Tangent, Normal and Binormal Vectors

Main idea

Anything moving along a curve carries its own set of directions: forward ( $\vec T$ ), the way it's turning ( $\vec N$ ), and the third direction perpendicular to both ( $\vec B = \vec T\times\vec N$ ). Think of a roller coaster car's forward, sideways and up.

{\htmlData{sym=0}{\vec T}} = \frac{\vec r\,'}{\|\vec r\,'\|}, \qquad {\htmlData{sym=1}{\vec N}} = \frac{{\htmlData{sym=0}{\vec T}}\,'}{\|{\htmlData{sym=0}{\vec T}}\,'\|}, \qquad {\htmlData{sym=2}{\vec B}} = {\htmlData{sym=0}{\vec T}}\times{\htmlData{sym=1}{\vec N}}

What the symbols mean

$\vec T$
Unit tangent
The direction of motion.
$\vec N$
Principal normal
Points toward the inside of the bend.
$\vec B$
Binormal
Perpendicular to the plane the curve is momentarily bending in.

Explanation

The tangent, normal and binormal vectors are simply three arrows that ride along with a moving object: "forward," "the way I'm turning" and "the third direction, perpendicular to both." Tangent means touching the curve, normal is the old geometry word for perpendicular, and binormal is a second normal. They're what roller coaster designers use to bank turns. The formulas look stacked, but each one is a small step: $\vec T$ is the velocity shrunk to length 1 (just the forward direction), $\vec N$ is how $\vec T$ is changing, also shrunk to length 1 (the direction of the turn), and $\vec B$ is the cross product of those two (the leftover direction perpendicular to both). Three arrows, each built from the one before. Bear with the notation; it's a very physical idea.

The last lesson gave us $\vec r\,'$ , which points along the curve. Its length is the speed, though, and for describing the shape of a path we don't care how fast you drive it. So divide by the length and keep only the direction. That's $\vec T$ , the unit tangent.

Why $\vec N$ is perpendicular to $\vec T$ . Now ask how $\vec T$ changes. On a straight road it doesn't. On a curve it swings around, and $\vec T\,'$ tells you which way. Here's the nice part: $\vec T$ always has length 1, so by the "constant length means perpendicular" fact from the last lesson, $\vec T\cdot\vec T\,' = 0$ . Since $\vec T$ can't get longer or shorter, it can only rotate, and the direction it's rotating toward is perpendicular to $\vec T$ itself. That's the direction of turning, and normalizing it gives $\vec N$ , which points toward the inside of the bend.

The osculating plane is spanned by $\vec T$ and $\vec N$ . It's the plane the curve is bending in at that moment, the one that hugs the curve best there ("osculating" means "kissing"). We already know how to get a normal to a plane spanned by two vectors: cross them. So $\vec B = \vec T\times\vec N$ is that plane's normal, and it's a unit vector automatically because $\vec T$ and $\vec N$ are perpendicular unit vectors.

Together $\vec T, \vec N, \vec B$ form a moving coordinate frame, the Frenet frame, that rides along the curve. There's an animation further down showing it on a helix.

See it

The T, N, B frame riding up a helix. Notice that N always points level, straight in toward the helix's axis, because that's the way the curve keeps turning. T tilts upward with the climb, and B stays tilted the other way, perpendicular to both. It's the forward, sideways and up that a roller coaster car would feel.

Worked examples

Example 1

Find $\vec T$ , $\vec N$ and $\vec B$ for the helix $\langle\cos t, \sin t, t\rangle$ .

1
$\vec r\,' = \langle -\sin t, \cos t, 1\rangle$ has length $\sqrt{\sin^2 t + \cos^2 t + 1} = \sqrt2$ , so $\vec T = \frac{1}{\sqrt2}\langle -\sin t, \cos t, 1\rangle$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Find the unit tangent of $\vec r(t) = \langle 3t, 4t, 0\rangle$ .

Problem 2+5 XP

For the circle $\langle\cos t, \sin t, 0\rangle$ , find $\vec T$ at $t = 0$ .

Problem 3+5 XP

Find $\vec N$ at $t = 0$ for the same circle.

Problem 4+5 XP

Find $\vec B = \vec T\times\vec N$ at $t = 0$ .

Problem 5+5 XP

For the helix $\langle\cos t, \sin t, t\rangle$ , what is the $z$ -component of $\vec T$ ?

Problem 6+5 XP

A self-driving car follows the path $\vec r(t) = \langle t, 0.1t^2\rangle$ (meters). Find its unit tangent vector, the direction it's heading, at $t = 5$ .

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

Find the unit binormal vector $\vec B$ of the twisted cubic $\vec r(t) = \langle t, t^2, t^3\rangle$ at $t = 1$ .

Where it's used

Roller-coaster and highway designers use the Frenet frame to bank turns; camera rigs in games ride along spline paths oriented by $\vec T$ , $\vec N$ and $\vec B$ .

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