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Main menu for Browse IS/STAG
Course info
KMA / G1
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Course description
Department/Unit / Abbreviation
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KMA
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G1
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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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Affine and Euclidean 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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0 / -
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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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43 / -
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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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Winter semester
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Semester taught
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Winter 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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KMA/G1-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 main aim of this course is to give students a thorough introduction to the analytic geometry in n-dimensional affine and Euclidean spaces. The course also aims at giving the student a firm understanding of analytic method in the visualization of mathematical concepts, it develops the student's skills to solve problems using the analytic method and finally it shows several applications not only in mathematical disciplines but also in other sciences.
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Requirements on student
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During semester, students have to write two tests (15 points each). Additional points (max. 10 points) may be awarded for continuous work during the semester. A total of at least 21 points is required. 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 chosen theoretical results and to use the analytic method on solving given problems.
Appendix in case of transition to online teaching:
With regard to the current situation, it is not possible to predict whether the whole semester will take place in full-time form, or it will be decided to change the contact form of teaching to distance or hybrid (partly distance). In that case, the conditions for completing the course would be updated and announced in a timely manner.
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Content
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Affine space, affine mappings and affine coordinate system. Affine subspaces and their parametric description. Equations of affine subspaces. Relative position of affine subspaces. Pencils of hyperplanes. Affine combination of points, barycentric coordinates. Vector spaces with scalar product. Euclidean space and Cartesian coordinate system. Orthogonality of Euclidean subspaces. Distances of Euclidean subspaces. Angles of Euclidean subspaces. Volumes of parallelotopes. Möbius extension of Euclidean space. Conics - ellipse, hyperbola, parabola. Intersections of the surface of a cone with a plane. Conic sections - quadratic curves in plane. Quadrics - quadratic surfaces in space. Selkected types of quadrics and their prroperties.
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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:
Lávička, M. a Vršek J. Geometrie 1: Úvod do afinních a eukleidovských prostorů. Plzeň, 2020.
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Extending:
Tarrida, R. Affine Mapsa, Euclidean Motions and Quadrics. London, 2011.
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Extending:
Bican, Ladislav. Lineární algebra a geometrie. Vyd. 1. Praha : Academia, 2000. ISBN 80-200-0843-8.
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Extending:
Mahel a kol. Sbírka úloh z lineární algebry a analytické geometrie. ČVUT, 1980.
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Recommended:
Horák, P. a Janyška, J. Analytická geometrie. Brno, 1997.
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Recommended:
Gallier, Jean. Geometric methods and applications : for computer science and engineering. New York : Springer, 2001. ISBN 0-387-95044-3.
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Recommended:
Sekanina, M. a kol. Geometrie I., Státní pedagogické nakladatelství. Praha, 1986.
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Recommended:
Audin, Michéle. Geometry. Berlin : Springer, 2003. ISBN 3-540-43498-4.
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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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Contact hours
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39
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Preparation for comprehensive test (10-40)
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10
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Preparation for formative assessments (2-20)
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10
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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: |
explain the basic lessons of plane, possibly spatial analytic geometry at the high school level |
describe and explain the basic procedures for solving geometric problems |
describe and explain the basic principles of linear algebra and vector calculus |
describe and explain the basic principles of calculus |
Skills - students are expected to possess the following skills before the course commences to finish it successfully: |
apply the acquired procedures to elementary geometric problems at the secondary school level |
calculate vectors, matrices and determinants and solve systems of linear and quadratic equations |
use the calculus apparatus for basic and intermediate problems |
Competences - students are expected to possess the following competences before the course commences to finish it successfully: |
N/A |
N/A |
N/A |
N/A |
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Learning outcomes
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Knowledge - knowledge resulting from the course: |
define an affine space and introduce a suitable system of coordinates, understand the problem of affine subspaces, derive their equations and determine their relative positions |
define Euclidean space, introduce the Cartesian coordinate system as a specialization of the general affine coordinate system, construct equations of orthogonal subspaces, determine distances and deviations of Euclidean subspaces |
define and classify conics in the Euclidean plane, convert their expressions into canonical forms and recognize them |
define and classify quadrics in three-dimensional Euclidean space, convert their expressions into canonical forms, and recognize and actively use them |
Skills - skills resulting from the course: |
actively use the analytical method in solving mathematical and application problems |
solve geometric problems with linear and quadratic objects |
actively use the analytical method to visualise a variety of mathematical concepts |
find and use application possibilities not only in geometry and other mathematical disciplines, but also in natural sciences, computer graphics, etc. |
solve geometric problems using the synthetic method |
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 |
Practicum |
Task-based study method |
Self-study of literature |
Skills - the following training methods are used to achieve the required skills: |
Lecture |
Lecture supplemented with a discussion |
Practicum |
Task-based study method |
Self-study of literature |
Competences - the following training methods are used to achieve the required competences: |
Lecture |
Lecture supplemented with a discussion |
Practicum |
Task-based study method |
Self-study of literature |
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