CiNii 図書 - Differential geometry of curves and surfaces

Students and professors of an undergraduate course in differential geometry will appreciate the clear exposition and comprehensive exercises in this book that focuses on the geometric properties of curves and surfaces, one- and two-dimensional objects in Euclidean space. The problems generally relate to questions of local properties (the properties observed at a point on the curve or surface) or global properties (the properties of the object as a whole). Some of the more interesting theorems explore relationships between local and global properties. A special feature is the availability of accompanying online interactive java applets coordinated with each section. The applets allow students to investigate and manipulate curves and surfaces to develop intuition and to help analyze geometric phenomena.

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

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Differential geometry of curves and surfaces

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