\begin{definition}[Definition:Osculating Circle/Definition 2]
Let $I, I' \subseteq \R$ be open subsets of real numbers.
Let $\gamma : I \to \R^2$ be a curve defined by a twice differentiable vector-valued function.
Let $C: I' \to \R$ be a circle.
Let both $\gamma$ and $C$ have the unit-speed parametrization.
Let $P$ be a point on $\gamma$.
Suppose $C$ is such that:
:$P \in C$
:$\map {\gamma'} P = \map {C'} P$
:$\map {\gamma''} P = \map {C''} P$
Then $C$ is called the '''osculating circle''' of $\gamma$ at $P$.
{{Stub|illustration}}
\end{definition}