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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 [email protected] ACMS Biennial Conference, Messiah College 1 June 2007. Shrinking Circle Problem. Let

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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

[email protected]

ACMS Biennial Conference, Messiah College

1 June 2007


Shrinking circle problem l.jpg
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.


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Visualization Algebra in the Analysis of the Shrinking Circle Problem


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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)


Geometric proof l.jpg

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 Proof


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Details 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/


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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?


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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


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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


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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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