The Geometry of Surfaces of Revolution
Andrew D. Hwang
College of the Holy Cross
November 13, 2002
Bailey Hall 201
A classical surface of revolution is an object having an axis of rotational symmetry, such as a sphere, cone, or a piece of furniture turned on a lathe. As Every Calculus Student Knows [tm], a surface of revolution is obtained by revolving the graph of a function about an axis. In this talk, we'll investigate "abstract" surfaces of revolution from the point of view of 2-dimensional beings who live inside such a "universe", with particular attention to the following questions:
- How could such beings prove that their universe is curved without reference to a third dimension?
- In what coordinate system is the curvature of their universe most simply described? (A convincing answer will be given.)
- Do all such universes arise as surfaces of revolution in the calculus sense? If not, can we visualize "exotic" surfaces?
The only prerequisites are geometric imagination and a bit of calculus.
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