The Elementary Differential Geometry Of Plane Curves

A precise account of the elementary differential properties of plane curves, this text provides a link between analysis and more complicated geometrical theorems, offering background to geometry students and practice for analysis students. It focuses on general plane curves, but it encompasses extensions to three dimensions in situations where the extension is immediate and forms a natural commentary on the two-dimensional work, and when special points of interest--such as Frenet’s formulae--are involved.
After a brief introduction, the text examines the elementary properties of tangents and normals, the curvature of plane curves, and the theories of contact and of envelopes. Additional topics include singular points of plane curves and asymptotes of plane curves. This treatment pursues the greatest generality in its enunciation of theorems; its primary consideration is the determination of the necessary and sufficient conditions for the truth of any theorem. In the proofs of the elementary theorems, where this procedure is adopted, the text is somewhat more laborious than any rigorous treatment; however, this procedure provides a connection between analysis and more complicated geometrical theorems, in which insistence on precise conditions is inappropriate and suitable sufficient conditions can always be tacitly assumed.