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Uses and Limitations of Dynamic Geometry and Computer Algebra in the Analysis of the Shrinking Circle ProblemPowerPoint Presentation

Uses and Limitations of Dynamic Geometry and Computer Algebra in the Analysis of the Shrinking Circle Problem

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### Uses and Limitations of Dynamic Geometry and Computer Algebra in the Analysis of the Shrinking Circle Problem

Douglas B. Meade

Department of Mathematics

University of South Carolina

ACMS Biennial Conference, Messiah College

1 June 2007

Shrinking Circle Problem Algebra in the Analysis of the Shrinking Circle Problem

Let

- C be the unit circle with center (1,0)
- Cr be the circle with radius r and center (0,0)
- P be the point (0, r)
- Q be the upper point of intersection of C and Cr
- R be the intersection of the line PQ and the x-axis.
What happens to R as Cr shrinks to the origin?

Stewart, Essential Calculus: Early Transcendentals,Thomson Brooks/Cole, 2007, p. 45, Exercise 56.

Visualization Algebra in the Analysis of the Shrinking Circle Problem

Proofs Algebra in the Analysis of the Shrinking Circle Problem

- Numeric
- Extremely sensitive to floating-point cancellation

- Symbolic
- Indeterminate form(l’Hopital, or simply rationalize)

- Geometric
- Tom Banchoff (Brown University)

OP = OQ Algebra in the Analysis of the Shrinking Circle Problem

is isosceles

P O

SR=SQ < OS

Q O

OR = OS + SR < 2 OS

QS OS

OR = OS+SR = OS+QS 2OS

Geometric ProofDetails and Generalizations Algebra in the Analysis of the Shrinking Circle Problem

D.B. Meade and W-C Yang,Analytic, Geometric, and Numeric Analysis of the Shrinking Circle and Sphere Problems, Electronic Journal of Mathematics and Technology, vol. 1, issue 1, Feb. 2007, ISSN 1993-2823

- http://www.math.sc.edu/~meade/eJMT-Shrink/
- http://www.radford.edu/~scorwin/eJMT/Content/Papers/v1n1p4/

Generalized Shrinking Circle Problem Algebra in the Analysis of the Shrinking Circle Problem

Let C be a fixed curve and let Cr be the circle with center at the origin and radius r. P is the point (0, r), Q is the upper point of intersection of C and Cr, and R is the point of intersection of the line PQ and the x-axis.

What happens to R as Cr shrinks?

Lemma Algebra in the Analysis of the Shrinking Circle Problem

Let C be the circle with center (a, b) that includes the origin: (x − a)2 + (y − b)2 = a2 + b2. Let Cr, P, Q, and R be defined as in the Generalized Shrinking Circle Problem.

Then

R (4a, 0) if b = 0

R ( 0, 0 ) otherwise

Theorem Algebra in the Analysis of the Shrinking Circle Problem

Let C be a curve in the plane that includes the origin and is twice continuously differentiable at the origin. Define Cr, P, Q, and R as in the Generalized Shrinking Circle Problem. If the curvature at the origin, k, is positive, the osculating circle of C at the origin has radius r=1/k and center (a, b) where a2 + b2 = r2.

Then,

R (4r, 0 ) if b=0

R ( 0, 0 ) otherwise

Theorem (Coordinate-Free) Algebra in the Analysis of the Shrinking Circle Problem

Let O be a point on a curve C in the plane where the osculating circle to C at O exists. Let T and N be the unit tangent and normal vectors to C at O, respectively. Let k be the curvature of C at O.

(N is oriented so that O + 1/kN is the center of the osculating circle to C at O.)

For any r > 0, define

- Cr to be the circle with radius r centered at O,
- P = O +rT, the point at the “top” of Cr,
- Q to be the intersection of C and Cr, and
- R to be the point on the line through P and Q such that OR is parallel with N
Then, as r decreases to 0, R converges to the point R0 = O + 4N.

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