Dual quaternion: Difference between revisions
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{{ main | Wikipedia: Dual quaternion}} | {{ main | Wikipedia: Dual quaternion}} | ||
A dual quaternion can be written as <math>\mathbf{q} = \mathbf{q}_r + \mathbf{q}_d \ | A dual quaternion can be written as <math>\mathbf{q} = \mathbf{q}_r + \mathbf{q}_d \varepsilon</math>. | ||
Here, <math>\ | Here, <math>\varepsilon^2=0</math>. | ||
;Scalar Multiplication | ;Scalar Multiplication | ||
<math>s\mathbf{q} = s\mathbf{q}_r + s \mathbf{q}_d \ | <math>s\mathbf{q} = s\mathbf{q}_r + s \mathbf{q}_d \varepsilon</math> | ||
;Addition | ;Addition | ||
<math>\mathbf{q}_1 + \mathbf{q}_2 = \mathbf{q}_{r1} +\mathbf{q}_{r2} + (\mathbf{q}_{d1} + \mathbf{q}_{d2}) \ | <math>\mathbf{q}_1 + \mathbf{q}_2 = \mathbf{q}_{r1} +\mathbf{q}_{r2} + (\mathbf{q}_{d1} + \mathbf{q}_{d2}) \varepsilon</math> | ||
;Multiplication | ;Multiplication | ||
<math>\mathbf{q}_1 \mathbf{q}_2 = \mathbf{q}_{r1} \mathbf{q}_{r2} + (\mathbf{q}_{r1}\mathbf{q}_{d2} + \mathbf{q}_{d1} \mathbf{q}_{r2})\ | <math>\mathbf{q}_1 \mathbf{q}_2 = \mathbf{q}_{r1} \mathbf{q}_{r2} + (\mathbf{q}_{r1}\mathbf{q}_{d2} + \mathbf{q}_{d1} \mathbf{q}_{r2})\varepsilon</math>. | ||
;Conjugate | ;Conjugate | ||
<math>\mathbf{q}^* = \mathbf{q}_{r}^* + \mathbf{q}_{d}^*\ | <math>\mathbf{q}^* = \mathbf{q}_{r}^* + \mathbf{q}_{d}^*\varepsilon</math> | ||
;Magnitude | ;Magnitude | ||
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==Rotations and Translations== | ==Rotations and Translations== | ||
A translation is represented as: | A translation is represented as: | ||
<math>\mathbf{q}_t = [1,0,0,0][0, \frac{t_x}{2}, \frac{t_y}{2}, \frac{t_z}{2}] = 1 + \frac{\ | <math>\mathbf{q}_t = [1,0,0,0][0, \frac{t_x}{2}, \frac{t_y}{2}, \frac{t_z}{2}] = 1 + \frac{\varepsilon}{2}\mathbf{t}</math> | ||
A rotation is represented as: | A rotation is represented as: | ||
<math>\mathbf{q}_r = [\cos(\frac{\theta}{2}), \sin(\frac{\theta}{2})n_x, \sin(\frac{\theta}{2})n_y, \sin(\frac{\theta}{2})n_z][0,0,0,0] = \cos(\frac{\theta}{2}) + \sin(\frac{\theta}{2}) \mathbf{n}</math> | <math>\mathbf{q}_r = [\cos(\frac{\theta}{2}), \sin(\frac{\theta}{2})n_x, \sin(\frac{\theta}{2})n_y, \sin(\frac{\theta}{2})n_z][0,0,0,0] = \cos(\frac{\theta}{2}) + \sin(\frac{\theta}{2}) \mathbf{n}</math> | ||
These can be combined as <math>\mathbf{q} = \mathbf{q}_t * \mathbf{q}_r = \mathbf{q}_r + \frac{\ | These can be combined as <math>\mathbf{q} = \mathbf{q}_t * \mathbf{q}_r = \mathbf{q}_r + \frac{\varepsilon}{2}\mathbf{t}\mathbf{q}_r</math>. | ||
Applying the transformation to a point <math>\mathbf{v} \in \mathbb{R}^3</math> is: | Applying the transformation to a point <math>\mathbf{v} \in \mathbb{R}^3</math> is: | ||
<math>\mathbf{p}' = \mathbf{q}(1 + \ | <math>\mathbf{p}' = \mathbf{q}(1 + \varepsilon\mathbf{v})\mathbf{q}^*</math> | ||
==Resources== | ==Resources== | ||
* [https://cs.gmu.edu/~jmlien/teaching/cs451/uploads/Main/dual-quaternion.pdf A Beginners Guide to Dual-Quaternions by Ben Kenwright] | * [https://cs.gmu.edu/~jmlien/teaching/cs451/uploads/Main/dual-quaternion.pdf A Beginners Guide to Dual-Quaternions by Ben Kenwright] | ||
Latest revision as of 13:39, 16 October 2020
Dual quaternions are an 8-dimensional number system (i.e. isomorphic to \(\displaystyle \mathbb{R}^8\)) which can be used to jointly represent rotations and translations in 3D space. They can be used in place of the standard \(\displaystyle 4 \times 4\) homogeneous transformation matrices.
Background
A dual quaternion can be written as \(\displaystyle \mathbf{q} = \mathbf{q}_r + \mathbf{q}_d \varepsilon\).
Here, \(\displaystyle \varepsilon^2=0\).
- Scalar Multiplication
\(\displaystyle s\mathbf{q} = s\mathbf{q}_r + s \mathbf{q}_d \varepsilon\)
- Addition
\(\displaystyle \mathbf{q}_1 + \mathbf{q}_2 = \mathbf{q}_{r1} +\mathbf{q}_{r2} + (\mathbf{q}_{d1} + \mathbf{q}_{d2}) \varepsilon\)
- Multiplication
\(\displaystyle \mathbf{q}_1 \mathbf{q}_2 = \mathbf{q}_{r1} \mathbf{q}_{r2} + (\mathbf{q}_{r1}\mathbf{q}_{d2} + \mathbf{q}_{d1} \mathbf{q}_{r2})\varepsilon\).
- Conjugate
\(\displaystyle \mathbf{q}^* = \mathbf{q}_{r}^* + \mathbf{q}_{d}^*\varepsilon\)
- Magnitude
\(\displaystyle \Vert \mathbf{q} \Vert = \mathbf{q}\mathbf{q}^*\)
Rotations and Translations
A translation is represented as:
\(\displaystyle \mathbf{q}_t = [1,0,0,0][0, \frac{t_x}{2}, \frac{t_y}{2}, \frac{t_z}{2}] = 1 + \frac{\varepsilon}{2}\mathbf{t}\)
A rotation is represented as:
\(\displaystyle \mathbf{q}_r = [\cos(\frac{\theta}{2}), \sin(\frac{\theta}{2})n_x, \sin(\frac{\theta}{2})n_y, \sin(\frac{\theta}{2})n_z][0,0,0,0] = \cos(\frac{\theta}{2}) + \sin(\frac{\theta}{2}) \mathbf{n}\)
These can be combined as \(\displaystyle \mathbf{q} = \mathbf{q}_t * \mathbf{q}_r = \mathbf{q}_r + \frac{\varepsilon}{2}\mathbf{t}\mathbf{q}_r\).
Applying the transformation to a point \(\displaystyle \mathbf{v} \in \mathbb{R}^3\) is:
\(\displaystyle \mathbf{p}' = \mathbf{q}(1 + \varepsilon\mathbf{v})\mathbf{q}^*\)