Banchoff, Thomas; Lovett, Stephen Differential geometry of curves and surfaces. (English) Zbl 1200.53001 Natick, MA: A K Peters (ISBN 978-1-56881-456-8/hbk). xvi, 331 p. (2010). This textbooks covers the content of a one-semester undergraduate course in classical differential geometry, starting with planar curves and ending with the global version of the Gauss-Bonnet Theorem. The table of contents is given below.It seems to be the author’s understanding that an intuitive and visual introduction to the subject is beneficial in an undergraduate course. (The reviewer shares this point of view.) This attitude is reflected in the text. The authors spent quite some time on motivating particular concepts and discuss simple but instructive examples. At the same time, they do not neglect rigour and precision.This book is accompanied by [St. T. Lovett, Differential Geometry of Manifolds. Natick, MA: A K Peters (2010; Zbl 1205.53001)], a textbook that represents the modern view on differential geometry. According to the authors, neither book is a pre-requisite for the other but reading one will be helpful for reading the other. In fact, especially in the later parts of the book under review the gentle guidance towards differential geometry on manifolds is clearly visible. The second book in the pair is also often references when more advanced results from calculus or topology are needed. Each section concludes with a collection of exercises (but no solutions). Approximately 100 figures, a list of 31 bibliography items and an index complete the printed part of the book. As a distinguishing feature to other textbooks, there is an accompanying web-page http://www.akpeters.com/DiffGeo/ containing numerous interactive Java applets. Whenever a definition or an example is illustrated or visualized by an applet, this is indicated by a special symbol in the outer margin. The applets are well-suited for use in classroom teaching or as an aid to self-study.We conclude this review with the book’s table of contents:Preface Acknowledgements1. Plane Curves: Local PropertiesParameterizationsPosition, Velocity, and AccelerationCurvatureOsculating Circles, Evolutes, and InvolutesNatural Equations2. Plane Curves: Global PropertiesBasic PropertiesRotation IndexIsoperimetric InequalityCurvature, Convexity, and the Four-Vertex Theorem3. Curves in Space: Local PropertiesDefinitions, Examples, and DifferentiationCurvature, Torsion, and the Frenet FrameOsculating Plane and Osculating SphereNatural Equations4. Curves in Space: Global PropertiesBasic PropertiesIndicatrices and Total CurvatureKnots and Links5. Regular SurfacesParametrized SurfacesTangent Planes and Regular SurfacesChange of CoordinatesThe Tangent Space and the Normal VectorOrientable Surfaces6. The First and Second Fundamental FormsThe First Fundamental FormThe Gauss MapThe Second Fundamental FormNormal and Principal CurvaturesGaussian and Mean CurvatureRuled Surfaces and Minimal Surfaces7. The Fundamental Equations of SurfacesTensor NotationGauss’s Equations and the Christoffel SymbolsCodazzi Equations and the Theorema EgregiumThe Fundamental Theorem of Surface Theory8. Curves on SurfacesCurvatures and TorsionGeodesicsGeodesic CoordinatesGauss-Bonnet Theorem and ApplicationsIntrinsic GeometryBibliography Reviewer: Hans-Peter Schröcker (Innsbruck) Cited in 3 ReviewsCited in 10 Documents MSC: 53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry 53A04 Curves in Euclidean and related spaces 53A05 Surfaces in Euclidean and related spaces Keywords:differential geometry; computer graphics; Java applet Citations:Zbl 1205.53001 PDFBibTeX XMLCite \textit{T. Banchoff} and \textit{S. Lovett}, Differential geometry of curves and surfaces. Natick, MA: A K Peters (2010; Zbl 1200.53001)