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Published online by Cambridge University Press: 05 June 2012
Investigations, in which the directions of various straight lines in space are to be considered, attain a high degree of clearness and simplicity if we employ, as an auxiliary, a sphere of unit radius described about an arbitrary center, and suppose the different points of the sphere to represent the directions of straight lines parallel to the radii ending at these points.
C. F. Gauss (8 October 1827)In Chapter 8, surfaces in ℝ3 were introduced and some of their basic structure, such as the first fundamental form, was defined. We now define analogs of curvature and torsion for surfaces in ℝ3. The goal is to describe how a surface “curves” at a point. The first measure we introduce is naive – it will depend on curves in the surface. Later in the chapter, we associate a more appropriate, two-dimensional measure to the same task. Based on these ideas, the generalization for surfaces of the fundamental theorems for curves is realized in Chapter 10. Following the historical path of the subject, we begin with Euler's work.
Euler's work on surfaces
Suppose p is a point in a surface S and N(p) is a choice of unit normal. Let be a unit tangent vector in Tp(S). In ℝ3 take the right-handed frame to define a plane. Translate the plane so that the origin is at p and consider the intersection of this plane with S. This gives a curve on S near p.
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