# Definition:Osculating Circle

## Definition

### 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 $P$.

### 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 $P$.