2016/17 Undergraduate Module Catalogue

MATH3232 Transformation Geometry

15 creditsClass Size: 50

Module manager: Dr Kevin Houston
Email: K.Houston@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2016/17

Pre-requisite qualifications

MATH2020 or MATH2022 or MATH2080, or equivalent.

This module is approved as a discovery module

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 disc model of non-Euclidean geometry. A basic idea is to use 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, Desargues, Menelaus, Pascal and Pappus.

Objectives

To develop abstract ideas of geometry based on considering the transformations that respect the various geometrical constructs.

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 Moeobius transformations.
e) do calculations either in Poincaré's disc model or in spherical geometry.

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, Desargues' Theorem, Pappus' Theorem, Pascal's Theorem, projective conics, cross ratio.
5. Inversions in circles, Mobius transformations of the extended complex plane, either Poincaré's disc model of non-Euclidean geometry or, an introduction to spherical geometry.

Teaching methods

Delivery type	Number	Length hours	Student hours
Lecture	33	1.00	33.00
Private study hours			117.00
Total Contact hours			33.00
Total hours (100hr per 10 credits)			150.00

Private study

Studying and revising of course material.
Completing of assignments and assessments.

Opportunities for Formative Feedback

Regular problem solving assignments

Methods of assessment

Exams

Exam type	Exam duration	% of formal assessment
Standard exam (closed essays, MCQs etc)	2 hr 30 mins	100.00
Total percentage (Assessment Exams)		100.00

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

Reading list

The reading list is available from the Library website

Last updated: 08/04/2016

Disclaimer

Browse Other Catalogues

• Undergraduate module catalogue
• Taught Postgraduate module catalogue
• Undergraduate programme catalogue
• Taught Postgraduate programme catalogue

Errors, omissions, failed links etc should be notified to the Catalogue Team.PROD