## MATH3232 Transformation Geometry

### 15 creditsClass Size: 100

Module manager: Professor J. Truss
Email: J.K.Truss@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2008/09

### Pre-requisite qualifications

MATH2200 or MATH2080, or equivalent.

This module is approved as an Elective

### Module summary

Two-dimensional Euclidean geometry is the study of isometries of the plane. Extending the group of transformations leads us first to affine geometry, and then to projective geometry and Poincaré's model of non-Euclidean geometry. A basic idea is to appropriate transformations to turn a complicated geometrical problem to a simple special case. In this way we give proofs of some classical geometrical theorems due to Ceva, Desargue, Menelaus, Pascal and Pappus.

### Objectives

On completion of this module, students should be able to:
a) Use affine transformations to prove appropriate theorems of Euclidean Geometry
b) Use projective coordinates to prove theorems of projective geometry.
c) Express a conic in standard form for affine, Euclidean and projective geometry.
d) Do calculations using inversion and Möbius transformations.

### Syllabus

1. The Kleinian view of geometry.
2. Isometries and Euclidean geometry.
3. Affine transformations, affine geometry, Ceva's Theorem, Menelaus' Theorem, affine classification of conics.
4. Projective geometry, projective coordinates for the plane, projective transformations, Desargue's Theorem, Pappus' Theorem, Pascal's Theorem, projective conics, cross ratio.
5. Inversions in circles, Möbius transformations of the extended complex plane, Poincaré's non-Euclidean Geometry.

### Teaching methods

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

 Delivery type Number Length hours Student hours Example Class 7 1.00 7.00 Lecture 26 1.00 26.00 Private study hours 117.00 Total Contact hours 33.00 Total hours (100hr per 10 credits) 150.00

### Methods of assessment

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

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 3 hr 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated