Next: MC248 Further Real Analysis
Up: Year 2
Previous: MC243 Aspects of Linear
MC244 Geometry
Credits: 10 
Convenor: Dr. J.F. Watters 
Semester: 1 (weeks 7 to 12) 
Prerequisites: 
essential: MC144, MC145, MC147 

Assessment: 
Coursework: 20% 
One and a half hour exam in January: 80% 
Lectures: 
18 
Classes: 
5 
Tutorials: 
none 
Private Study: 
52 
Labs: 
none 
Seminars: 
none 
Project: 
none 
Other: 
none 
Total: 
75 


Explanation of Prerequisites
The module MC144 provides the mapping terminology
and ideas, e.g. bijection, whilst MC145 covers the
basic work on complex numbers used in this module. A number of examples of
groups arise in this module and that concept is introduced in MC145.
Module MC147 discusses
matrices and these are used here to define mappings with invertible matrices
being particularly significant.
Course Description
The course covers four main topics. Isometries and similarities of the
Euclidean (complex) plane are defined, classified, and represented by
mappings, e.g. of the form . The extended complex plane
(inversive plane) is defined
and properties of bilinear transformations, e.g.
mappings of the sort , are explored using
stereographic projection. The final topic is a brief discussion of the affine
plane using matrices and here the basic idea is to examine mappings of the
plane which are required simply to map lines to lines.
Aims
This module provides concrete examples of mappings and so aims to deepen
understanding of the concept. It also provides examples of classification
and invariants so giving experience of these approaches to mathematical ideas.
Through exercises and examples the module shows how transformations can be
used to solve geometrical problems.
The module aims to develop the group concept through specific examples.
Another main aim of the module is to provide an opportunity to increase the
facility to work with complex numbers and to look at some elementary complex
functions. Finally, using affine transformations the module examines
affine properties of the plane and deals with the affine
classification of conics.
Objectives
To identify transformations in Euclidean and nonEuclidean plane geometries.
To construct transformations with specific properties.
To be familiar with some geometric invariants of groups of transformations of
the plane.
To understand stereographic projection and its mapping properties.
To have an understanding of bilinear transformations of the extended complex
plane.
To use transformations to solve geometric problems.
To be able to identify the affine nature of a conic.
Transferable Skills
Developing facility to work with complex numbers.
Developing understanding of mappings and some grouptheoretical ideas.
Transformation as a problem solving technique.
Written presentation of logical argument in geometric settings.
Syllabus
Basic complex algebra.
Isometries as transformations of the complex plane; products of reflections;
properties invariant under isometries; classification of isometries.
Apply isometric transformations to solve geometric construction problems.
Similarities as transformations of the complex plane;
properties invariant under similarities; classification of similarities.
Apply similarity transformations to solve geometric construction problems.
Define and recognise groups of similarities (isometries).
Define inversion and bilinear transformations of the complex plane;
stereographic projection and the extended complex plane;
properties invariant under bilinear transformations (e.g. crossratio).
Apply inversion to solve geometric construction problems.
Define affine transformations of the plane;
properties invariant under affine transformations (e.g. ratio).
Obtain affine equivalent forms of conics.
Reading list
Recommended:
H. S. M. Coxeter,
Introduction to Geometry, 2nd edition,
Wiley.
M. Jeger,
Transformation Geometry,
Allen and Unwin.
P. J. Ryan,
Euclidean and NonEuclidean Geometry 
an Analytic Approach,
Cambridge University Press.
Details of Assessment
Coursework will be set weekly and marks gained will provide the coursework
component for assessment .Examination  one and a half hours duration with four questions, all of equal
weight, three to
be answered for full marks.
Next: MC248 Further Real Analysis
Up: Year 2
Previous: MC243 Aspects of Linear
S. J. Ambler
11/20/1999