Quaternion: Difference between revisions

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==Double Quaternions==

Note that double quaternions are different from dual quaternions.

Double quaternions are written as <math>q_1 + \epsilon q_2</math> with <math>\epsilon^2 = 1</math>.

Double quaternions are written as <math>q_1 + \epsilon q_2</math> with <math>\epsilon^2 = 1</math> and applied to a quaternion <math>p</math> representing a point as <math>q_1 p \bar{q}_2</math>.

Dual quaternions are written as <math>q_1 + \varepsilon q_2</math> with <math>\varepsilon^2 = 0</math>.

Dual quaternions are written as <math>q_1 + \varepsilon q_2</math> with <math>\varepsilon^2 = 0</math> and applied to a dual quaternion <math>p</math> representing a point as <math>(q_1 + \varepsilon q_2) p (\bar{q}_1 + \varepsilon \bar{q}_2)</math>

===Cayley Factorization===

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Any 4D rotation matrix can be decomposed into a right and a left isoclinic rotation matrix:

<math>R^R</math> and <math>R^L</math> can be ~~seen~~ as a single double quaternion.

<math>R^R</math> and <math>R^L</math> can be viewed as a matrix representation of single double quaternion <math>(q_1, q_2)</math>.

For a double quaternion, the 4D rotation written is <math>x' = q_1 x q_2</math>.

The product of left and right isoclinic rotation matrices commute.

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Thus, <math>R_1 R_2 = (R_1^L R_1^R) (R_2^L R_2^R) = (R_1^L R_2^L) (R_1^R R_2^R) = R</math>.

This shows that the composition of two double quaternions will be a double quaternion.

===Approximating 3D Translations using Double quaternions===

See Ge ''et. al.''<ref name="ge1998double"></ref>

<!-- {{hidden | Algebraic Verification |

Suppose our point is <math>\mathbf{x} \in \mathbb{R}^3</math> and our translation has direction <math>\mathbf{d} \in \mathbb{R}^3</math> with distance <math>d</math>.

Let <math>q=1+\frac{d}{2}\mathbf{n}</math> and <math>q^* = 1-\frac{d}{2}\mathbf{n}</math>.

<math>

\begin{aligned}

q(1+\mathbf{x})(q^*)^* &= (1+\frac{d}{2}\mathbf{n})(1+\mathbf{x})(1+\frac{d}{2}\mathbf{n})\\

&= [(1 - \frac{d}{2} \mathbf{n} \cdot \mathbf{x}) + (\frac{d}{2}\mathbf{n} + \mathbf{x} + \frac{d}{2}\mathbf{n} \times \mathbf{x})](1+\frac{d}{2}\mathbf{n})\\

&= \left(1 - \frac{d}{2} \mathbf{n} \cdot \mathbf{x} - ((\frac{d}{2}\mathbf{n} + \mathbf{x} + \frac{d}{2}\mathbf{n} \times \mathbf{x})) \cdot \frac{d}{2} \mathbf{n}\right) + \left(\frac{d}{2}\mathbf{n} + \mathbf{x} + \frac{d}{2}\mathbf{n} \times \mathbf{x} +\frac{d}{2}\mathbf{n} + (\frac{d}{2}\mathbf{n} + \mathbf{x} + \frac{d}{2}\mathbf{n} \times \mathbf{x})\times(\frac{d}{2}\mathbf{n})\right)\\

&= c + \left(d\mathbf{n} + \mathbf{x} + \frac{d}{2}\mathbf{n} \times \mathbf{x} + (\frac{d}{2}\mathbf{n}\times(\frac{d}{2}\mathbf{n}) + \mathbf{x}\times(\frac{d}{2}\mathbf{n}) + (\frac{d}{2}\mathbf{n} \times \mathbf{x})\times(\frac{d}{2}\mathbf{n}))\right)\\

&= c + \left(d\mathbf{n} + \mathbf{x} + \frac{d}{2}\mathbf{n} \times \mathbf{x} + (0 + \mathbf{x}\times(\frac{d}{2}\mathbf{n}) + 0)\right)\\

&= c + \left(d\mathbf{n} + \mathbf{x} + \frac{d}{2}\mathbf{n} \times \mathbf{x} - \frac{d}{2}\mathbf{n}\times \mathbf{x}\right)\\

&= c + \left(\mathbf{x} + d\mathbf{n}\right)\\

\end{aligned}

</math>

}} -->

==Dual Quaternions==

Dual quaternions are used to represent 3D rotations and translations together.

==Combined Point + Frame Alignment Problem==

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==Resources==

* [https://arxiv.org/pdf/1804.03528.pdf The Quaternion-Based Spatial Coordinate and Orientation Frame Alignment Problems by Andrew Hanson] [https://journals.iucr.org/a/issues/2020/04/00/ib5072/ib5072.pdf Edited Paper]

* [https://www.researchgate.net/publication/265550132_Approaching_Dual_Quaternions_From_Matrix_Algebra Approaching Dual Quaternions

* [https://www.researchgate.net/publication/265550132_Approaching_Dual_Quaternions_From_Matrix_Algebra Approaching Dual Quaternions From Matrix Algebra by Federico Thomas]

From Matrix Algebra by Federico Thomas]

* Double Quaternions for Motion Interpolation by Q.J. Ge, Amitabh Varshney, Jai P. Menon, Chu-Fei Chang

* [https://probablydance.com/2017/08/05/intuitive-quaternions/ Less Weird Quaternions by Malte Skarupke]

** Derives rotations as a sequence of reflection and quaternion algebra from wedge products.

** Presents quaternions as a sequence of (90 deg rotation + scaling) operations in 3D space.

==References==

{{reflist|refs=

<ref name="ge1998double">Q.J. Ge, Amitabh Varshney, Jai P. Menon, Chu-Fei Chang (1998) Double Quaternions for Motion Interpolation</ref>

}}

@@ Line 113: / Line 113: @@
 ==Double Quaternions==
 Note that double quaternions are different from dual quaternions.
-Double quaternions are written as <math>q_1 + \epsilon q_2</math> with <math>\epsilon^2 = 1</math>.
+Double quaternions are written as <math>q_1 + \epsilon q_2</math> with <math>\epsilon^2 = 1</math> and applied to a quaternion <math>p</math> representing a point as <math>q_1 p \bar{q}_2</math>.
-Dual quaternions are written as <math>q_1 + \varepsilon q_2</math> with <math>\varepsilon^2 = 0</math>.
+Dual quaternions are written as <math>q_1 + \varepsilon q_2</math> with <math>\varepsilon^2 = 0</math> and applied to a dual quaternion <math>p</math> representing a point as <math>(q_1 + \varepsilon q_2) p (\bar{q}_1 + \varepsilon \bar{q}_2)</math>
 ===Cayley Factorization===
@@ Line 121: / Line 121: @@
 Any 4D rotation matrix can be decomposed into a right and a left isoclinic rotation matrix:
 <math>R = R^L R^R = R^R R^L</math>
-<math>R^R</math> and <math>R^L</math> can be seen as a single double quaternion.
+<math>R^R</math> and <math>R^L</math> can be viewed as a matrix representation of single double quaternion <math>(q_1, q_2)</math>.
+For a double quaternion, the 4D rotation written is <math>x' = q_1 x q_2</math>.
 The product of left and right isoclinic rotation matrices commute.
@@ Line 127: / Line 128: @@
 Thus, <math>R_1 R_2 = (R_1^L R_1^R) (R_2^L R_2^R) = (R_1^L R_2^L) (R_1^R R_2^R) = R</math>.
 This shows that the composition of two double quaternions will be a double quaternion.
+===Approximating 3D Translations using Double quaternions===
+See Ge ''et. al.''<ref name="ge1998double"></ref>
+<!-- {{hidden | Algebraic Verification |
+Suppose our point is <math>\mathbf{x} \in \mathbb{R}^3</math> and our translation has direction <math>\mathbf{d} \in \mathbb{R}^3</math> with distance <math>d</math>.
+Let <math>q=1+\frac{d}{2}\mathbf{n}</math> and <math>q^* = 1-\frac{d}{2}\mathbf{n}</math>.
+<math>
+\begin{aligned}
+q(1+\mathbf{x})(q^*)^* &= (1+\frac{d}{2}\mathbf{n})(1+\mathbf{x})(1+\frac{d}{2}\mathbf{n})\\
+&= [(1 - \frac{d}{2} \mathbf{n} \cdot \mathbf{x}) + (\frac{d}{2}\mathbf{n} + \mathbf{x} + \frac{d}{2}\mathbf{n} \times \mathbf{x})](1+\frac{d}{2}\mathbf{n})\\
+&= \left(1 - \frac{d}{2} \mathbf{n} \cdot \mathbf{x} - ((\frac{d}{2}\mathbf{n} + \mathbf{x} + \frac{d}{2}\mathbf{n} \times \mathbf{x})) \cdot \frac{d}{2} \mathbf{n}\right) + \left(\frac{d}{2}\mathbf{n} + \mathbf{x} + \frac{d}{2}\mathbf{n} \times \mathbf{x} +\frac{d}{2}\mathbf{n} + (\frac{d}{2}\mathbf{n} + \mathbf{x} + \frac{d}{2}\mathbf{n} \times \mathbf{x})\times(\frac{d}{2}\mathbf{n})\right)\\
+&= c + \left(d\mathbf{n} + \mathbf{x} + \frac{d}{2}\mathbf{n} \times \mathbf{x} + (\frac{d}{2}\mathbf{n}\times(\frac{d}{2}\mathbf{n}) + \mathbf{x}\times(\frac{d}{2}\mathbf{n}) + (\frac{d}{2}\mathbf{n} \times \mathbf{x})\times(\frac{d}{2}\mathbf{n}))\right)\\
+&= c + \left(d\mathbf{n} + \mathbf{x} + \frac{d}{2}\mathbf{n} \times \mathbf{x} + (0 + \mathbf{x}\times(\frac{d}{2}\mathbf{n}) + 0)\right)\\
+&= c + \left(d\mathbf{n} + \mathbf{x} + \frac{d}{2}\mathbf{n} \times \mathbf{x} - \frac{d}{2}\mathbf{n}\times \mathbf{x}\right)\\
+&= c + \left(\mathbf{x} + d\mathbf{n}\right)\\
+\end{aligned}
+</math>
+}} -->
 ==Dual Quaternions==
 {{main | Dual quaternion}}
+Dual quaternions are used to represent 3D rotations and translations together.
 ==Combined Point + Frame Alignment Problem==
@@ Line 144: / Line 168: @@
 ==Resources==
 * [https://arxiv.org/pdf/1804.03528.pdf The Quaternion-Based Spatial Coordinate and Orientation Frame Alignment Problems by Andrew Hanson] [https://journals.iucr.org/a/issues/2020/04/00/ib5072/ib5072.pdf Edited Paper]
-* [https://www.researchgate.net/publication/265550132_Approaching_Dual_Quaternions_From_Matrix_Algebra Approaching Dual Quaternions
+* [https://www.researchgate.net/publication/265550132_Approaching_Dual_Quaternions_From_Matrix_Algebra Approaching Dual Quaternions From Matrix Algebra by Federico Thomas]
-From Matrix Algebra by Federico Thomas]
+* Double Quaternions for Motion Interpolation by Q.J. Ge, Amitabh Varshney, Jai P. Menon, Chu-Fei Chang
+* [https://probablydance.com/2017/08/05/intuitive-quaternions/ Less Weird Quaternions by Malte Skarupke]
+** Derives rotations as a sequence of reflection and quaternion algebra from wedge products.
+** Presents quaternions as a sequence of (90 deg rotation + scaling) operations in 3D space.
+==References==
+{{reflist|refs=
+<ref name="ge1998double">Q.J. Ge, Amitabh Varshney, Jai P. Menon, Chu-Fei Chang (1998) Double Quaternions for Motion Interpolation</ref>
+}}