Calculations with quaternions

Quaternions are an expansion of the concept of complex numbers on structures with four (instead of two) components. A quaterion \(h\) can be written as a vector or in the form of \(h = h_0 + ih_1 + j h_2 + kh_3\), where \(i, j\) and \(k\) are related to the \(i\) in complex numbers. Accordingly \(h_0\) is often called real part and h_1, h_2, h_3 are called imaginary part of a quaternion.

For \(i, j\) and \(k\) the following rules are applied:

\(i^2 = j^2 = k^2 = -1\) and \(ijk=-1\)

Basic rules

From these rules follows:

\begin{align} ij &= k\\ ji &= -k\\ jk &= i\\ kj &= -i\\ ki &= j\\ ik &= -j \end{align}

(proof)

Multiplication

Now, obviously quaternions multiplication is not commutative: \(ij = k \neq -k = ji\)

But for numbers in \(\mathbb{R}\), it is commutative (proof).

The multiplication is:

\begin{align} x y &=( x_0 y_0 - x_1 y_1 - x_2 y_2 - x_3 y_3)\\ &+( x_0 y_1 + x_1 y_0 + x_2 y_3 - x_3 y_2) \mathrm i\\ &+( x_0 y_2 - x_1 y_3 + x_2 y_0 + x_3 y_1) \mathrm j\\ &+( x_0 y_3 + x_1 y_2 - x_2 y_1 + x_3 y_0) \mathrm k \end{align}

Calculating multiplicative inverse

This means, when you're given an element \(x = x_0 + x_1 \mathrm i + x_2 \mathrm j + x_3 \mathrm k\) its inverse \(y\) can be calculated by solving the following linear system of equations:

\begin{align} x_0 y_0 - x_1 y_1 - x_2 y_2 - x_3 y_3 &= 1\\ x_0 y_1 + x_1 y_0 + x_2 y_3 - x_3 y_2 &= 0\\ x_0 y_2 - x_1 y_3 + x_2 y_0 + x_3 y_1 &= 0\\ x_0 y_3 + x_1 y_2 - x_2 y_1 + x_3 y_0 &= 0\\ \end{align}

which can be written as:

$$\left(\begin{array}{cccc|c} x_0 & -x_1 & -x_2 & -x_3 & 1\\ x_1 & x_0 & -x_3 & x_2 & 0\\ x_2 & x_3 & x_0 & -x_1 & 0\\ x_3 & -x_2 & x_1 & x_0 & 0 \end{array}\right).$$

According to mathworks it is

\(y = \frac{x_0 - \mathrm i x_1 - \mathrm j x_2 - \mathrm k x_3}{x_0^2 + x_1^2 + x_2^2 + x_3^2}\)

More

Conjugate

Just like the complex conjugate is defined as

\(\overline{z} = a - ib\) with \(z=a+ib\)

the conjugate of quaternions is defined as

\(\bar x := x_0-x_1\mathrm i-x_2\mathrm j-x_3\mathrm k\) with \(x=x_0+x_1\mathrm i+x_2\mathrm j+x_3\mathrm k\)

Rotating points

Quaternions can be used to rotate points. It works like this:

\(x' = q x \overline{q}\)

Pretty simple, isn't it?

For example, if you had the point (2,0,0) and you wanted to rotated it by \(q = (\frac{\sqrt{2}}{2}, 0, \frac{\sqrt{2}}{2}, 0)\) you would transform (2,0,0) to (2i+0k+0j) and calculate

\begin{align} x' &= q x \overline{q}\\ &= (\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} \mathrm j) \cdot 2 \mathrm i \cdot (\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \mathrm j)\\ &= (\sqrt{2} \mathrm i + \sqrt{2} \mathrm{ji}) \cdot (\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \mathrm j)\\ &= (\sqrt{2} \mathrm i - \sqrt{2} \mathrm k) \cdot (\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} \mathrm j)\\ &= \mathrm i - \mathrm{ij} - \mathrm k + \mathrm{kj}\\ &= \mathrm{i - k -k - i}\\ &= -2 \mathrm k\\ &\Rightarrow x' = (0,0,-2) \end{align}

Rotation matrix to quaternion

Let \(M\) be a rotation matrix and \(m_{ij}\) be an entry of this matrix.

General rule

\begin{align} s &= \frac{1}{2} \sqrt{1 + \sum_{i=1}^3 m_{ii}}\\ x &= \frac{m_{32} - m_{23}}{4s}\\ y &= \frac{m_{13} - m_{31}}{4s}\\ z &= \frac{m_{21} - m_{12}}{4s} \end{align}

resulting in the quaternion \((s, x, y, z)\).

Special rule

A rotation matrix

$$R_y = \begin{pmatrix} \cos(\theta) & 0 & \sin(\theta)\\ 0 & 1 & 0\\ -\sin(\theta) & 0 & \cos(\theta) \end{pmatrix}$$

can be transformed to a quaternion

\(q = (\cos(\frac{\theta}{2}), \vec u \sin (\frac{\theta}{2}))\)

where \(\vec u\) describes the axis you rotate by.

In this case \(R_y\) is the y-axis, so

$$\vec u = \begin{pmatrix}0\\1\\0\end{pmatrix}$$

.

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