# Definition:Osculating Circle/Definition 2

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## 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 $P$.

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## Also see

## Sources

- 2018: John M. Lee:
*Introduction to Riemannian Manifolds*(2nd ed.) ... (previous) ... (next): $\S 1$: What Is Curvature? The Euclidean Plane