info prev up next book cdrom email home

Binormal Vector

$\displaystyle {\bf\hat B}$ $\textstyle \equiv$ $\displaystyle {\bf\hat T}\times{\bf\hat N}$ (1)
  $\textstyle =$ $\displaystyle {{\bf r}'\times {\bf r}''\over \vert{\bf r}'\times{\bf r}''\vert},$ (2)

where the unit Tangent Vector ${\bf T}$ and unit ``principal'' Normal Vector ${\bf N}$ are defined by
$\displaystyle {\bf\hat T}$ $\textstyle \equiv$ $\displaystyle {{\bf r}'(s)\over \vert{\bf r}'(s)\vert}$ (3)
$\displaystyle {\bf\hat N}$ $\textstyle \equiv$ $\displaystyle {{\bf r}''(s)\over \vert{\bf r}''(s)\vert}$ (4)

Here, ${\bf r}$ is the Radius Vector, $s$ is the Arc Length, $\tau$ is the Torsion, and $\kappa$ is the Curvature. The binormal vector satisfies the remarkable identity
\begin{displaymath}[\dot{\bf B},\ddot{\bf B},\raise7.5pt\hbox{.}\mkern0mu\raise7...
...15mu{\bf B}]=\tau^5 {d\over ds}\left({\kappa\over\tau}\right).
\end{displaymath} (5)

See also Frenet Formulas, Normal Vector, Tangent Vector


Kreyszig, E. ``Binormal. Moving Trihedron of a Curve.'' §13 in Differential Geometry. New York: Dover, p. 36-37, 1991.

© 1996-9 Eric W. Weisstein