2005/06 Undergraduate Module Catalogue

MATH3232 Transformation Geometry

15 creditsClass Size: 100

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2005/06

Pre-requisites

MATH2200 or MATH2080 or equivalent.

This module is approved as an Elective

Objectives

To develop some basic geometrical ideas used in applications as diverse as navigation and relativity. On completion of this module, students should be able to: a) use transformations to obtain projective theorems from particular cases of Euclidean theorems; b) calculate and use cross-ratios; c) compute time contraction in relativity; d) compute navigational problems using spherical geometry; e) express a conic in standard form for affine, Euclidean and projective geometry.

Syllabus

This module uses linear algebra to develop the geometry of groups of transformations and their invariants. A basic idea is to transform a complicated geometrical problem to a simple special case such that the essential features of the problem are invariant under the transformation. Affine, Euclidean, projective, Lorentz and spherical geometries will be studied. The topics covered are: 1. Transformation groups and invariants. 2. Affine group and ratio. 3. Euclidean group and Euclidean distance, congruent triangles. 4. Orthogonal group acting on S2 , area of spherical triangles. 5. Projective group, duality, Desargues? Theorem, cross-ratio. 6. Classification of conics for affine, Euclidean and projective geometries. 7. Lorentz group and light lines.

Teaching methods

Lectures: 26 hours. 7 examples classes.

Methods of assessment

One 3 hour examination at end of semester (100%).

Reading list

The reading list is available from the Library website

Last updated: 13/05/2005

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