The word
"osculate" means "to kiss." A succession of circles that converge
to snuggly "kiss" a curve are said to be osculating circles. They
are also known as the Circle of
Curvature.
The
tangent to a point on a curve was applied by the earliest
investigators to study the derivative and thus Calculus. Just as
the tangent line
approximates a curve at a point, the osculating circle is an even
better approximation by capturing the curvature.

Note that at the point of inflection (second derivative), the tangent
must jump across the curve; thus, the circle also jumps across this
curve.
Replay the animation to watch the
jump at (0,0).
In our MATHEMATICA^{®} animations, both the tangent
and normal lines are drawn at the point of contact of the curve and the
circle.

Our
animations feature several
Classic
Curves
with their osculating circles.
We ask you to
recall the construction for circumscribing a circle about a triangle.
A minimum of
three points on the curve are
needed to determine, first the perpendicular bisectors, and then the
center and radius of the osculating circle. As the three
points on a given curve move closer together  converge to a single
point  the radius of the circle becomes the normal to the tangent at
the point.
Play this animation.
As the three
points on the parabola converge to the vertex, the circle of curvature
takes on a radius and center containing the normal to the point of
tangency.


y
= sin (x),
a tangent, normal, and osculating circle.

In our MATHEMATICA^{®} animations, both
the tangent and normal lines are drawn at the point of contact of
the curve and the circle.
Useful Links and Books


Eves, Howard, AN
INTRODUCTION TO THE HISTORY OF MATHEMATICS, 6th ed.,
Saunders College Publishing, 1992, p. 405.
Leibniz
"defined the
osculating circle and showed its importance in the study of curves."
A large statue of Leibniz is at the Royal
Academy of Arts in the heart of fashionable London. Note the
English chose to spell his name as
. . . .
.

Gray, Alfred, Modern Differential Geometry of
Curves and Surfaces with MATHEMATICA^{®},
2nd ed., CRC Press, 1998, pp. 111115.

Wagon, Stan, MATHEMATICA^{®}IN
ACTION, W. H. Freeman and Co., 1991. ISBN
0716722291 or ISBN 071672202X
(pbk.)
Wagon, Stan, MATHEMATICA® IN ACTION, 2nd ed.,
SpringerVerlag, 2000. ISBN 0387986847 for other animations.

Yates, Robert C., Curves and Their Properties,
NCTM, 1952, pp. 6063.



MATHEMATICA^{®}
Code and animation contributed by
Gustavo Gordillo
2005.


