The binormal vector $\mathbf{B}\=\mathbf{T}\times \mathbf{N}$ is a unit vector orthogonal to the osculating plane, the plane formed by the vectors T and N. (See Example 2.6.5 for a verification that $∥\mathbf{B}∥\=1$.) This author remembers the definition of the binormal vector by recalling the word button (BuTtoN). In fact, he would tell his students that if they remembered that word, they would be "right on the button" in their recollection of this construct. It got groans, but it worked!

The three vectors T, N, and B form the TNB-frame, and are called the Frenet (or Frenet-Serret) vectors.

The vector $\mathbf{B}\prime \left(s\right)$ is in the direction of N (see Example 2.6.6), so that $\mathbf{B}\prime \left(s\right)\= -\mathrm{\τ} \mathbf{N}$, where the scalar of proportionality $\mathrm{\τ}$ is called the torsion. Some texts omit the minus sign. The definition given here makes the torsion positive when, as T advances with increasing arc length, the osculating plane rotates clockwise around the tangent line, the rotation being viewed in the direction of T.

Table 2.6.1 lists formulas for the torsion $\mathrm{\τ}$. The formulas on the left are more on the order of definitions because in them, the vectors are considered functions of the arc length $s$, and the primes denote differentiation with respect to arc length. The formulas on the right are more practical, since in them the vectors are considered to be functions of a parameter $p$, other than arc length. The over-dot is used to denote differentiation with respect to this other parameter.