Abstract
This chapter focuses on frame fields. Geometry begins with the measurement of distances and angles. The geometry of Euclidean space can be derived from the dot product, the natural inner product on Euclidean space. The chapter reviews the geometry of curves in Euclidean 3-space E3. A curve in E3 is discussed by assigning at each point a certain frame—that is, a set of three orthogonal unit vectors. The rate of change of these vectors along the curve is then expressed in terms of the vectors themselves by the Frenet formulas. The chapter uses the “method of moving frames” to study a surface in E3. The general idea is to think of a surface as a kind of two-dimensional curve and follow the Frenet approach as closely as possible. The derivation of mathematical measurements of the turning and twisting of a curve in E3 is reviewed, and the unit-speed curves are dealt with in the chapter.
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