Quaternion Basics and Quaternion Operations

Quaternions have ushered in a significant paradigm shift in the representation and computation of rotations. This post is part of a series dedicated to exploring Quaternions and their practical applications in the context of rotation. In this particular post, we will delve into the Quaternion basics, including essential quaternion operations for working with them. Subsequent posts will explore more advanced applications of quaternions in expressing various types of rotations.

We commence by discussing the motivation behind the study of quaternions. It is crucial to establish why one should delve into this topic, and a glimpse into its historical context can greatly enhance our understanding. In particular, exploring the history of an ingenious invention like quaternions is an essential section. Following this historical exploration, we will delve into the fundamental operations that can be applied to quaternions. These foundational concepts are vital for comprehending how quaternions are commonly employed to represent rotations

Quaternion Basics

The orientation of any object can be described using various methods:

1. Euler Angles: This method employs three angles around independent axes to express orientation.
2. Orientation Direction Cosine Matrices: It utilizes a 3×3 matrix to represent a rotation.
3. Quaternion: This method employs a 4D vector to represent orientation.

Each of these methods has its own advantages and disadvantages. In this discussion, our primary focus is on Quaternions. When using quaternions to describe rotations, it offers several advantages compared to the other methods:

• Clear Axis Representation: Quaternions make it very straightforward to identify the axis around which the rotation occurs.
• Smooth Interpolation: Interpolating between two orientations is easy with quaternions. They do not suffer from issues like Gimbal Lock, which can be problematic in some other representations.

However, the only drawback of the quaternion representation is that it may lack intuitiveness. In this system, a single entity expresses rotation. Compared to Euler Angles or Direction Cosine Matrices, this can be less immediately understandable.

History

William Rowan Hamilton developed quaternions in 1843. At that time, he was immersed in complex analysis as his primary area of interest. Complex analysis had not yet matured entirely, making his discovery a significant advancement. He found appeal in the notion that complex numbers could undergo rotations in a plane through multiplication with another complex number. Similarly, he harbored a keen interest in applying the same concept to vectors in three-dimensional space.

Purely speaking mathematically, his contribution was in “Field Extension of the complex field \(\mathbb{C}\) to 4 dimensional space of quaternions represented by \(\mathbb{H}\).” As this field is not Abelian, that is, it does not have the properties of being commutative, Hamilton received a lot of backlash. People considered studying such mathematical structures not very useful in those days. But in the current times it is no longer the case. A lot of non abelian groups like matrices are of very large importance.

His contributions had a large impact on the mathematical community. As the American mathematician Howard Eves described the significance of Hamilton’s new approach by saying that

it opened the floodgates of modern abstract algebra. By weakening or deleting various postulates of common algebra, or by replacing one of more of the postulates by others, which are consistent with the remaining postulates, an enormous variety of systems can be studied. As some of these systems we have groupoids, quasigroups, loops, semi-groups, monoids, groups, rings, integral domains, lattices, division rings, Boolean rings, Boolean algebras, fields, vector spaces, Jordan algebras, and Lie algebras, the last two being examples of non-associative algebras.

— Howard Eves

Quaternions also picked up interest in the computer science and graphics community due to its ease of representing rotations. In addition, Quaternions find applications in quantum physics, where they can represent spin states.

Quaternion Operations

Quaternions is an extension of the 2D complex numbers to 3d. Willain Rowan Hamilton invented it in1843. He was attempting to come up with a technique for multiplying triplets when he arrived at the quaternions. Just as two complex numbers in 2D easily represent rotations through multiplication, quaternions simplify the computation of rotations in 3D.

Quaternions represent themselves as a vector of 4 elements in \(R^4\)

$$
\begin{bmatrix}
a \\
b \\
c \\
d \\
\end{bmatrix} \epsilon R^4
$$

This also breaks down into scalar and vector parts

a = scalar
(b,c,d) = vector

This is also represented as $$q =j\alpha_1 + i\alpha_2 + j\alpha_3 + k\alpha_4$$
The Quaternions are governed by the following rules for multiplication

$$
i^2 = j^2 = k^2 = ijk = -1
$$

Addition

Addition in quaternion is same like addition in the case of complex numbers where the numbers are added element wise.

Two quaternions given by

$$
q_1 = a + b\hat{i} + c\hat{j} + d\hat{k}$$ and
$$ q_2 = e + f\hat{i} + g\hat{j} + h\hat{k}$$

gives a resultant sum

$$
q_1 + q_2 = (a + e) + (b + f)\hat{i} + (c+g)\hat{j} + (d+h)\hat{k}
$$

Multiplication

The multiplication table is given as

	i	j	k
i	-1	k	-j
j	-k	-1	i
k	j	-i	-1

Using these rules the two quaternions \(q_1\) and \(q_2\) given by

$$
q_1 = a + b\hat{i} + c\hat{j} + d\hat{k}$$ and
$$ q_2 = e + f\hat{i} + g\hat{j} + h\hat{k}$$

when mulplied using the distributive rule and applyting the multiplication table gives

$$
q_1 q_2 = (ae – bf – cg – dh) + \hat{i}(af + be + ch – dg) + \hat{j}(ag-bh+ce+df) + \hat{k}(ah+bg-cf+de)
$$

The other way of looking at this multiplication when viewed as scalar and vector combination is given as

$$
q_1 = (s_1, \vec{v_1})$$ and
$$ q_2 = (s_2, \vec{v_2})$$

where

$$
s1 = a, s2 = e, \vec{v_1} = (b,c,d), \vec{v2} = (f,g,h)
$$

hence the product is given as

$$
q_1q_2 = (s_1s_2 – \vec{v_1}.\vec{v2}, s_1\vec{v_2} + s_2\vec{v_1}+\vec{v_1}\times\vec{v_2})
$$

Conjugation

Conjugation is the same as conjugation in the case of the complex numbers except that all the components are inverted in sign

$$
\overline{q_1} = a – b\hat{i} – c\hat{j} – d\hat{k}
$$

Magnitude

Magnitude of a quaternion is given as the square root of sum of square of each element

$$
|{q}| = \sqrt{a^2 + b ^ 2 + c^2 +d^2}
$$

Inverse

Inverse is given as conujugate divided by the square of the magnitude

$$
q^{-1}=\frac{\overline{q}}{|q|^2}
$$

Conclusion

This post covered the history of the quaternion, the motivations around it. The quaternion basics sections also covers the basic quaternion operations. These will gear us to understand the applications of quaternion in the following post.