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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Douglas B. Meade
Department of Mathematics
University of South Carolina
ACMS Biennial Conference, Messiah College
1 June 2007
What happens to R as Cr shrinks to the origin?
Stewart, Essential Calculus: Early Transcendentals,Thomson Brooks/Cole, 2007, p. 45, Exercise 56.
OP = OQ Algebra in the Analysis of the Shrinking Circle Problem
SR=SQ < OS
OR = OS + SR < 2 OS
OR = OS+SR = OS+QS 2OSGeometric Proof
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
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?
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.
R (4a, 0) if b = 0
R ( 0, 0 ) otherwise
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.
R (4r, 0 ) if b=0
R ( 0, 0 ) otherwise
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
Then, as r decreases to 0, R converges to the point R0 = O + 4N.