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Course Code : PMAT 42443
Title : Advanced Geometry
Pre-requisites : PMAT 22293
Learning Outcomes:
After the completion of this course unit, the student will be able to:
1. develop an intuitive understanding of the nature of general equation of second degree
2. apply transformations and use symmetry to analyze mathematical situations
3. define conic sections and their properties
4. explain and use various techniques for calculating tangents, normal, pair of tangents, pole and polar of
conic sections
5. compute lines, plane, cone, sphere, ellipsoid and hyperboloid in three dimensions
6. analyze characteristics and properties of two dimensional and three-dimensional geometric shapes and
develop mathematical arguments about geometric relationships
7. demonstrate the knowledge of geometry and its applications in the real world.
8. develop knowledge and intuitive understanding on projective geometry.
Course Contents:
Analytical Geometry in Two Dimensions:
Pairs of straight lines, General equation of second degree, Translations and rotations of axes, Invariants of transformations.
Conic sections:
Classifying general equation of second degree into Parabolas, Ellipses, Hyperbolas, eccentricity. Equation of tangent, Pairs of tangents and chords of contact, Harmonic conjugates, Pole and Polar, Parametric treatment, Degenerate conic, Properties of conic.
Analytical Geometry in Three Dimension:
Coordinate system, Direction cosines and direction ratios of a line, Angle between two lines, Parallel lines, Perpendicular lines, Plane and Straight line, Shortest distance between two-non intersecting lines, Skewed lines, General Equation of the second degree, Sphere, Cone, Ellipsoid and hyperboloid, Tangent plane, Normal, Pole and Polar.
Projective Geometry:
Projective Spaces; Definitions and Properties, Hyperplane at Infinity, Projective lines; Projective Transformation of Projective Line Space, The Cross Ratio, Introduction to projective plane.
Method of Teaching and Learning : A combination of lectures, tutorial discussions and presentations.
Assessment : Based on tutorials, tests, presentations and end of course examination.
Recommended Reading:
1. Chatterjee, D., (2008) Analytical Geometry, Alpha Science International.
2. Thomas, G.B. and Finney, R.L., (2008) Calculus and Analytic Geometry.
3. Maxwell, E.A., (1962) Elementary Coordinate Geometry, Oxford University press.
4. Jain, P.K. & Ahmad, K. (1994) Analytical Geometry of Two Dimensions, Wiley.
5. Kishan, H. (2006) Coordinate Geometry of Two Dimensions. Atlantic Publishers & Dist.
6. Jain, P.K. (2005) A Textbook of Analytical Geometry Of Three Dimensions. New Age International.
7. McCrea, W.H. (2006). Analytical Geometry of Three Dimensions, Dover Publications, INC.
8. Richter-Gebert J. (2011) Perspectives on Projective Geometry: A Guided Tour Through Real and
Complex Geometry, Springer.
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