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Osculating Curves

\begin{figure}\begin{center}\BoxedEPSF{OsculatingCurves.epsf}\end{center}\end{figure}

An osculating curve to $f(x)$ at $x_0$ is tangent at that point and has the same Curvature. It therefore satisfies

\begin{displaymath}
y^{(k)}(x_0)=f^{(k)}(x_0)
\end{displaymath}

for $k=0$, 1, 2. The point of tangency is called a Tacnode. The simplest example of osculating curves are $x^2$ and $x^4$, which osculate at the point $x_0=0$.

See also Tacnode




© 1996-9 Eric W. Weisstein
1999-05-26