= Osculating plane
The osculating plane of a regular space <curve> at a point of nonzero <curvature> is the affine plane through the curve point spanned by its <unit tangent vector> $T$ and <principal normal vector> $N$. Its normal is the <binormal vector> $B=T\times N$, so the plane is described by
$$
(x-\gamma(t_0))\cdot B(t_0)=0.
$$
Equivalently, any regular parameter gives the span of $\gamma'(t_0)$ and $\gamma''(t_0)$ when these vectors are linearly independent. The second-order <Taylor expansion> then lies in this plane, which explains the name: it captures the curve's instantaneous bending.
A planar circle has its own containing plane as its osculating plane everywhere. For the <helix> $\gamma(t)=(\cos t,\sin t,t)$ at $t=0$, the point is $(1,0,0)$ and a normal is $(0,-1,1)$, giving $z=y$. At zero curvature the ordinary <Frenet frame> does not define a unique osculating plane.
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