Applications of dual quaternions to 2D geometry

Planar quaternion multiplication<table><tbody><tr><th>                    ×              {\displaystyle \times }  </th><th>                    1              {\displaystyle 1}  </th><th>                    i              {\displaystyle i}  </th><th>                    ε        j              {\displaystyle \varepsilon j}  </th><th>                    ε        k              {\displaystyle \varepsilon k}  </th></tr><tr><th>                    1              {\displaystyle 1}  </th><td>                    1              {\displaystyle 1}  </td><td>                    i              {\displaystyle i}  </td><td>                    ε        j              {\displaystyle \varepsilon j}  </td><td>                    ε        k              {\displaystyle \varepsilon k}  </td></tr><tr><th>                    i              {\displaystyle i}  </th><td>                    i              {\displaystyle i}  </td><td>                    −        1              {\displaystyle -1}  </td><td>                    ε        k              {\displaystyle \varepsilon k}  </td><td>                    −        ε        j              {\displaystyle -\varepsilon j}  </td></tr><tr><th>                    ε        j              {\displaystyle \varepsilon j}  </th><td>                    ε        j              {\displaystyle \varepsilon j}  </td><td>                    −        ε        k              {\displaystyle -\varepsilon k}  </td><td>                    0              {\displaystyle 0}  </td><td>                    0              {\displaystyle 0}  </td></tr><tr><th>                    ε        k              {\displaystyle \varepsilon k}  </th><td>                    ε        k              {\displaystyle \varepsilon k}  </td><td>                    ε        j              {\displaystyle \varepsilon j}  </td><td>                    0              {\displaystyle 0}  </td><td>                    0              {\displaystyle 0}  </td></tr></tbody></table>
<p>In this article, certain applications of the <a href="/facts/Dual_quaternion/3pDCq0gN">dual quaternion</a> algebra to 2D geometry are discussed. At this present time, the article is focused on a 4-dimensional subalgebra of the dual quaternions which will later be called the <i>planar quaternions</i>.
</p><p>The planar quaternions make up a four-dimensional <a href="/facts/Algebra_over_a_field/8T8ulWJb">algebra</a> over the <a href="/facts/Real_number/R02gw5Pb">real numbers</a>. Their primary application is in representing <a href="/facts/Rigid_body_motion/THezBlq2">rigid body motions</a> in 2D space.
</p><p>Unlike multiplication of <a href="/facts/Dual_number/DntuRu3F">dual numbers</a> or of <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>, that of planar quaternions is <a href="/facts/Non-commutative/WaYxJLxd">non-commutative</a>.
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Applications of dual quaternions to 2D geometry open-in-new

Applications of dual quaternions to 2D geometry