Osculating Circle

$\begin{figure}\begin{center}\BoxedEPSF{OsculatingCircle.epsf scaled 700}\end{center}\end{figure}$

The Circle which shares the same Tangent as a curve at a given point. The Radius of Curvature of the osculating circle is

$\begin{displaymath} \rho(t) = {1\over\vert\kappa(t)\vert}, \end{displaymath}$

(1)

where $\kappa$ is the Curvature, and the center is

$\displaystyle x$	$\textstyle =$	$\displaystyle f-{(f'^2+g'^2)g'\over f'g''-f''g'}$	(2)
$\displaystyle y$	$\textstyle =$	$\displaystyle g+{(f'^2+g'^2)g'\over f'g''-f''g'},$	(3)

i.e., the centers of the osculating circles to a curve form the Evolute to that curve.

$\begin{figure}\begin{center}\BoxedEPSF{OsculatingCirclePoints.epsf scaled 750}\end{center}\end{figure}$

In addition, let denote the Circle passing through three points on a curve with . Then the osculating circle is given by

$\begin{displaymath} C=\lim_{t_1, t_2, t_3\to t} C(t_1, t_2, t_3) \end{displaymath}$

(4)

(Gray 1993).

References

Gardner, M. ``The Game of Life, Parts I-III.'' Chs. 20-22 in Wheels, Life, and other Mathematical Amusements. New York: W. H. Freeman, pp. 221, 237, and 243, 1983.

Gray, A. ``Osculating Circles to Plane Curves.'' §5.6 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 90-95, 1993.