# Lesson 4: Meet the quaternion: four numbers for one turn

Source: https://quaternion.steppeschool.com/what-is-a-quaternion/ (part of "Quaternions, Visually" by Steppeschool, free)

A quaternion is four numbers, [w, x, y, z]. For a rotation, they are just the axis and the angle from lesson 2, written in a clever way. In this lesson we learn to read them.

## 4.1 The formula

A quaternion has one real part and three imaginary parts:

$$q = w + x\,i + y\,j + z\,k \quad\text{or simply}\quad q = [\,w,\ x,\ y,\ z\,]$$

**A simple case first.** In lesson 3, a turn by θ in the plane was the number cos θ + i sin θ. A turn about the z axis is the same kind of turn, in the x-y plane. Its quaternion looks almost the same, with k for the z axis and half the angle (we see why in 4.3):

$$\text{turn by } \theta \text{ about } z: \qquad q = \cos\tfrac{\theta}{2} + k\,\sin\tfrac{\theta}{2} = \big[\cos\tfrac{\theta}{2},\ 0,\ 0,\ \sin\tfrac{\theta}{2}\big]$$

> **Interactive visual: A turn about z, as a quaternion.** **Try it:** move θ. Above, the object turns about z. Below, the two numbers w = cos(θ/2) and z = sin(θ/2) are drawn as a point on a circle. The object turns by θ, and the point by only half of it.

About x it would be i instead of k, about y it would be j. **For any axis** $\mathbf n = (n_x, n_y, n_z)$ we take the mix $n_x i + n_y j + n_z k$:

$$q = \Big[\ \cos\tfrac{\theta}{2},\ \ n_x\sin\tfrac{\theta}{2},\ \ n_y\sin\tfrac{\theta}{2},\ \ n_z\sin\tfrac{\theta}{2}\ \Big]$$

w holds the angle. The vector part (x, y, z) points along the axis.

> **Interactive visual: Any axis: the four numbers.** **Try it:** pick an axis and move θ. Watch the four bars: w goes from 1 down to 0 at 180° and to −1 at 360°. The x, y, z bars share sin(θ/2) in the proportions of the axis. Press **0 → 720°** and follow the two curves on the chart. Then tick **show −q**: every number flips its sign, and the object does not move.

## 4.2 How to read a quaternion

- **w close to 1**: a small turn. The object is almost where it started.

- **w = 0**: a half turn, 180°.

- **(x, y, z)**: the direction of the axis. Its length is $\sin\frac{\theta}{2}$.

- The length of a quaternion is $\sqrt{w^2 + x^2 + y^2 + z^2}$. For a rotation it is always 1, because $\cos^2 + \sin^2 = 1$. We call it a **unit quaternion**.

- Four numbers, but the length must be 1, so only three are free. That’s the same count as in lesson 2.

- [1, 0, 0, 0] | no turn at all

- [0.707, 0, 0, 0.707] | 90° about z: a quarter turn to the left

- [0.707, 0, 0.707, 0] | 90° about y: the front points straight down

- [0, 1, 0, 0] | 180° about x: upside down

## 4.3 Why half the angle?

Many people find this part confusing, so let’s be clear. In lesson 6 we will see that turning a vector uses q twice, once on each side of the vector, and each use turns it by half the angle. Half plus half gives the full angle θ.

**Important:** Because of the half angle, 360° gives $w = \cos 180° = -1$, so q = [−1, 0, 0, 0]. The object is back where it started, but q is not [1, 0, 0, 0]. Only after 720° do we get the same numbers again. So **q and −q describe the same orientation**. In code, this matters when we compare or blend two quaternions. We come back to it in lesson 10.

> **Interactive visual: q and −q are the same orientation.** **Try it:** press **0 → 720°**. After 360° the object is back, but the point q has only gone half-way round, to where −q started. q and −q always sit on opposite sides of the circle and show the same orientation.

**Remember:**
- A rotation of θ about axis n is q = [cos(θ/2), n·sin(θ/2)].

- A rotation quaternion always has length 1.

- q and −q are the same orientation.

## ✓ Check yourself

**Q: Which rotation is q = [0.5, 0.5, 0.5, 0.5]?**

w = cos(θ/2) = 0.5, so θ/2 = 60° and θ = 120°. The axis is (0.5, 0.5, 0.5) made unit length: (1, 1, 1)/√3. So 120° about the diagonal.

**Q: Write the quaternion for 180° about z.**

[cos 90°, 0, 0, sin 90°] = [0, 0, 0, 1].

**Q: Your program prints q = [−0.98, 0.1, 0, −0.17]. Is the object turned by a lot?**

No. −q = [0.98, −0.1, 0, 0.17] has w close to 1, so it is a small turn, about 23°.

## Sources and further reading

- J. Solà, "Quaternion kinematics for the error-state Kalman filter", 2017 <https://arxiv.org/abs/1711.02508>: chapter 1 and 2: definition, the half angle and the double cover
- G. Sanderson and B. Eater, "Visualizing quaternions: an explorable video series" <https://eater.net/quaternions>: interactive videos on what the four numbers do