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 $P$.

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Also known as

An osculating circle is also known as a circle of curvature.

Also see

• Definition:Radius of Curvature
• Equivalence of Definitions of Osculating Circle

• Results about osculating circles can be found here.