Tangent and Normal Vectors | Sahithyan's Notes

Let $C$ be a curve traced by a differentiable vector-valued function $\boldsymbol{r}(t)$ with $\boldsymbol{r}'(t) \ne \vec{0}$ , where $t$ is the parameter. $s$ denotes the arc-length parameter of $C$ .

Unit Tangent Vector

The unit tangent vector at $t$ is

T(t) = \frac{\boldsymbol{r}'(t)}{\lVert \boldsymbol{r}'(t) \rVert}

$T(t)$ is tangent to $C$ and points in the direction of increasing $t$ .

If $\boldsymbol{r}$ is parametrized by arc length $s$ , $\lVert \boldsymbol{r}'(s) \rVert = 1$ , so

T(s) = \boldsymbol{r}'(s)

Unit Normal Vector

If $T'(t)$ exists and $T'(t) \ne \vec{0}$ , the principal unit normal vector at $t$ is

N(t) = \frac{T'(t)}{\lVert T'(t) \rVert}

$N(t)$ is normal to $C$ and points in the direction of $T'(t)$ .

If $T'(t)$ exists and is non-zero, $N(t) \perp T(t)$ .

In two-dimensional space, 2 unit vectors are orthogonal to $T(t)$ . In three-dimensional space, infinitely many unit vectors are orthogonal to $T(t)$ . $N(t)$ is the one in the direction of $T'(t)$ .

$T$ points in the direction of motion. $N$ points in the direction the moving object is turning. For a plane curve, $N$ points toward the concave side of the trajectory.

If $\boldsymbol{r}$ is parametrized by arc length $s$ and $\boldsymbol{r}''(s) \ne \vec{0}$ , then $T(s) = \boldsymbol{r}'(s)$ gives $T'(s) = \boldsymbol{r}''(s)$ , so

N(s) = \frac{\boldsymbol{r}''(s)}{\lVert \boldsymbol{r}''(s) \rVert}

Binormal Vector

For $C$ in three-dimensional space, with $T(t)$ and $N(t)$ defined, the binormal vector is

B(t) = T(t) \times N(t)

$B(t)$ is orthogonal to both $T(t)$ and $N(t)$ , oriented by the right-hand rule, and is a unit vector since $\lVert T(t) \times N(t) \rVert = \lVert T(t) \rVert \lVert N(t) \rVert \sin\frac{\pi}{2} = 1$ .

If $\boldsymbol{r}'(t)$ and $\boldsymbol{r}''(t)$ exist and $\boldsymbol{r}'(t) \times \boldsymbol{r}''(t) \ne \vec{0}$ ,

B(t) = \frac{\boldsymbol{r}'(t) \times \boldsymbol{r}''(t)}{\lVert \boldsymbol{r}'(t) \times \boldsymbol{r}''(t) \rVert}

If $\boldsymbol{r}$ is parametrized by arc length $s$ and $\boldsymbol{r}''(s) \ne \vec{0}$ ,

B(s) = \frac{\boldsymbol{r}'(s) \times \boldsymbol{r}''(s)}{\lVert \boldsymbol{r}''(s) \rVert}

TNB Frame

Aka. Frenet frame. The moving orthonormal frame determined by $T(t)$ , $N(t)$ , $B(t)$ . Origin is the current point in the curve, and changes with $t$ .

At each point on $C$ , $T(t)$ , $N(t)$ , $B(t)$ determine 3 mutually perpendicular planes:

$TB$ -plane
Rectifying plane. $C$ does not bend within it; to second order, $C$ projected onto this plane is a straight line, i.e. “rectified”.
$TN$ -plane
Osculating plane. The plane that best approximates $C$ near the point.
$NB$ -plane
Normal plane. Perpendicular to $T$ . Contains all normals to $C$ at the point.