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

The unit complex numbers and are isomorphic. To see this clearly, recall that complex numbers can be represented in polar form as ; a unit complex number is simply . A bijective mapping can be made between 2D rotation matrices and unit complex numbers by letting 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 , followed by . In polar form, the complex numbers are multiplied to yield , which clearly represents a rotation of . If the unit complex number is represented in Cartesian form, then the rotations corresponding to and are combined to obtain . 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 can be converted into the equivalent rotation matrix

Recall that only one independent parameter needs to be specified because . Hence, it appears that the set of unit complex numbers is the same manifold as , which is the circle (recall, that ``same'' means in the sense of homeomorphism).

Steven M LaValle 2012-04-20