## 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

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Lectures: 26 hours. 7 examples classes.

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

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