differential geometry
The Analysis of SurfacesIn the analysis of surfaces, points on a surface may be described not only with respect to the threedimensional coordinates of the space in which the surface is considered but also with respect to an intrinsic coordinate system defined in terms of a system of curves on the surface itself. The curves on the surface that locally represent the shortest distances between points on the surface are called geodesics; geodesics on a plane are straight lines. Tangent and normal vectors are also defined for a surface, but the relationships between them are more complex than for a space curve (e.g., a surface has a whole circle of unit vectors tangent to it at a given point). The results of the theory of surfaces are expressed most easily in the notation of tensors. It is found that the total, or Gaussian, curvature of a surface is a bending invariant, i.e., an intrinsic property of the surface itself, independent of the space in which the surface may be considered. Of particular importance are surfaces of constant curvature; planes, cylinders, cones, and other socalled developable surfaces have zero curvature, while the elliptic and hyperbolic planes of nonEuclidean geometry are surfaces of constant positive and negative curvature, respectively. Sections in this article:
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