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Next: MC248 Further Real Analysis Up: Year 2 Previous: MC243 Aspects of Linear

MC244 Geometry

MC244 Geometry

Credits: 10 Convenor: Dr. J.F. Watters Semester: 1 (weeks 1 to 6)

Prerequisites: essential: MC144, MC145, MC147
Assessment: Coursework: 20% One and a half hour exam: 80%

Lectures: 18 Classes: none
Tutorials: 6 Private Study: 51
Labs: none Seminars: none
Project: none Other: none
Total: 75

Explanation of Pre-requisites

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 $z \mapsto az + b$. The extended complex plane (inversive plane) is defined and properties of bilinear transformations, e.g. mappings of the sort $z \mapsto 1/z$, 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 non-Euclidean 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 group-theoretical ideas.

Transformation as a problem solving technique.

Written presentation of logical argument in geometric settings.

Syllabus

Definition of the complex numbers. Modulus and conjugate and their properties. Polar form. Definition of ez and $re^{i\theta}$ form for z. Equation of circle.

Define a transformation of the plane as a bijective mapping. Define an isometry; distinguish direct and opposite. $z \mapsto \bar{z}$ is reflection in the real axis. $z \mapsto z + b $ is a translation. $z \mapsto az, \vert a\vert = 1$ is the rotation about O through arg a. Prove that $z \mapsto az + b, ( z \mapsto a\bar{z} + b)$ is an isometry iff |a| = 1. Isometries map lines to lines and preserve parallelism and angles. $z \mapsto a\bar{z}, \vert a\vert = 1$ is a reflection and find its axis. Obtain $z \mapsto a\bar{z} + b, \vert a\vert = 1$ form for reflection in line thorugh given point in given direction. Find criteria under which the transformation $z \mapsto a\bar{z} + b, \vert a\vert = 1$ is a reflection. Describe the properties of the transformation $z \mapsto a\bar{z} + b, \vert a\vert = 1$ when $a\bar{b} + b \neq 0$. Define a glide reflection. Prove that every isometry is uniquely determined by its effect on three non-collinear points. Classify direct and opposite isometries. Classify the various types of isometry by their complex representations. Describe the product of two reflections. Describe the product of three reflections, distinguishing reflection and glide reflection cases. Prove that every isometry is a product of at most three reflections. Describe products of rotations and translations as a rotation identifying centre and angle. Apply transformations to solve geometric construction problems. Identify positional relationships between points and lines via equations involving the corresponding half-turns and reflections and vice-versa.

Define similarity transformations and central dilations. Describe $z \mapsto a\bar{z}$ where a is non-zero complex number. Show that central dilations are given by $z \mapsto az + b$ where a is non-zero real number $\neq 1, b \in {\bf C}$. Show that all similarities are given by either $z \mapsto az + b$ or $z \mapsto a\bar{z} + b$ where a is non-zero complex number, $b \in {\bf C}$. Prove that every similarity is the product of a central dilation and an isometry. Similarities map lines to lines, preserve parallelism and angles. Know that there is a (unique) similarity mapping one triangle onto a given similar triangle. Define dilative rotations and dilative reflections recognising centre and axes. Express dilative rotations and dilative reflections as products of central dilations and rotations or reflections. Compute similarities (isometries) given two or three points and their images. Define and recognise groups of similarities (isometries).

Define the operation of inversion in a given circle as a transformation of the extended complex (inversive) plane. Give inversion in the unit circle as a complex transformation and relate this to the reciprocal map. Define bilinear transformations. Express bilinear transformations as products of the reciprocal map and similarities. Compute bilinear transformations given two or three points and their images. Define stereographic projection and be able to relate points on the sphere to points in the plane by their coordinates. Show that stereographic projection is conformal. Show that the reciprocal map and all bilinear maps are conformal. Understand the relationship between various circles on the unit sphere and lines and circles (circlines) in the plane. Know that lines through the centre of inversion are fixed. Know that lines not through the centre of inversion are mapped to circles through the centre and conversely. Know that circles not through the centre of inversion are mapped to circles of the same type. Use inversion to change circular problems into linear problems. Know that bilinear transformations are uniquely determined by their effect on three distinct points. Define the cross-ratio of four numbers in the extended complex plane. Show that bilinear transformations preserve cross-ratio. Prove that four points are collinear or concyclic if and only if their cross-ratio is real.

Define affine transformations. Prove that every affine transformation is a collineation and know that converse is true. Know that affine transformations are determined by action on three non-collinear points. Show that (collineations) affine transformations form a group of transformations. Construct affine transformations to map one triangle to another. Shoe that segment division ratios are preserved by affine transformations. Know that collinearity, parallelism, and segment division ratios, in particular mid-points, are affine properties. Show that the standard conics in the Euclidean plane are affine equivalent to the unit circle (ellipse), rectangular hyperbola xy = 1 (hyperbola), and y2 = x (parabola). Use affine equivalent forms to describe properties of conics. Use completion of squares to reduce general equation of the second degree in x,y to a standard form.

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 Non-Euclidean Geometry - an Analytic Approach, Cambridge University Press.

Details of Assessment

Coursework - six pieces of work, each of equal weight;
Examination - one and a half hours duration with four questions, all to be answered for full marks, all of equal weight. The Casio FX82 is the only calculator allowed in this examination.

next up previous
Next: MC248 Further Real Analysis Up: Year 2 Previous: MC243 Aspects of Linear
Roy L. Crole
10/22/1998