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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

\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 $C(t_1, t_2, t_3)$ denote the Circle passing through three points on a curve $(f(t),g(t))$ with $t_1<t_2<t_3$. Then the osculating circle $C$ is given by

C=\lim_{t_1, t_2, t_3\to t} C(t_1, t_2, t_3)
\end{displaymath} (4)

(Gray 1993).

See also Curvature, Evolute, Radius of Curvature, Tangent


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.

© 1996-9 Eric W. Weisstein