# Definition:Osculating Circle/Definition 1

## Definition

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 $P$.