abbreviation
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KMA
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G2
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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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Projective 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,
4
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]
Tutorial
1
[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, English
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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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9 / -
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0 / -
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0 / -
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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, English
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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 |
Yes
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Fundamental course |
Yes
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Fundamental theoretical course |
Yes
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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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KMA/AGE, KMA/AGM
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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 give students a thorough introduction to geometry of quadrics in n-dimensional projective, affine and Euclidean spaces, motivated by the Felix Klein's classification of geometries via group of geometric transformations and studying of corresponding absolute quadrics. The course also aims at giving the student a firm understanding of projective geometry and projective transformations and it enables the student to apply the methods in other areas of science, e.g. in geometric modelling and computer graphics.
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Requirements on student
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*Confirmation of fulfillment of course requirements*
It is stipulated by the course guarantor that the credits earned for the prevevious successful fulfillment of course requirements are not accepted.
During semester, students have to write two assignments (10 points each) - it is necessary to obtain at least 11 points in sum from both. In the case of a distance learning order, the alternative conditions will be specified on CourseWare.
*Examination*
The final examination is in the form of a written exam (70% of the grade) which is supplemented by an oral examination (30% of the grade). All assessment tasks will assess the learning outcomes, especially, the ability to provide logical and coherent proofs of theoretical results and to analyze and solve specific problems related to projective geometry, geometry of quadrics and their applications.
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Content
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Affine mappings in affine spaces and their classification. Isometries and similarities in Euclidean spaces. Composition of transformations. Projective embedding, projective space and its subspaces, Principle of duality. Projective mappings and transformations. Quadrics (especially conic sections in the plane and quadrics in the 3-dimensional space), their projective, affine and metric classification. Geometry and group theory, classification of geometries. Klein-Cayley's geometries.
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Activities
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Fields of study
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Guarantors and lecturers
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Guarantors:
Prof. RNDr. Miroslav Lávička, Ph.D. (100%),
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Lecturer:
Doc. RNDr. Michal Bizzarri, Ph.D. (100%),
Prof. RNDr. Miroslav Lávička, Ph.D. (100%),
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Tutorial lecturer:
Doc. RNDr. Michal Bizzarri, Ph.D. (100%),
Prof. RNDr. Miroslav Lávička, Ph.D. (100%),
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Literature
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Basic:
Lávička, M. Geometrie 2: Projektivní prostory, geometrická zobrazení a kvadriky. Plzeň, 2021.
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Basic:
Pomocné studijní texty na KMA/G2, sekce Materiály pro studenty
(Lávička, M.)
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Extending:
Casas-Alvero, E. Analytic Projective Geometry. Zürich, 2014. ISBN 978-3-03719-138-5.
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Extending:
Reid., M., Szebdroi, B. Geometry and topology. Cambridge University Press, 2005. ISBN 9780521848893.
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Recommended:
Sekaninová, A. a Janyška, J. Analytická teorie kuželoseček a kvadrik. Alfa, Bratislava, 1984.
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Recommended:
Sekanina, M. a kol. Geometrie. 1. díl..
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Recommended:
Sekanina, M. a kol. Geometrie. 2. díl.. 1. vyd. Praha : Státní pedagogické nakladatelství, 1988.
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Recommended:
Audin, Michéle. Geometry. Berlin : Springer, 2003. ISBN 3-540-43498-4.
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Recommended:
Čižmár, J. Grupy geometrických transformácií. 1. vyd. Bratislava : Alfa, 1984.
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Recommended:
Boček, L., Šedivý, J. Grupy geometrických zobrazení. SPN Praha, 1980.
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Recommended:
Coxeter, Harold Scott MacDonald. The real projective plane : with an appendix for mathematica by George Beck : Macintosh version. 3rd ed. New York : Springer, 1993. ISBN 0-387-97889-5.
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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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Contact hours
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39
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Preparation for formative assessments (2-20)
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20
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Preparation for an examination (30-60)
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50
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Total
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109
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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: |
to describe and explain advanced principles from linear algebra and vector calculus |
to describe and explain selected procedures for solving problems of affine and Euclidean geometry |
to understand basic concepts of group theory |
Skills - students are expected to possess the following skills before the course commences to finish it successfully: |
to apply adopted methods to selected geometric problems in n-dimensional affine and Euclidean spaces |
to use the apparatus of linear algebra for mid-level problems |
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: |
to understand affine, isometric and similar mappings, derive their equations, analyse their properties and applicability and decide whether a given mapping is affine, isometric or similar |
to define projective space and its subspaces, understand their mutual relations and work with them using basic methods of projective geometry (especially using the Principle of Duality) |
to define and classify quadrics in n-dimensional projective, affine and Euclidean space, convert their expressions into canonical forms, recognize them and use them actively |
to classify projective mappings and understand the structure of the projective group |
Skills - skills resulting from the course: |
to provide logical proofs of selected important theorems of the studied theory |
to compare and relate different types of geometry (e.g. projective, affine, Euclidean, hyperbolic, elliptic, Möbius) |
to analyze the basic characteristics of quadrics and use their properties to solve selected problems based on specific situations in real life and engineering practice |
to demonstrate the fundamental propositions of an abstract theory using an appropriate combination of examples and counterexamples, look for analogies and make generalizations |
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 |
Test |
Seminar work |
Skills - skills achieved by taking this course are verified by the following means: |
Combined exam |
Test |
Seminar work |
Competences - competence achieved by taking this course are verified by the following means: |
Combined exam |
Test |
Seminar work |
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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 |
Practicum |
Task-based study method |
Self-study of literature |
Discussion |
Textual studies |
Skills - the following training methods are used to achieve the required skills: |
Lecture |
Lecture with visual aids |
Interactive lecture |
Task-based study method |
Textual studies |
Discussion |
Self-study of literature |
Competences - the following training methods are used to achieve the required competences: |
Lecture |
Lecture with visual aids |
Interactive lecture |
Practicum |
Task-based study method |
Textual studies |
Self-study of literature |
Discussion |
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