Using complex numbers to represent SO(2)

Before introducing quaternions to parameterize 3D rotations, consider using complex numbers to parameterize 2D rotations. Let the term unit complex number refer to any complex number, $a + bi$, for which $a^2 + b^2 = 1$.

The set of all unit complex numbers forms a group under multiplication. It will be seen that it is ``the same'' group as $SO(2)$. This idea needs to be made more precise. Two groups, $G$ and $H$, are considered ``the same'' if they are isomorphic, which means that there exists a bijective function $f: G \rightarrow H$ such that for all $a, b \in G$, $f(a) \circ f(b) = f(a \circ b)$. This means that we can perform some calculations in $G$, map the result to $H$, perform more calculations, and map back to $G$ without any trouble. The sets $G$ and $H$ are just two alternative ways to express the same group.

The unit complex numbers and $SO(2)$ are isomorphic. To see this clearly, recall that complex numbers can be represented in polar form as $r e^{i \theta}$; a unit complex number is simply $e^{i \theta}$. A bijective mapping can be made between 2D rotation matrices and unit complex numbers by letting $e^{i \theta}$ correspond to the rotation matrix (3.31).

If complex numbers are used to represent rotations, it is important that they behave algebraically in the same way. If two rotations are combined, the matrices are multiplied. The equivalent operation is multiplication of complex numbers. Suppose that a 2D robot is rotated by $\theta_1$, followed by $\theta_2$. In polar form, the complex numbers are multiplied to yield $e^{i \theta_1} e^{i \theta_2} = e^{i (\theta_1 + \theta_2)}$, which clearly represents a rotation of $\theta_1 + \theta_2$. If the unit complex number is represented in Cartesian form, then the rotations corresponding to $a_1 + b_1 i$ and $a_2 + b_2 i$ are combined to obtain $(a_1 a_2 - b_1 b_2) + (a_1 b_2 + a_2 b_1) i$. Note that here we have not used complex numbers to express the solution to a polynomial equation, which is their more popular use; we simply borrowed their nice algebraic properties. At any time, a complex number $a + bi$ can be converted into the equivalent rotation matrix

$$R(a,b) = \begin{pmatrix}a & -b b & a \end{pmatrix} .$$ (4.18)

Recall that only one independent parameter needs to be specified because $a^2 + b^2 = 1$. Hence, it appears that the set of unit complex numbers is the same manifold as $SO(2)$, which is the circle ${\mathbb{S}}^1$ (recall, that ``same'' means in the sense of homeomorphism).