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

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 meade@math.sc.edu. 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

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  1. 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 meade@math.sc.edu ACMS Biennial Conference, Messiah College 1 June 2007

  2. 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.

  3. Visualization

  4. Proofs • Numeric • Extremely sensitive to floating-point cancellation • Symbolic • Indeterminate form(l’Hopital, or simply rationalize) • Geometric • Tom Banchoff (Brown University)

  5. OP = OQ is isosceles P  O SR=SQ < OS Q  O OR = OS + SR < 2 OS QS  OS OR = OS+SR = OS+QS  2OS Geometric Proof

  6. Details and Generalizations 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/

  7. Generalized 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?

  8. Lemma 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

  9. Theorem 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

  10. Theorem (Coordinate-Free) 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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