Author: R. H. Fowler

Publisher: Forgotten Books

Published: 2015-06-12

Total Pages: 128

ISBN-13: 9781330044407

Excerpt from The Elementary Differential Geometry of Plane Curves This tract is intended to present a precise account of the elementary differential properties of plane curves. The matter contained is in no sense new, but a suitable connected treatment in the English language has not been available. As a result, a number of interesting misconceptions are current in English text books. It is sufficient to mention two somewhat striking examples, (a) According to the ordinary definition of an envelope, as the locus of the limits of points of intersection of neighbouring curves, a curve is not the envelope of its circles of curvature, for neighbouring circles of curvature do not intersect. (b) The definitions of an asymptote - (1) a straight line, the distance from which of a point on the curve tends to zero as the point tends to infinity; (2) the limit of a tangent to the curve, whose point of contact tends to infinity - are not equivalent. The curve may have an asymptote according to the former definition, and the tangent may exist at every point, but have no limit as its point of contact tends to infinity. The subjects dealt with, and the general method of treatment, are similar to those of the usual chapters on geometry in any Cours d' Analyse, except that in general plane curves alone are considered. At the same time extensions to three dimensions are made in a somewhat arbitrary selection of places, where the extension is immediate, and forms a natural commentary on the two dimensional work, or presents special points of interest (Frenet's formulae). To make such extensions systematically would make the tract too long. The subject matter being wholly classical, no attempt has been made to give full references to sources of information; the reader however is referred at most stages to the analogous treatment of the subject in the Cours or Traite d' Analyse of de la Vallée Poussin, Goursat, Jordan or Picard, works to which the author is much indebted. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.