April 2016
Intermediate to advanced
430 pages
8h 58m
English
Content preview from Differential Geometry of Curves and Surfaces, 2nd Edition
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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 for a regular plane curve ...
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