Principal unit normal vector

The unit vector $\mathbf{N}$ pointing the direction $\mathbf{T}$ is turning toward.

The principal unit normal vector $\mathbf{N}(t) = \mathbf{T}'(t)/|\mathbf{T}'(t)|$ points in the direction $\mathbf{T}$ is turning toward, and is only defined where $\kappa\ne0$ — a straight line isn't turning at all, so it has no turning direction to point toward. $\mathbf{N}$ always points toward the inside of whatever bend the curve is making, and is always perpendicular to $\mathbf{T}$ .

Examples

For the helix $\mathbf{r}(t) = \langle 3\cos t, 3\sin t, 4t\rangle$ , $\mathbf{N}(t) = \langle -\cos t, -\sin t, 0\rangle$ , always pointing straight back toward the helix's central axis.