5,337
edits
Dual quaternion: Difference between revisions
→Rotations and Translations
| Line 24: | Line 24: | ||
==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] | ||