Definition:Osculating Circle/Definition 2

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 $P$.

Also see

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