Titlebook: Elementary Differential Geometry; Andrew Pressley Textbook 20011st edition Springer-Verlag London 2001 Curves and Surfaces.Euclidean Geome

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How Much Does a Curve Curve?,curve is not contained in a straight line (so that straight lines have zero curvature), and the torsion measures the extent to which a curve is not contained in a plane (so that plane curves have zero torsion). It turns out that the curvature and torsion together determine the shape of a curve.

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Global Properties of Curves,e ‘global’ shape of the curve. In this chapter, we discuss some global results about curves. The most famous, and perhaps the oldest, of these is the ‘isoperimetric inequality’, which relates the length of certain ‘closed’ curves to the area they contain.

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Surfaces in Three Dimensions,rface patch, is all that is needed for most of the book, it does not describe adequately most of the objects that we would want to call surfaces. For example, a sphere is not a surface patch, but it can be described by gluing two surface patches together suitably. The idea behind this gluing procedu

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The First Fundamental Form,rface. Of course, this will usually be different from the distance between these points as measured by an inhabitant of the ambient three dimensional space, since the straight line segment which furnishes the shortest path between the points in .. will generally not be contained in the surface. The

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Curvature of Surfaces,t (see Theorem 10.4) that a surface patch is determined up to a rigid motion of .. by its first and second fundamental forms, just as a unit-speed plane curve is determined up to a rigid motion by its signed curvature.