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Geometry with Applications, 6 ECTS credits.
/Geometri med tillämpningar/
For:
Mat
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Prel. scheduled
hours: 56
Rec. self-study hours: 104
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Area of Education: Science
Main field of studies: Mathematics, Applied Mathematics
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Advancement level
(G1, G2, A): G1
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Aim:
The course presents methods and concepts in modern geometry, i.e.
it is based on geometrical transformations. The course treats Euclidean and non-euclidean geometry, and real and finite projective geometry. By generalization of Euclidean transformation one obtains projective geometries. These geometries form the mathematical basis for computer graphics, latin squares and error-correcting codes.
Students should be able to:
use the concept of group to study different geometries
classify and to determine the different (Euclidean)
transformations of the plane.
study frieze and wallpaper patterns with the help of transformations
know of hyperbolic and elliptic geometry.
work with the projective plane and its transformations:
collineations and projectivities
use collineations and projectivities to explain the foundations of
computer graphics
recognise finite projective geometries and their applications
to coding theory and configurations.
apply quaternions to computer animations
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Prerequisites: (valid for students admitted to programmes within which the course is offered)
First courses in Linear algebra and Discrete mathematics (desirable)
Note: Admission requirements for non-programme students usually also include admission requirements for the programme and threshhold requirements for progression within the programme, or corresponding.
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Supplementary courses:
Linear Algebra, honours course. Combinatorics
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Organisation:
Lectures and tutorials.
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Course contents:
Groups: cyclic and dihedral groups. Quaternions. Stereographic projection. Euclidean plane geometry: isometries, reflections, direct and inverse isometries. Frieze and wallpaper patterns. Three-dimensional isometries. Hyperbolic and elliptic geometries. Projective plane: harmonic sets, perspectivity, projectivity, conics, cross ratios, collineations and polarity. Application in computer graphics Finite projective planes. Applications to error-correcting codes, configurations, design and latin squares.
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Course literature:
J. N. Cederberg: A course in Modern Geometries (Undergraduate Texts in Mathematics.)
Handouts.
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Examination: |
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Hand-in assignments
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6 ECTS
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Course language is Swedish/English.
Department offering the course: MAI.
Director of Studies: Jesper Thorén
Examiner: Milagros Izquierdo Barrios
Link to the course homepage at the department
Course Syllabus in Swedish
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