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

Dual Number

Introduction
Recently, I read the "Spherical Skinning with Dual-Quaternions and QTangents" from Crytek. It raised my interest on the topic of "Dual Number" (which is related to Dual Quaternion). Dual number, just like imaginary number, has the form of:

where the real number a is called real part and the real number b is called dual part.


Arithmetic operations
Dual number can perform the arithmetic operations as below:

Addition:

Multiplication:

Division:

Finding derivative using Dual Number
The interesting part of dual number is when it is applied to Taylor Series. When substituting a dual number into a differentiable function using the Taylor Series:

This gives a very nice property that we can find the first derivative, f'(a), by consider the dual part of f(a+bε), which can be evaluated using dual number arithmetic.
For example, given a function

we want to find the first derivative of f(x) at x = 2, i.e. f'(2). We can find it by using dual number arithmetic where f'(2) will equals to the dual part of  f(2+ε) according to Taylor Series.

Therefore, f'(2)= 8/9, you can verify this by finding f'(x) and substitute 2 into it, which will give the same answer.

Conclusion
By using dual number, we can find the derivative of a function using dual arithmetic. Hence, we can also find the tangent to an arbitrary point, p, on a given parametric curve which is equals to the normalized dual part of the point p. For those who are interested in finding out more about dual number, I recommend to read the presentation "Dual Numbers: Simple Math, Easy C++ Coding, and Lots of Tricks" by Gino van den Bergen in GDC Europe 2009.

Reference:
[1] http://en.wikipedia.org/wiki/Dual_number
[2] http://www.gdcvault.com/play/10103/Dual-Numbers-Simple-Math-Easy