Differential Geometry of Curves and Surfaces

Thomas F. Banchoff is a geometer and has been a professor at Brown University since 1967. Banchoff was president of the MAA from 1999-2000. He is published widely and known to a broad audience as editor and commentator on Abbotts Flatland. He has been the recipient of such awards as the MAA National Award for Distinguished College or University Teaching of Mathematics and most recently the 2007 Teaching with Technology Award. Stephen Lovett is an associate professor of mathematics at Wheaton College in Illinois. Lovett has also taught at Eastern Nazarene College and has taught introductory courses on differential geometry for many years. Lovett has traveled extensively and has given many talks over the past several years on differential and algebraic geometry, as well as cryptography.

Preface


Acknowledgements








Plane Curves: Local Properties



Parameterizations


Position, Velocity, and Acceleration


Curvature


Osculating Circles, Evolutes, and Involutes


Natural Equations






Plane Curves: Global Properties



Basic Properties


Rotation Index


Isoperimetric Inequality


Curvature, Convexity, and the Four-Vertex Theorem






Curves in Space: Local Properties



Definitions, Examples, and Differentiation


Curvature, Torsion, and the Frenet Frame


Osculating Plane and Osculating Sphere


Natural Equations






Curves in Space: Global Properties



Basic Properties


Indicatrices and Total Curvature


Knots and Links






Regular Surfaces



Parametrized Surfaces


Tangent Planes and Regular Surfaces


Change of Coordinates


The Tangent Space and the Normal Vector


Orientable Surfaces






The First and Second Fundamental Forms



The First Fundamental Form


The Gauss Map


The Second Fundamental Form


Normal and Principal Curvatures


Gaussian and Mean Curvature


Ruled Surfaces and Minimal Surfaces






The Fundamental Equations of Surfaces



Tensor Notation


Gauss’s Equations and the Christoffel Symbols


Codazzi Equations and the Theorema Egregium


The Fundamental Theorem of Surface Theory






Curves on Surfaces





Curvatures and Torsion


Geodesics


Geodesic Coordinates


Gauss-Bonnet Theorem and Applications


Intrinsic Geometry


Bibliography