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- Douglas B. Meade
- Department of Mathematics
- University of South Carolina
- meade@math.sc.edu
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- 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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- Numeric
- Extremely sensitive to floating-point cancellation
- Symbolic
- Indeterminate form
(l’Hopital, or simply rationalize)
- Geometric
- Tom Banchoff (Brown University)
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- D.B. Meade and W-C Yang,
Analytic, Geometric, and Numeric Analysis of the Shrinking Ci=
rcle
and Sphere Problems, Electronic Journal of Mathematics and Technolog=
y,
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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- Let C be a fixed curve and let Cr be the circle with cen=
ter
at the origin and radius r. P is the point (0, r), Q is the upper po=
int
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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- Let C be the circle with center (a, b) that includes the origin: (x
− a)2 + (y − b)2 =3D a2=
+ b2.
Let Cr, P, Q, and R be defined as in the Generalized
Shrinking Circle Problem.
- Then
- R à (4a, 0)=
if b =3D
0
- R à ( 0, 0 )
otherwise
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- 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 curvatu=
re
at the origin, k, is positive, the osculating circle of C at the ori=
gin
has radius r=3D1/k and=
center (a,
b) where a2 + b2 =3D r2.
- Then,
- R à (4r, 0 =
) if b=3D0
- R à ( 0, 0 =
) otherwise
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- Let O be a point on a curve C in the plane where the osculating cir=
cle
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/k N is the center of the osculat=
ing
circle to C at O.)
- For any r > 0, define
- Cr to be the circle with radius r centered at O,
- P =3D O + r T, 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 paral=
lel
with N
- Then, as r decreases to 0, R converges to the point R0 =
=3D O +
4 N.
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