Curves

Abstract

In preceding chapters, we have discussed some special types of curves, such as the ellipse, the circle, and the straight line. But, these are only a few of the infinite variety of curves that can be imagined. The path of a bird flying through the air, the instantaneous shape of a swinging chain held at its top, the outlines of petals, the forms of arches and suspension bridges-all provide physical examples of curves. We wish to describe such general curves mathematically and to explore their essential properties. To do so, we will utilize the concepts and theoretical tools that have been introduced in earlier chapters. Our aim is to proceed from intuitive notions about curves to a clear, abstract definition. This process - the clarification of ideas - is really one of the most important activities of the mathematician.