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

Abstract
The Shrinking Circle Problem is an example of a simple-to-state geometry problems can be 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 dimensions: the Shrinking Sphere Problem. We show why traditional numerical attempts to answer even the simplest problem is futile. The general results were first formulated based on visual evidence produced by dynamic geometry software. Only then was it possible to utilize symbolic computation tools to put together the complete proofs. All supplemental materials that accompany this paper can be found at the URL
http://www.math.sc.edu/~meade/eJMT-Shrink.
http://www.radford.edu/~scorwin/eJMT/Content/Papers/v1n1p4/

Complete Paper [PDF]

Complete Web Materials [ZIP]

Supplemental Resources by Section

Video	Sketch Pad	Geom Expr	Maple 10	HTML
						§1 - Introduction
				HTML			Geometric Solution of the Original Shrinking Circle Problem
						§2 - The Original Shrinking Circle Problem
				HTML			Analytic Solution of the Original Shrinking Circle Problem
				HTML			Numeric Solution of the Original Shrinking Circle Problem (Table 1)
						§3 - The Generalized Shrinking Circle Problem
				HTML			Lemma 1
							Derivations and Animations for Other Fixed Curves
				HTML				Ellipse
				HTML				Line
				HTML				Parabola
							Theorem 2
						§4 - The Simplest Shrinking Sphere Problems
				HTML			Lemma 4 (including Figure 4)
						§5 - The Generalized Shrinking Sphere Problems
							Derivations and Animations for Other Fixed Surfaces
				HTML				Ellipsoid
				HTML				Cone
				HTML				Paraboloid
						§6 - Discussion of Numerical Limits and Visualization
							Numerical Difficulties for the Original Shrinking Circle Problem
				HTML			Visualizing the Generalized Shrinking Sphere with Discrete Samples (Figure 5)
							Visualizing the Generalized Shrinking Sphere with Geometric Objects (Figure 6)
						§7 - Conclusion

List of Software

Icon	Name	Web Address
	Cabri 3D	http://www.cabri.com/
	ClassPad Manager	http://www.classpad.net/
	Geometry Expressions	http://www.geometryexpressions.com/
	Geometer's Sketchpad	http://www.dynamicgeometry.com/
	Maple	http://www.maplesoft.com/
	Mathematica	http://www.wolfram.com/