The TNB frame

The natural frame of reference for a parametric curve

At any point on a curve in space presented in a parametric form, its unit tangent vector, principal unit normal vector, and the unit binormal vector form particular useful coordinate system that encodes important geometric (and kinematic) information. This is known as the Frenet frame of reference, a.k.a. the "TNB frame". In this lecture, we will gain a geometric understanding of this important construction.

Parametric space curve

Throughout this discussion, we will use the notation \[ \mathbf{r}(t) = \begin{bmatrix} x(t) \\ y(t) \\ z(t) \end{bmatrix} \] to represent a vector-value function. For simplicity, we assume it is defined and differentiable for all $t$ (over the entire real line).

It may be useful to consider $t$ as time and imagine the function $\mathbf{r}(t)$ describe the trajectory of an object moving in space, so that we can talk about the velocity and acceleration vectors.

From this point of view, we actually consider the function values $\mathbf{r}(t)$ to be points in space, and the image of $\mathbf{r}$ will form a curve.

The unit tangent vector

For any given $t$ value such that $\| \mathbf{r}'(t) \| \ne 0$, we define the (principal) unit tangent vector to be \[ \mathbf{T}(t) = \frac{\mathbf{r}'(t)}{ \| \mathbf{r}'(t) \| }. \]

We can see that the unit tangent vector is not defined when $\mathbf{r}'(t)$ is the zero vector. This is a minor issue that we will fix later.

It means exact what you think it means: it is a unit vector that is tangent to the curve that $\mathbf{r}(t)$ represents.
If we consider $\mathbf{r}(t)$ as the trajectory of a moving object, then $\mathbf{T}(t)$ is simply the normalized version of the velocity vector.

The principal unit normal vector

For a given $t$ value where $\mathbf{T}'(t)$ is defined and nonzero, the principal unit normal vector at $t$ is \[ \mathbf{N}(t) = \frac{ \mathbf{T}'(t) }{ \| \mathbf{T}'(t) \|}. \]

By this definition, $\mathbf{N}(t)$ is undefined whenever $\mathbf{T}'(t) = \mathbf{0}$. This is indeed intentional, and it's a useful feature rather than a bug.

From this construction, it's not hard to verify that

  • $\mathbf{N}(t)$ is a unit vector.
  • $\mathbf{N}(t)$ is orthogonal to $\mathbf{T}(t)$.

The unit binormal vector

For a given $t$ value where the unit tangent vector $\mathbf{T}(t)$ and the principal unit normal vector $\mathbf{N}(t)$ are both defined, we define the binormal vector at $t$ to be \[ \mathbf{B}(t) = \mathbf{T}(t) \times \mathbf{N}(t). \]

We can see that $\mathbf{B}(t)$ is orthogonal to both $\mathbf{T}(t)$ and $\mathbf{N}(t)$.

Moreover, since $\mathbf{T}(t)$ and $\mathbf{N}(t)$ are both unit vector and they are orthogonal to each other, \[ \| \mathbf{B}(t) \| = \| \mathbf{T}(t) \times \mathbf{N}(t) \| = \| \mathbf{T}(t) \| \, \| \mathbf{N}(t) \| \, | \sin \theta | = 1. \] That is, $\mathbf{B}(t)$ is indeed a unit vector, which is why we usually call it the unit binormal vector.

The "TNB" frame

Assuming $\mathbf{r}(t)$ describes a smooth curve, then at any given $t$ value where the vectors $\mathbf{T}(t)$, $\mathbf{N}(t)$, and $\mathbf{B}(t)$ are all defined, these three vector are orthogonal to one another, and they form a natural frame of reference of the 3-dimensional space.

This is the Frenet frame of reference, and it is also known as the TNB frame since it is constructed from the vectors $\mathbf{T}(t)$, $\mathbf{N}(t)$, and $\mathbf{B}(t)$. You should pause and think about the meaning of these vectors.