Geometric differentiation for the intelligence of curves and surfaces. (English) Zbl 0806.53001

Porteous, Ian

Cambridge: Cambridge University Press. xiii, 301 p. (1994).

This is a delightful and very useful introduction to geometry based on the interrelationship between a curve and its evolute, and a surface and its evolute surfaces. On the way, the author introduces all that is needed for the understanding of differential geometry and singularity classification, as well as applications of singularity theory. It is most highly recommended as an introduction to modern views of nonmetric differential geometry.
The first chapter deals with curves and their evolutes, introduces the singularities possible and impossible on evolutes, studies offset (parallel) curves, and ends with the four-vertices theorem. The next chapter teaches some technical details and methodical background material. Then comes a quick and elegant treatment of plane kinematics. A chapter on curves on the unit sphere leads to one on general space curves, their parallel curves and focal surfaces. The next two chapters are technical, one on quadratic and cubic forms, and one on normal forms (“probes”) of jets. A chapter on contact and contact equivalence leads to the study of surfaces and their lines of curvature, and then to the singularities, ridges and ribs, of the focal surfaces. The following chapters, containing the modern geometry, start out with Darboux’s and other classifications of umbilics [the half-integer nature of the index at the umbilic being taken for granted], a detailed study of the parabolic line as introduction to Gauss curvature, and the study of the relations between the two sheets of the focal surface if some regularity condition on the surface is violated. [To appreciate the modern mathematics, this chapter should be compared to Chapter IX in L. Bianchi, Lezioni di geometria differenziale, vol. I, parte II, Zanichelli (Bologna 1927)]. A final chapter applies the theory to several types of concrete surfaces. A few misprints are easily corrected. The name of the Russian master of singularities is Арнольд, not Ариопд!

Reviewer: H.Guggenheimer (West Hempstead)

MSC:

53A04	Curves in Euclidean and related spaces
53A05	Surfaces in Euclidean and related spaces
53-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
58C25	Differentiable maps on manifolds
58K99	Theory of singularities and catastrophe theory

Keywords:

evolute; evolute surfaces; singularity theory; four-vertices theorem; plane kinematics; focal surfaces; jets; contact equivalence; classifications of umbilics