M.Ed. Semester II Course
Math Ed. 527: Projective Geometry
An axiomatic study of incidence structures, plane projections, collineations, and conics in Pappian geometry
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.
General Objectives
- 1 Familiarize the concepts of incidence structure and prove its basic results.
- 2 Apply basic results of projection in geometric problem solving.
- 3 Analyze relation between Desarguesian and Pappian plane properties.
- 4 State and prove theorems on conics in Pappian projective planes.
- 5 Investigate relation between projective planes and projective spaces.
Syllabus Breakdown
Click on any unit to view its specific learning objectives, contents, and web links.
Specific Objectives
- To define incidence structure and its examples.
- To define plane, affine plane, projective plane, and prove 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 on principal sub-planes.
- To explain Homogeneous coordinates and define order of plane and prove related theorems.
Learning Content
- 1.1. Incidence structure ↗
- 1.2. Plane, affine plane and projective plane
- 1.3. Isomorphism ↗
- 1.4. Duality ↗
- 1.5. Configuration ↗
- 1.6. Embedded plane ↗
- 1.7. Homogeneous coordinate and Order of plane
Specific Objectives
- To define Perspectivity, derive its equation, and solve related problems.
- 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.
Learning Content
- 2.1. Perspectivity ↗
- 2.2. Projectivity ↗
- 2.3. Collineation ↗
- 2.4. Matrix induced collineation, central collineation and automorphic collineation
Specific Objectives
- To define Desarguesian plane and prove related theorems.
- To exemplify homogeneous coordinates for Desarguesian planes.
- To define Quadrangular set and prove related theorems.
- To define Pappian plane and prove related theorems.
- To exemplify homogeneous coordinates for Pappian planes.
- To state and prove fundamental and Uniqueness theorems.
- To define cross ratio and prove related theorems.
Learning Content
- 3.1. Desarguesian Plane
- 3.2. Quadrangular set and related theorems
- 3.3. Pappian plane and related theorems
- 3.4. Fundamental and uniqueness theorem
- 3.5. Cross-ratio
Specific Objectives
- To define point conic and line conic in Pappian 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.
Learning Content
- 4.1. Conics in Pappian plane
- 4.2. The projective conic and related theorem
- 4.3. Intersection of a range and a point conic
- 4.4. Closed projective plane and related theorems
- 4.5. Pascal’s Theorem and its converse
Specific Objectives
- To define projective space and prove related theorems.
- To define projective subspace and prove related theorems.
- To define spanning set and apply it in problem solving and theorem proofs.
- To state and prove Desargues’s theorem in projective space.
Learning Content
- 5.1. Projective space
- 5.2. Projective subspace
- 5.3. Theorems on spanning set
- 5.4. Desargues’s theorem
* Note: The figures in the parentheses indicate approximate teaching hours allocated for respective units.
Instructional Techniques
The instructor will select the methods most suitable for each topic. A combination of general and specific techniques will be adopted to foster deep conceptual understanding.
General Techniques
Lecture, Demonstration, Discussion, and Group Work.
Specific Instructional Techniques
| Unit | Activity and Instructional Techniques | Allocated Hours |
|---|---|---|
| Unit I | Multimedia presentation • Project work | 12 Hours |
| Unit II | Multimedia presentation • Project Work | 10 Hours |
| Unit III | Project work and presentation | 10 Hours |
| Unit IV | Multimedia presentation | 8 Hours |
| Unit V | Multimedia presentation • Project work • Group Discussion | 8 Hours |
| Total Teaching Hours | 48 Hours | |
Internal Evaluation (40%)
Continuous evaluation conducted by the subject teacher based on course activities.
| Evaluation Component | Marks |
|---|---|
| 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 Internal Marks | 40 Marks |
External Examination (60%)
Final written examination conducted by the Office of the Dean, Faculty of Education.
| Question Type | Marks Allocation |
|---|---|
| Objective questions (multiple choice) | (10 x 1) 10 Marks |
| Short answer questions (6 items with 2 OR questions) | (6 x 5) 30 Marks |
| Long answer questions (2 items with 1 OR question) | (2 x 10) 20 Marks |
| Total External Marks | 60 Marks |
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