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  • Print publication year: 1990
  • Online publication date: June 2012

3 - Curves: Part 1

Summary

In this chapter and in Chapter 5 we shall present many mathematical applications of computer graphics. In order to draw the line somewhere (pardon the pun) we shall restrict ourselves to the mathematics associated with the plane and in particular with curves in the plane. There is a whole other realm, just as fascinating, connected with objects, such as curves, polyhedra and surfaces, in three-dimensional space. Nevertheless what the computer actually draws is, in these cases too, a curve or system of curves in the plane – that of the screen. The extra complications come from taking a three-dimensional object and associating with it a curve or system of curves – for example its outline when seen from a distance, or a sequence of such outlines or a sequence of plane sections of the object. We touch on this in a discussion of the swallowtail surface in Section 4.14.

The computer can (with our help) draw curves and collections of curves which are just too complicated to attempt by hand. For some purposes a rough sketch of a curve does very well – you will probably have drawn many such sketches by hand, and we are certain that the art of curve-sketching by hand is still an art well worth acquiring. However, for some other purposes, such as the illustration or discovery of facts connected with the differential geometry of curves and families of curves (not to mention surfaces), accurate drawings are essential.