\begin{definition}[Definition:Osculating Circle/Definition 1]
Let $I \subseteq \R$ be an open subset of real numbers.
Let $\gamma : I \to \R^2$ be a curve defined by a twice differentiable vector-valued function.
Let $P$ be a point on $\gamma$.
Let $\kappa_P$ be the curvature of $\gamma$ at $P$.
Suppose $C$ is a circle such that:
:$P \in C$
:the tangent to $\gamma$ at $P$ is also a tangent to $C$ at $P$
:the radius of $C$ at $P$ is equal to $\dfrac 1 {\size {\kappa_P} }$
:the center of $C$ is on the inner (concave) side of $\gamma$.
Then $C$ is called the '''osculating circle''' of $\gamma$ at $P$.
{{Stub|illustration}}
\end{definition}