Analytic, Geometric, and Numeric Analysis of the Shrinking Circle and Sphere Problems.pdf

The Electronic Journal of Mathematics and Technology, Volume 1, Issue 1, ISSN 1933-2823 Analytic, Geometric, and Numeric Analysis of the Shrinking Circle and Sphere Problems Douglas B. Meade∗ Wei-Chi Yang e-mail: meade@ e-mail: wyang@ Department of Mathematics Department of Mathematics and Statistics University of South Carolina Radford University Columbia, SC 29208 USA Radford, VA 24142 USA 31 January 2007 Abstract The Shrinking Circle Problem is an example of a simple-to-state geometry problem that is visually appealing yet quite challenging to solve. A combination of geometry and analysis is used to completely solve the general problem in the plane, and its extension to three dimen- sions: the Shrinking Sphere Problem. We show why traditional numerical attempts to an- swer even the simplest problem is futile. The original problem was generalized based on vi- sual evidence produced by dynamic geometry software. Only with this insight was it possible to utilize symbolic computation tools to put together the complete proofs. All supplemen- tal materials that accompany this paper can be found online at either of the folowing URLs: http: // www. math. sc. edu/ ~meade/ eJMT-Shrink/ http: // www. radford. edu/ ~scorwin/ eJMT/ Content/ Papers/ v1n1p4 . 1 Introduction The following problem appears, with slightly different notation, as an exercise in [Stewart, 2007, p. 45, Exercise 56]. Problem 1 (The Original Shrinking Circle Problem) Figure 1 shows a fixed circle C with equation (x − 1)2