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## MATH3232 Transformation Geometry

### 15 creditsClass Size: 45

Module manager: Dr Alexandr Buryak
Email: a.buryak@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2019/20

### Pre-requisite qualifications

MATH2020 or MATH2022 or MATH2080, or equivalent.

This module is not approved as a discovery module

### Module summary

In Transformation Geometry we study geometrical properties that are preserved after we apply diff erent types of maps. So, for example, if we rotate or reflect a shape, or stretch it in a certain direction, which properties, such as angle, length, area, parallelism of lines, are unchanged? We shall study transformations in Euclidean,affine and projective geometries as these are the most basic for the world around us and are used in applications as diverse as computer games and in Google street view.A major theme of the course is to transform a complicated geometrical problem into a special case that is extremely easy to solve. With this approach we will prove some classic theorems of geometry due to Ceva, Desargues, Menelaus, Pascal and Pappus. Along the way we see why the most famous map of the world is misleading, how to manipulate photos from 360 degree cameras for fun and art, and we will even make parallel lines meet!

### 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 involving conformality and using inversion and Mobius transformations.
e) apply projections to relate spheres and planes.

### Syllabus

1. Isometries and Euclidean geometry.
2. Affine transformations, affine geometry, Ceva's Theorem, Menelaus' Theorem, affine classification of conics.
3. Projective geometry, projective coordinates for the plane, projective transformations, Desargues' Theorem, Pappus' Theorem, Pascal's Theorem, projective conics, cross ratio.
4. Conformality. Map projections.
5. Mobius transformations of the extended complex plane. Transformations of spherical pictures.

### 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 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

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) 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