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
Objectives
|
Unit
|
|
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
1.10.Configurations
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
Unit
|
Activity and Instructional Techniques
|
1
|
Experiences will be
shared between groups with a seminar
|
2
|
The Demonstration
method will be involve both giving task to students and
showing their task
|
3
|
Project assignment
on some theorems
|
4
|
Group discussion
with sharing
|
5
|
Guided Discussion
|
Evaluation
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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