Dual Quaternion

Introduction
Continue to the last post which introduce Dual Number, it can be extended to Dual Quaternion. Dual Quaternion is similar to Quaternion with all the real numbers replaced by dual numbers:

Dual quaternion can also be regrouped into the following form by:

which is in terms of 2 quaternions (real part and dual part).

Arithmetic operations
Dual quaternion can perform arithmetic operations as below:

Addition:

Multiplication:

Conjugate:

Norm:

note that the norm of a dual quaternion is a dual number.

Dual Quaternion as Rigid Transform
Unit Dual Quaternion can represent a rigid transformation (i.e. rotation & translation) like matrix. Like ordinary quaternion, dual quaternion can represent a rotation using ordinary quaternion with the dual part equals to zero:

To representation a translation (tx, ty, tz), the following dual quaternion can be used:

Then, we can combine the above 2 transform into 1 dual quaternion (which is also unit dual quaternion) to represent a rotation followed by a translation

With the above arithmetic operations, we can transform a point p(px, py, pz), using unit dual quaternion like ordinary unit quaternion:

Dual Quaternion - Matrix Conversion
Sometimes it is convenient to have methods to convert between Unit Dual Quaternion and Matrix. Assume we have ordinary quaternion - matrix conversion functions.

Converting Dual Quaternion to Matrix:
Given any unit dual quaternion which is composed of a rotation followed by a translation, we have:

Therefore, we can find the rotation matrix by considering the real part of the dual quaternion while the translation (i.e. tx, ty, tz) can be solved by using system of linear equations by equating coefficients.

Converting Matrix to Dual Quaternion:
Given a rigid transform matrix, we can decompose the rotation matrix into real quaternion as usual and the translation dual quaternion can also be obtained by:

then multiply the translation dual quaternion with rotation dual quaternion will give the answer.

Blending Rigid Transform using Dual Quaternion
Using dual quaternion to represent transform is better when blending multiple transformations, which can be applied to skinning a mesh. From that paper, it suggests using an fast approximation for blending called Dual quaternion Linear Blending (DLB):

note that the norm in the denominator is a dual number, which can be treated as 1 divided by the norm and gives another dual number (using dual number division) so that it can be multiply by a dual quaternion.

Conclusion
Dual quaternion is an alternative way to represent a rigid transform other than matrix. And it may gives a faster accumulation of transformation if joint inferences per vertex is large enough according to "Spherical Skinning with Dual-Quaternions and QTangents".



Reference:
[1] Dual Quaternion for Rigid Transformation Blending: http://www.jarmilakavanova.cz/ladislav/papers/sdq-i3d07/sdq-i3d07.pdf
[2] Spherical Skinning with Dual-Quaternions and QTangents: http://www.crytek.com/sites/default/files/izfrey_siggraph2011.pdf
[3] Estimating 3-D location parameters using dual number quaternions: https://pwww2.cse.tamu.edu/volzfest/proceedings/paper-dedications/Shao_Lejun.pdf
[4] Dual quaternion as a tool for rigid body motion: http://www.ingegneriameccanica.org/papers/pennestrivalentini_paper.pdf