Differential Geometry of Curves and Surfaces, 2nd Edition

CHAPTER 9

Curves and Surfaces in n-dimensional Euclidean Space

Up to this point, this text emphasized curves and surfaces in ℝ³. However, our only reason to do so was based on a discrimination by dimensionality: As beings living in three-dimensional space, we are accustomed to visualizing in ℝ³. In this chapter, we generalize a number of definitions and theorems about curves and surfaces to Euclidean n-dimensional space.

9.1 Curves in n-dimensional Euclidean Space

In Chapters 1 and 3, we studied local properties of curves with an emphasis on notions of velocity, acceleration, speed, curvature, and torsion. Along the way, we introduced the natural orthonormal frames attached to curves at a point, namely the (T→,U→) for a regular plane curve ...