Introducing dual quaternions
A dual quaternion combines linear and rotational transformations together into one variable. This single variable can be interpolated, transformed, and concatenated. A dual quaternion can be represented with two quaternions or eight floating-point numbers.
Dual numbers are like complex numbers. A complex number has a real part and an imaginary part, and a dual number has a real part and a dual part. Assuming 
is the dual operator, a dual number can be represented as 
, where 
and 
.
Operations on dual numbers are done as imaginary numbers, where the dual components and real components must be acted on separately. For example, dual quaternion addition can be expressed in the following way:
Notice how the real and dual parts are added independently.
Important note
If you are interested in the more formal mathematics behind dual quaternions, check out A Beginner's Guide to Dual-Quaternions by Ben Kenwright, at https://cs.gmu.edu/~jmlien...
