Dualquaternions are a mathematical tool used in computer graphics and robotics to represent transformations, such as rotations and translations, in threedimensional space. They are an extension of quaternions, which are themselves an extension of complex numbers. Just as complex numbers have a real part and an imaginary part, quaternions have a real part and three imaginary parts. Similarly, dualquaternions have a real part and a dual part.
Like Acrobatic Gymnastics  dualquaterniosn can be connected together in unique ways to create innovative and original ways of solving problems.
Dualquaternions are make up of 8 float point numbers (i.e., two quaternions  as each quaternion is 4 floating point numbers).
With dualquaternions, you can pack lots of information into one neat package. It's like having a single ticket for a ride that includes dancing, spinning and shifting around in space!
A dualquaternion is represented as:
\[
q = q_r + \varepsilon q_d
\]
where \(q_r\) is the real part, \(q_d\) is the dualpart, and \(\varepsilon\) is the dualunit, such that \(\varepsilon^2 = 0\) and \(\varepsilon \neq 0\).
The real part of a dualquaternion represents the rotation, while the dual part represents the translation. Combining these two aspects allows dualquaternions to represent rigid body transformations in a compact and efficient manner.
To understand the math behind dualquaternions, it's crucial to grasp their algebraic properties, such as addition, multiplication, and normalization. Addition and multiplication of dualquaternions follow certain rules, similar to those for complex numbers and quaternions, but with adaptations to incorporate the dualpart.
Dualquaternions are particularly useful in robotics for representing the motion of rigid bodies, as they can efficiently handle both rotational and translational transformations in a single mathematical entity. They have applications in areas such as animation, computeraided design (CAD), and robot kinematics. Despite their complexity, dualquaternions provide a powerful and elegant framework for handling spatial transformations in threedimensional environments.
An analogy of something like dualquaternions are 'matrices' or 'vectors'  we use a single vector 'v' to represent a direction or a position (composed of an x, y and z), or a quaternion 'q' to represent an axis and angle. The dualquaternion `$$\zeta$$` is a single numer for representing both rotation and translation.
