Geometry with Applications, 6 credits

Geometri med tillämpningar, 6 hp

TATA49

Main field of study

Mathematics Applied Mathematics

Course level

First cycle

Course type

Programme course

Examiner

Milagros Izquierdo Barrios

Director of studies or equivalent

Jesper Thorén

Education components

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

Available for exchange students

Yes
Possible deviation from course syllabus/programme syllabus
ECV = Elective / Compulsory / Voluntary
Course offered for Semester Period Timetable module Language Campus ECV
6KMAT Mathematics 5 (Autumn 2017) 1, 2 4, 4 Swedish/English Linköping, Valla C

Main field of study

Mathematics, Applied Mathematics

Course level

First cycle

Advancement level

G1X

Course offered for

  • 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

      UPG1Hand-in assignments6 creditsU, 3, 4, 5

      Grades

      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

      Additional literature

      Books

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

      Compendia

    • Code Name Scope Grading scale
    UPG1 Hand-in assignments 6 credits U, 3, 4, 5

    Regulations (apply to LiU in its entirety)

    The university is a government agency whose operations are regulated by legislation and ordinances, which include the Higher Education Act and the Higher Education Ordinance. In addition to legislation and ordinances, operations are subject to several policy documents. The Linköping University rule book collects currently valid decisions of a regulatory nature taken by the university board, the vice-chancellor and faculty/department boards.

    LiU’s rule book for education at first-cycle and second-cycle levels is available at http://styrdokument.liu.se/Regelsamling/Innehall/Utbildning_pa_grund-_och_avancerad_niva. 

    Student rights and obligations (login required)
    Linköping University common rules and regulations (in Swedish)

    Additional literature

    Books

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

    Compendia

    Download

    Note: The course matrix might contain more information in Swedish.

    I = Introduce, U = Teach, A = Utilize
    I U A Modules Comment
    1. DISCIPLINARY KNOWLEDGE AND REASONING
    1.1 Knowledge of underlying mathematics and science (G1X level)
    X
    X
    X
    UPG1
    		 
    
                                		
                        
    
                        
    
                            
                            1.2 Fundamental engineering knowledge (G1X level)
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                            
                            
                                
    
                                		
                        
    
                        
    
                            
                            1.3 Further knowledge, methods, and tools in one or several subjects in engineering or natural science (G2X level)
                            
                                
    X
    
                                		
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                            
                            
                                
    
                                		
                        
    
                        
    
                            
                            1.4 Advanced knowledge, methods, and tools in one or several subjects in engineering or natural sciences (A1X level)
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                            
                            
                                
    
                                		
                        
    
                        
    
                            
                            1.5 Insight into current research and development work
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                            
                            
                                
    
                                		
                        
    
                        
                    
                        
    
                            
                                2. PERSONAL AND PROFESSIONAL SKILLS AND ATTRIBUTES
                            
                        
    
                        
                    
                        
    
                            
                            2.1 Analytical reasoning and problem solving
                            
                                
     
    
                                		
                            
                                
    X
    
                                		
                            
                                
    X
    
                                		
                            
                                  
    UPG1
    
                                		
                            
                                
    
                                		
                        
    
                        
    
                            
                            2.2 Experimentation, investigation, and knowledge discovery
                            
                                
     
    
                                		
                            
                                
    X
    
                                		
                            
                                
    X
    
                                		
                            
                                  
    UPG1
    
                                		
                            
                                
    
                                		
                        
    
                        
    
                            
                            2.3 System thinking
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                                
    X
    
                                		
                            
                            
                            
                                
    
                                		
                        
    
                        
    
                            
                            2.4 Attitudes, thought, and learning
                            
                                
     
    
                                		
                            
                                
    X
    
                                		
                            
                                
    X
    
                                		
                            
                                  
    UPG1
    
                                		
                            
                                
    
                                		
                        
    
                        
    
                            
                            2.5 Ethics, equity, and other responsibilities
                            
                                
    X
    
                                		
                            
                                
     
    
                                		
                            
                                
    X
    
                                		
                            
                            
                            
                                
    
                                		
                        
    
                        
                    
                        
    
                            
                                3. INTERPERSONAL SKILLS: TEAMWORK AND COMMUNICATION
                            
                        
    
                        
                    
                        
    
                            
                            3.1 Teamwork
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                                
    X
    
                                		
                            
                            
                            
                                
    
                                		
                        
    
                        
    
                            
                            3.2 Communications
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                                
    X
    
                                		
                            
                                  
    UPG1
    
                                		
                            
                                
    
                                		
                        
    
                        
    
                            
                            3.3 Communication in foreign languages
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                                
    X
    
                                		
                            
                            
                            
                                
    
                                		
                        
    
                        
                    
                        
    
                            
                                4. CONCEIVING, DESIGNING, IMPLEMENTING AND OPERATING SYSTEMS IN THE ENTERPRISE, SOCIETAL AND ENVIRONMENTAL CONTEXT
                            
                        
    
                        
                    
                        
    
                            
                            4.1 External, societal, and environmental context
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                            
                            
                                
    
                                		
                        
    
                        
    
                            
                            4.2 Enterprise and business context
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                            
                            
                                
    
                                		
                        
    
                        
    
                            
                            4.3 Conceiving, system engineering and management
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                            
                            
                                
    
                                		
                        
    
                        
    
                            
                            4.4 Designing
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                            
                            
                                
    
                                		
                        
    
                        
    
                            
                            4.5 Implementing
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                            
                            
                                
    
                                		
                        
    
                        
    
                            
                            4.6 Operating
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                            
                            
                                
    
                                		
                        
    
                        
                    
                        
    
                            
                                5. PLANNING, EXECUTION AND PRESENTATION OF RESEARCH DEVELOPMENT PROJECTS WITH RESPECT TO SCIENTIFIC AND SOCIETAL NEEDS AND REQUIREMENTS
                            
                        
    
                        
                    
                        
    
                            
                            5.1 Societal conditions, including economic, social, and ecological aspects of sustainable development for knowledge development
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                            
                            
                                
    
                                		
                        
    
                        
    
                            
                            5.2 Economic conditions for knowledge development
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                            
                            
                                
    
                                		
                        
    
                        
    
                            
                            5.3 Identification of needs, structuring and planning of research or development projects
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                            
                            
                                
    
                                		
                        
    
                        
    
                            
                            5.4 Execution of research or development projects
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                            
                            
                                
    
                                		
                        
    
                        
    
                            
                            5.5 Presentation and evaluation of research or development projects
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                                
     
    
                                		
                            
                            
                            
                                
    
                                		
                        
    
                        
            
    
        
    

    

            
    
        
    
            

    
        
    
            
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