# Definition:Osculating Circle/Definition 2

## Definition

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 $P$.