Multivariable Lesson 1.3: The TNB Frame

We can expand upon our use of the derivative to characterize the motion of space curves. In this section, we will associate three unit vectors, $\mathbf{T}$, $\mathbf{N}$, and $\mathbf{B}$, with every point on a space curve. These will form a “frame” of perpendicular vectors which will slide along the space curve indicating at each point where it’s going as well as where it’s turning.

First, we define the unit tangent vector $\mathbf{T}(t)$ as the derivative of $\mathbf{r}(t)$ divided by its magnitude: \mathbf{T}(t)=\frac{\mathbf{r}'(t)}{\vert\mathbf{r}'(t)\vert} Because it points in the direction of the derivative, this vector is tangent to the curve at that point.

Next, we define the unit normal vector $\mathbf{N}(t)$. Though we have an infinite number of vectors which satisfy that criterion, we will make the sensible choice of \begin{displaymath}\mathbf{N}(t)=\frac{\mathbf{T}'(t)}{\vert\mathbf{T}'(t)\vert}. It can be proven fairly simply that $\mathbf{T}(t)\perp\mathbf{T}'(t)$, but perhaps sound reasoning is enough in this case: since $\vert\mathbf{T}(t)\vert$ is constant (equalling one) for all $t$, its derivative may never alter that magnitude. Any parallel component of the derivative would change its magnitude, but a strictly perpendicular derivative only changes direction and not magnitude.

Finally, we define the unit binormal vector $\mathbf{B}(t)$ as \mathbf{B}(t)=\mathbf{T}(t)\times\mathbf{N}(t). Notice we don't need to divide by the magnitude: the cross product of two perpendicular unit vectors must also have magnitude one (since $1\cdot1\cdot\sin(\pi/2)=1$).

Image of our favorite helix with the TNB frame drawn at a point

What are these vectors good for? If we draw them on our favorite helix, as we have done on the right, we notice that $\mathbf{T}$ is indeed showing the direction the curve is going at that point, $\mathbf{N}$ is showing the direction in which the curve is turning, and $\mathbf{B}$ is providing an orientation of sorts for the curve.

The plane containing $\mathbf{N}$ and $\mathbf{B}$ is called the normal plane. If you visualize that plane, the curve should be piercing it directly perpendicular (because $\mathbf{T}$ is a normal vector to the plane).

We will use the TNB frame to generate one more way to describe space curves, but you'll have to look in the next lesson for that.

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