Projective Geometry

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Projective Geometry

Course title: Projective Geometry Nature of the course: Theory
Course no.: Math Ed.527 Credit hours: 3
Level: M.Ed. Teaching hours: 48
Semester: Second

1. Course Description

This course is designed to provide wider knowledge and skills on axiomatic system in geometry for math educators. It comprises a range of skills varied from introductory projective geometry to projective space. This course edifies axiomatic structure that remain unchanged under projection. Incidence structure, perspectivity and projectivity are the beauty of this course. This course is divided into five major units. It starts with incidence geometry and then discusses collineations, Desarguesian and Pappian planes. Finally, the course focuses on projective space.

2. The General objectives

The general objectives of this course are as follows:

1. To familiarize the concepts of incidence structure and prove its basic results
2. To apply basic results of projection in problem solving
3. To analyze relation between Desarguesian and Papian plane
4. To prove theorems on conics in Papian plane
5. To investigate relation between projective plane and projective space

3. Specific Objectives and Contents

Unit I: Incidence Geometry (12)

• To define incidence structure and its examples
• To define plane, affine plane, projective plane, and prove its related theorems
• To define isomorphism and prove related theorem
• To define duality, its principle and prove related theorem
• To define configuration and prove related theorems
• To define embedded plane and prove theorems principle sub-planes
• To explain Homogeneous coordinate and define order of plane and prove related theorems

1. 1.1. Incidence structure
2. 1.2. Plane, affine plane and projective plane
3. 1.3. Isomorphism
4. 1.4. Duality
5. 1.5. Configuration
6. 1.6. Embeded plane
7. 1.7. Homogeneous coordinate and Order of plane

Unit II: Collineation (10)

• To define Perpectivity, derive its equation, and related problem solving
• To define projectivity, and prove related theorems
• To define collineation and prove related theorems
• To define Matrix induced collineation, central collineation and automorphic collineation and prove related theorems

1. 2.1 Perspectivity
2. 2.2 Projectivity
3. 2.3 Collineation
4. 2.4 Matrix induced collineation, central collineation and automorphic collineation

UnitIII: Desarguesian and Papian Plane (10)

• To define Desarguesian plane and prove related theorems
• To exemplify homogeneous coordinate for Desarguesian plane
• To define Quadrangular set and prove related theorems
• To define Pappian plane and prove related theorems
• To exemplify homogeneous coordinate for Papian plane
• To state and prove fundamental and Uniqueness theorem
• To define cross ratio and prove related theorem

1. 3.1 Desarguesian Plane
2. 3.2 Quadrangular set and related theorems
3. 3.3 Papian plane and related theorems
4. 3.4 Fundamental and uniqueness theorem
5. 3.5 Cross-ratio

Unit IV: Conics in Papian Plane (8)

• To define point conic and line conic in Papian plane
• To prove conics related theorems
• To define intersection of a range and a point conic and prove related theorems
• To prove closed projective plane related theorems
• To state and prove Pascal’s theorem and its converse

1. 4.1 Conics in Pappian plane
2. 4.2 The projective conic and related theorem
3. 4.3 Intersection of a range and a point conic
4. 4.4 Closed projective plane and related theorems
5. 4.5 Pascal’s Theorem and its converse

UnitV: Projective Space (8)

• To define projective space and prove related theorems
• To define projective subspace and prove related theorem
• To define spanning set and apply it in problem solving and theorem proof
• To state and prove Desargues’s theorem

1. 5.1 Projective space
2. 5.2 Projective subspace
3. 5.3 Theorems on spanning set
4. 5.4 Desargues’s theorem

Note: The figures in the parentheses indicate approximate teaching hours allocated for respective units.

4. Instructional Techniques

The instructor will select the method or methods of instruction most suitable for a topic. It is quite acceptable to select more than one method and combine them into a single period of instruction whenever it is needed. The general and specific instructional techniques are described below.

• 4.1 General Techniques:
  The following general method of instruction will be adopted according to the need and nature of the lesson:
  • Lecture
  • Demonstration
  • Discussion
  • Group Work
• 4.2 Specific Instructional Techniques

Unit	Activity and Instructional Techniques	48
1	Multimedia presentation • Project work	12
2	Multimedia presentation • Project Work	10
3	Project work and presentation	10
4	Multimedia presentation	8
5	Multimedia presentation • Project work • Group Discussion	8

4. Evaluation

• 5.1 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 (assignment)	10 marks
  Third assessment (assignment)	10 marks
  Total	40 marks

• 5.2 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 x 1) 10 marks
  Short answer questions 6 with 2 OR questions	(6 x 5) 30 marks
  Long answer questions 2 with 1 OR question	(2 x 10) 20 marks
  Total	60 marks

Recommended and References

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