2 - Classical differential geometry of space-curves

Abstract

This chapter focuses on the classical differential geometry of space-curves. It focuses on the smooth curves in E3 specified in terms of rectangular cartesian coordinates x, y, z (or y1, y2, y3). Such curves are generated by three smooth functions of a single real parameter so that the position vector r of points on the curve relative to some origin O is given by r = r(t) = x(t)i + y(t)j + z(t)k. While discussing the serret-frenet equations, the chapter explains that given a unit tangent vector t and a unit principal normal n at a point on a curve in E3, one can define a third unit vector b, called the unit binormal vector, orthogonal to both of them, such that b =t x n. . It may be instructive to derive the Serret-Frenet formulas. In the process, the chapter introduces the concept of intrinsic differentiation. In the context of grid generation, space-curves appear as boundaries of surfaces and as edges of three-dimensional blocks, and it is convenient to map a given finite length of space-curve onto an interval of the ξ-axis. A uniformly spaced set of points in the ξ -interval will then map to a set of points along the curve.