Elementary geometry of differentiable curves: an undergraduate introduction (Q2715720)

This genuine introduction to the differential geometry of plane curves is designed as a first text for undergraduates in mathematics, or postgraduates and researchers in the engineering and physical sciences. The algebraic viewpoint was developed in the author's book [`Elementary Geometry of Algebraic Curves', Cambridge University Press (1998; Zbl 0997.14500)]. The present text is intended as a companion volume, representing the differentiable viewpoint. The basic concepts are illustrated by named curves, of historical and scientific significance, leading to the central idea of curvature. The singular viewpoint is represented by a study of contact with lines and circles, leading to an understanding of exceptional points on curves such as cusps, inflexions and vertices. NEWLINENEWLINENEWLINEThe singular theme is continued via a discussion of envelopes. As a serious application, two chapters are devoted to caustics by reflexion, via the central concepts of evolute and orthotomic, an attractive area of potential application to several areas of the physical sciences, which deserves to be better known. Caustics are already of considerable importance in geometric optics. However, their significance in acoustics is not so well established, and there is little doubt that they play a key role in understanding the mysteries of radio propagation. NEWLINENEWLINENEWLINEThe final chapters introduce the core material of classical kinematics, developing the geometry of trajectories via the ideas of roulettes and centrodes, and culminating in the inflexion circle and cubic of stationary curvature. The book assumes only foundational year mathematics: it is well illustrated, and contains several hundred worked examples and exercises (many culled from the older literature), making it suitable for adoption as a course text. NEWLINENEWLINENEWLINEFrom the table of contents: The Euclidean plane, Parametrized curves, Classes of special curves, Arc length, Curvature, Existence and uniqueness, Contact with lines, Contact with circles, vertices, envelopes, orthotomics, Caustics by reflexion, Planar kinematics, Centrodes, Geometry of trajectories.