Projective Geometry_Syllabus


Course Title     : Projective Geometry                         Nature of the Course: Theoretical
Course No.      : Math Ed.                                            Credit Hours: 3 CH               
Semester          : Second                                              Teaching Hours: 48   

Course Description

Projective Geometry examines those properties of geometric figures that remain unchanged by a central projection. Perspective in art, images of conic section under projection analyzed through point at infinity and duality are the beauty of Projective geometry.

The General Objectives of the Course

The aim of the course is to provide students with an introduction to axiomatic system Projective geometry. The course will begin by looking at incidence structure and end with Projective space.
On successful completion of the course the students will be able to:
·         to let students; understand the concept incidence structure and prove basic results of planes
·         to enable students' to understand the basic results of Projective transformation
·         to help students' to analyze and describe connection on Desarguesian and Papian plane 
·         to acquaint students' with the knowledge of Projective space 

Course Outlines


  • ·         Define Incidence Structure
  • ·        Define Plane
  • ·         Define Affine plane and prove related theorems (four theorems)
  • ·         Define Projective plane and prove related theorems (four theorems)
Unit 1: Incidence Geometry (12)
1.1.    Incidence Structure and Planes
1.2.    Affine plane
1.3.    Projective plane
1.4.    Real affine plane
1.5.    Proportionality class and Homogeneous coordinates
1.6.    Real projective plane
1.7.    Isomorphism   
1.8.    Duality
1.9.    Principal of duality
1.11.Tactical configuration
1.12.Theorems on configuration    
1.13.Embeded plan, Subplane and Principal subplane
1.14.Theorem on subplane
1.15.Order of plane

·      Define perspectivity, projectivity and collineation
·      Use and apply perspectivity, projectivity and collineation  in proving theorems
·      Define extended collineation and use them in proving theorems
Unit 2: Collineation (9)
2.1        Perspectivity
2.1.1         Equation of perspectivity
2.2        Projectivity
2.3        Collineation
2.3.1     Matrix induced collineation
2.3.2     Central collineations
2.4        Automorphic collineation

·      Define Desarguesian plane and develop their theorems
·      Use projecrtivities to prove theorems  in Desarguesian planes
·      Define pappian plane and develop their theorems
·      Use projecrtivities to prove theorems  in Pappian planes

Unit 3: Desarguesian and Papian Plane (10)
3.1        Desarguesian plane and related theorems
3.1.1   Desargues triangle theorem
3.1.2   Couple
3.1.3   Central couple
3.1.4   Axial couple                  
3.2        Quadrangular set and related theorems
3.3        Projectivities in desarguesian plane
3.4        Papian plane and related theorems
3.4.1         Pappus theorem          
3.5        The perspectivity theorem (concept only)
3.6        Fundamental theorem
3.7        The permutation theorem (concept only)
3.8        Cross-ratio      

·      Define conic from projective view point  and develop their theorems
·      Describe and Derive Desargesian and Pascals theorem for conics

Unit 4: Conics in Papian Plane (10)
4.1           Conics
4.2           Perspectivity and projectivity
4.3           The projective  conic and related theorem
4.4           Intersection of a range and a point conic
4.5           Closed projective plane
4.6           Conics in closed plane 
4.7           Involution
4.8           Desargesian conic theorem
4.9           Pascal’s theorem
4.10       Converse of pascal’s theorem

·      Define Projective space 
·      Define subspace and prove related theorems
·      Define spanning set and prove related theorems
·      Define independent set and prove related theorems
·      Define dimension and prove related theorems
·      Define homomorphism and prove related theorem
Unit 5: Projective Space (7)
5.1         Projective space and related theorems
5.1.1 subspace
5.1.2 spanning set
5.1.3        Properties of spans
5.1.4        Dimension
    5.2     Desargues’s theorem           
    5.3       Homomorphism

Instructional Techniques

General Instructional Techniques

Following instructional techniques will be adopted.
·         Lecturers
·         Discussion
·         Question Answer

Specific Instructional Techniques

Activity and Instructional Techniques
Experiences will be shared between groups with a seminar
The Demonstration method will be involve both giving task to students and
showing their task
Project assignment on some theorems
Group discussion with sharing
Guided Discussion


Internal Evaluation (40%)

Internal evaluation will be conducted by course teacher based on following activities:
·         Attendance                                                                           5 marks
·         Participation in learning activities                                        5 marks
·         First assessment       (assignment)                                        10 marks
·         Second assessment (written test)                                         10 marks
·         Third assessment (written test)                                            10 marks
Total                                                                                     40 marks

External Examination (60%)

Examination Division, Office of the Dean, Faculty of Education will conduct final examination at the end of the semester. The number of questions and their types along with their marks allocated to each type will be as follows:

  • ·         Objective questions (multiple choice) (10 ´ 1)                      10 marks
  • ·         Short answer questions 6 with 2 OR questions (6  ´ 5)         30 marks
  • ·         Long answer questions 2 with 1 OR question (2´ 10)           20 marks
           Total                                                                                       60 marks

Recommended and References
Recommended Books
Garner, L. E., (1981). An outline of projective geometry. New York: North Holand Oxford. (Units 1 -5)
Koirala S. P., Dhakal B. P., (2075). Introductory projective geometry. Read Publication: Kalimati, Nepal (Units 1 -5)
Reference Books
Coxeter, H. S. M., (1973). Projective geometry. New York: Springer-Verlag, London. (Units 1 -3)


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