Frenet frame TNBFrame 2002-02-02 17:02:19 2006-12-09 09:03:02 Definition osculating plane normal plane rectifying plane unit normal unit tangent binormal \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand{\reals}{\mathbb{R}} Let $I\subset \reals$ be an interval and let $\gamma:I\to\reals^3$ be a parameterized space curve, assumed to be \PMlinkname{regular}{SpaceCurve} and free of points of inflection. We interpret $\gamma(t)$ as the trajectory of a particle moving through 3-dimensional space. The moving trihedron (also known as the Frenet frame, the Frenet trihedron, the rep\`ere mobile, and the moving frame) is an orthonormal basis of 3-vectors $T(t),N(t),B(t),$ defined and named as follows: \begin{align*} T(t) &= \displaystyle \frac{\gamma'(t)}{\Vert \gamma'(t) \Vert}\, , && \text{the unit tangent;}\\ N(t) &= \displaystyle \frac{T'(t)}{\Vert T'(t) \Vert} \, ,&& \text{the unit normal;}\\ \\ B(t) &= T(t)\times N(t) \, ,&& \text{the unit binormal.}\\ \end{align*} A straightforward application of the chain rule shows that these definitions are covariant with respect to reparameterizations. Hence, the above three vectors should be conceived as being attached to the point $\gamma(t)$ of the oriented space curve, rather than being functions of the parameter $t$. Corresponding to the above vectors are 3 planes, passing through each point $\gamma(t)$ of the space curve. The \emph{osculating plane} at the point $\gamma(t)$ is the plane spanned by $T(t)$ and $N(t)$; the \emph{normal plane} at $\gamma(t)$ is the plane spanned by $N(t)$ and $B(t)$; the rectifying plane at $\gamma(t)$ is the plane spanned by $T(t)$ and $B(t)$.