It is worth saying something about the background to this book, since it is linked to a sea change in the teaching of university mathematics, namely the renaissance in undergraduate geometry, following a postwar decline. There is little doubt that the enormous progress made in studying non-linear phenomena by geometric methods has rekindled interest in the subject. However, that is not the only reason for seeking change, as I pointed out in the preface to Elementary Geometry of Algebraic Curves:

‘For some time I have felt there is a good case for raising the profile of undergraduate geometry. The case can be argued on academic grounds alone. Geometry represents a way of thinking within mathematics, quite distinct from algebra and analysis, and so offers a fresh perspective on the subject. It can also be argued on purely practical grounds. My experience is that there is a measure of concern in various practical disciplines where geometry plays a substantial role (engineering science for instance) that their students no longer receive a basic geometric training. And thirdly, it can be argued on psychological grounds. Few would deny that substantial areas of mathematics fail to excite student interest: yet there are many students attracted to geometry by its sheer visual content.’

Background

A good starting point in developing undergraduate geometry is to focus on plane curves. They comprise a rich area, of historical significance and increasing relevance in the physical and engineering sciences.