Preface

This book details a fairly traditional view of articulatory phonetics, and some related aspects of phonology. Our focus throughout is on English phonetics, as English is the language of instruction, and the one with which all readers will therefore be familiar. Aspects of general phonetic theory are illustrated using examples from English, and supported by other languages where appropriate. We begin in Section 1 with a concentration on individual speech sounds, think about how sounds combine into words in Section 2, and finish in Section 3 with phenomena that occur when words are combined into longer stretches of speech.

The book is aimed at students with no prior knowledge of phonetics or linguistics; therefore, new terminology is emboldened and explained when it is first introduced. The book is suitable for first-year undergraduates studying subjects such as linguistics or speech and language therapy, and may also be used for revision by more advanced students. It would certainly be possible for students to teach themselves a good deal of phonetics using this coursebook. However, as phonetics is the study of speech, discussion with a tutor, who can demonstrate particular sounds and clarify any variant aspects of pronunciation, is sometimes recommended in the text. The book may also be used in class, with students working through the exercises either before or during contact hours. Whether used alone, with a tutor or in a class, the units should be attempted in order. Each unit builds on the last, and it is assumed that all previous units have been completed at each stage.

Geometry is one branch of mathematics that has an obvious relevance to the ‘real world’. Earlier, we studied some results in Euclidean geometry and we described the group of Euclidean transformations, the isometries. We saw that the Euclidean transformations preserve distances and angles, and have a definite physical significance.

In this chapter we study projective geometry, a very different type of geometry, that has important but less obvious applications. It was discovered through artists' attempts over many centuries to paint realistic-looking pictures of scenes composed of objects situated at differing distances from the eye. How can three-dimensional scenes be represented on a two-dimensional canvas? Projective geometry explains how an eye perceives ‘the real world’, and so explains how artists can achieve realism in their work.

In Section 3.1, we look at the development of perspective in Art and explain the concept of a perspectivity. We describe Desargues' Theorem, which concerns a curious property of two triangles whose vertices are in perspective from a single point, and so explain that perspective can play a key role in the statement and the proof of theorems in mathematics.

In Section 3.2, we define the term projective point (or Point) and call the set of all such Points the projective plane, which we denote by ℝℙ2. We also define a projective line (or Line). To enable us to tackle problems in projective geometry algebraically, we introduce homogeneous coordinates to specify the Points in ℝℙ2.