= Osculating circle
{title2=$c=r+\kappa^{-1}N_p$}
{wiki}
At a point of a twice continuously differentiable <regular curve> with positive <curvature> $\kappa$, its osculating circle has <radius> $1/\kappa$ and center $c=r+N_p/\kappa$, where $N_p$ is the <principal normal vector>. It matches the point, <tangent line> and <curvature>. For a <space curve>, it lies in the plane spanned by the <unit tangent vector> and <principal normal vector>. The <parabola> example under <curvature of a plane curve> gives center $(0,1/2)$ and <radius> $1/2$ at the origin.
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