Elementary Geometry of Differentiable Curves: An Undergraduate Introduction

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.

Table of Contents:
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.

About the Author :
Chris Gibson received an honours degree in Mathematics from St Andrews University in 1963, and later the degrees of Drs Math and Dr Math from the University of Amsterdam, returning to England in 1967 to begin his 35 year mathematics career at the University of Liverpool. His interests turned towards the geometric areas, and he was a founder member of the Liverpool Singularities Group until his retirement in 2002 as Reader in Pure Mathematics, with over 60 published papers in that area. In 1974 he co-authored the significant Topological Stability of Smooth Mappings, presenting the first detailed proof of Thom's Topological Stability Theorem. In addition to purely theoretical work in singularity theory, he jointly applied singular methods to specific questions about caustics arising in the physical sciences. His later interests lay largely in the applications to theoretical kinematics, and to problems arising in theoretical robotics. This interest gave rise to a substantial collaboration with Professor K. H. Hunt in the Universities of Monash and Melbourne, and produced a formal classification of screw systems. At the teaching level his major contribution was to pioneer the re-introduction of undergraduate geometry teaching. The practical experience of many years of undergraduate teaching was distilled into three undergraduate texts published by Cambridge University Press, now widely adopted internationally for undergraduate (and graduate) teaching.

Review :
'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)