Uses and Limitations of Dynamic Geometry and Computer Algebra in the Analysis of the Shrinking Circle Problem Douglas B. Meade Department of Mathematics.

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Presentation transcript:

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 Let C be the unit circle with center (1,0) C r 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 C r R be the intersection of the line PQ and the x- axis. What happens to R as C r shrinks to the origin? Stewart, Essential Calculus: Early Transcendentals, Thomson Brooks/Cole, 2007, p. 45, Exercise 56.

Visualization Geometer’s Sketchpad MapleGeometry Expressions

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

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

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 Papers/v1n1p4/ Papers/v1n1p4/

Generalized Shrinking Circle Problem Let C be a fixed curve and let C r 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 C r, and R is the point of intersection of the line PQ and the x- axis. What happens to R as C r shrinks?

Lemma Let C be the circle with center (a, b) that includes the origin: (x − a) 2 + (y − b) 2 = a 2 + b 2. Let C r, 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 Let C be a curve in the plane that includes the origin and is twice continuously differentiable at the origin. Define C r, P, Q, and R as in the Generalized Shrinking Circle Problem. If the curvature at the origin, , is positive, the osculating circle of C at the origin has radius  and center (a, b) where a 2 + b 2 =  2. Then, R  (4r, 0 ) if b=0 R  ( 0, 0 ) otherwise

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  be the curvature of C at O. (N is oriented so that O + 1/  N is the center of the osculating circle to C at O.) For any r > 0, define C r to be the circle with radius r centered at O, P = O + r T, the point at the “top” of C r, Q to be the intersection of C and C r, 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 R 0 = O + 4 N.