# Geometry with Applications, 6 credits (TATA49)

Geometri med tillämpningar, 6 hp

### Main field of study

Mathematics Applied Mathematics

First cycle

Programme course

### Examiner

Milagros Izquierdo Barrios

Jesper Thorén

### Available for exchange students

Yes
Course offered for Semester Period Timetable module Language Campus VOF
6KMAT Mathematics 5 (Autumn 2017) 1, 2 4, 4 Swedish/English Linköping o

### Main field of study

Mathematics, Applied Mathematics

First cycle

G1X

• Mathematics

### Entry requirements

Note: Admission requirements for non-programme students usually also include admission requirements for the programme and threshold requirements for progression within the programme, or corresponding.

### Prerequisites

First courses in Linear algebra and Discrete mathematics (desirable)

### Intended learning outcomes

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

• ### Course content

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.

### Teaching and working methods

Lectures and tutorials.

### Examination

 UPG1 Hand-in assignments U, 3, 4, 5 6 credits

Four-grade scale, LiU, U, 3, 4, 5

### Other information

Supplementary courses: Linear Algebra, honours course. Combinatorics

### Department

Matematiska institutionen

### Director of Studies or equivalent

Jesper Thorén

### Examiner

Milagros Izquierdo Barrios

### Course website and other links

http://www.mai.liu.se/und/kurser/index-amne-tm.html

### Education components

Preliminary scheduled hours: 56 h
Recommended self-study hours: 104 h

### Course literature

##### Books
J. N. Cederberg, A course in Modern Geometries (Undergraduate Texts in Mathematics)

### Books

J. N. Cederberg, A course in Modern Geometries (Undergraduate Texts in Mathematics)