# Definition:Osculating Circle

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## 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 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 $P$.