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 byEquivalently, any regular parameter gives the span of ${\gamma }^{\prime }(t_0)$ and ${\gamma }^{\prime \prime }(t_0)$ 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.