Elementary Geometry of Differentiable Curves
An Undergraduate Introduction

This book is an introductory text on the differential geometry of plane curves.

C. G. Gibson (Author)

9780521011075, Cambridge University Press

Paperback, published 17 May 2001

238 pages, 40 b/w illus.
22.9 x 15.2 x 1.7 cm, 0.341 kg

'It is meant to be a genuine introduction to the differential geometry of plane curves and in fact it is … I can warmly recommend this booklet for students and scientists who have not yet gathered experience in differential geometry and who want to give themselves a treat.' J. Lang, IMN (Internationale Mathematische Nachrichten)

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 book assumes only foundational year mathematics: it is well illustrated, and contains several hundred worked examples and exercises, making it suitable for adoption as a course text. 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, illuminating the ideas of cusp, inflexion and vertex. There are two major physical applications. Caustics are discussed via the central concepts of evolute and orthotomic. The 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.

1. The Euclidean plane
2. Parametrized curves
3. Classes of special curves
4. Arc length
5. Curvature
6. Existence and uniqueness
7. Contact with lines
8. Contact with circles
9. Vertices
10. Envelopes
11. Orthotomics
12. Caustics by reflexion
13. Planar kinematics
14. Centrodes
15. Geometry of trajectories.

Subject Areas: Topology [PBP], Geometry [PBM], Mathematics [PB]