Geometric Differentiation: For the Intelligence of Curves and Surfaces
This is a revised and extended version of the popular first edition. Inspired by the work of Thom and Arnold on singularity theory, such topics as umbilics, ridges and subparabolic lines, all robust features of a smooth surface, which are rarely treated in elementary courses on differential geometry, are considered here in detail. These features are of immediate relevance in modern areas of application such as interpretation of range data from curved surfaces and the processing of magnetic resonance and cat-scan images. The author has included many exercises and examples to illustrate the results proved.
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centre of curvature Chapter coincides cubic form defined denote derivative determinant diffeomorphism differential ellipsoid elliptic equation equivalent everywhere example Figure finite-dimensional vector spaces flexcord focal curve focal line focal surface foliation form on IR2 Gauss Gauss map Gaussian curvature geodesic inflection Hessian hyperbolic implying intersection inverse involutes Jacobian kernel vector linear map linear subspace linearly independent lines of curvature map-germ matrix monstar multiple mutually orthogonal non-zero vector normal line ordinary cusp ordinary inflection origin parabolic line parametric parametrisation plane curve polynomial principal curvatures principal vector probe Proof quadratic form radius of curvature regular curve regular smooth surface regular space curve regular surface rhamphoid cusp ridge root lines second fundamental form sheet singularity smooth curve space curve space evolute stationary curvature submanifold subparabolic lines suppose surface in IR3 surjective symmetric tangent developable tangent line Theorem thrice linear form unit normal vector space vertices zero
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