Difference between revisions of "Geometric property"
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An element of a projective geometric algebra possesses the ''geometric property'' if and only if its [[bulk norm]] is a pure [[scalar]] and its [[weight norm]] is a pure [[antiscalar]]. The set of all elements possessing the geometric property is closed under both the [[geometric product]] and [[geometric antiproduct]]. | An element of a projective geometric algebra possesses the ''geometric property'' if and only if its [[bulk norm]] is a pure [[scalar]] and its [[weight norm]] is a pure [[antiscalar]]. The set of all elements possessing the geometric property is closed under both the [[geometric product]] and [[geometric antiproduct]]. | ||
| − | + | {| class="wikitable" | |
| − | + | ! Type !! Definition !! Requirement | |
| − | + | |- | |
| − | + | | style="padding: 12px;" | [[Point]] | |
| − | + | | style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ | |
| − | + | | style="padding: 12px;" | — | |
| − | + | |- | |
| − | + | | style="padding: 12px;" | [[Line]] | |
| − | + | | style="padding: 12px;" | $$\mathbf L = v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} + m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$ | |
| + | | style="padding: 12px;" | $$v_xm_x + v_ym_y + v_zm_z = 0$$ | ||
| + | |- | ||
| + | | style="padding: 12px;" | [[Plane]] | ||
| + | | style="padding: 12px;" | $$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$ | ||
| + | | style="padding: 12px;" | — | ||
| + | |- | ||
| + | | style="padding: 12px;" | [[Motor]] | ||
| + | | style="padding: 12px;" | $$\mathbf Q = r_x \mathbf e_{41} + r_y \mathbf e_{42} + r_z \mathbf e_{43} + r_w {\large\unicode{x1d7d9}} + u_x \mathbf e_{23} + u_y \mathbf e_{31} + u_z \mathbf e_{12} + u_w$$ | ||
| + | | style="padding: 12px;" | $$r_xy_x + r_yu_y + r_zu_z + r_wu_w = 0$$ | ||
| + | |- | ||
| + | | style="padding: 12px;" | [[Flector]] | ||
| + | | style="padding: 12px;" | $$\mathbf F = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 + s_w \mathbf e_4 + h_x \mathbf e_{234} + h_y \mathbf e_{314} + h_z \mathbf e_{124} + h_w \mathbf e_{321}$$ | ||
| + | | style="padding: 12px;" | $$s_xh_x + s_yh_y + s_zh_z + s_wh_w = 0$$ | ||
| + | |} | ||
== See Also == | == See Also == | ||
* [[Geometric Norm]] | * [[Geometric Norm]] | ||
Revision as of 02:07, 19 April 2021
An element of a projective geometric algebra possesses the geometric property if and only if its bulk norm is a pure scalar and its weight norm is a pure antiscalar. The set of all elements possessing the geometric property is closed under both the geometric product and geometric antiproduct.
| Type | Definition | Requirement |
|---|---|---|
| Point | $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ | — |
| Line | $$\mathbf L = v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} + m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$ | $$v_xm_x + v_ym_y + v_zm_z = 0$$ |
| Plane | $$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$ | — |
| Motor | $$\mathbf Q = r_x \mathbf e_{41} + r_y \mathbf e_{42} + r_z \mathbf e_{43} + r_w {\large\unicode{x1d7d9}} + u_x \mathbf e_{23} + u_y \mathbf e_{31} + u_z \mathbf e_{12} + u_w$$ | $$r_xy_x + r_yu_y + r_zu_z + r_wu_w = 0$$ |
| Flector | $$\mathbf F = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 + s_w \mathbf e_4 + h_x \mathbf e_{234} + h_y \mathbf e_{314} + h_z \mathbf e_{124} + h_w \mathbf e_{321}$$ | $$s_xh_x + s_yh_y + s_zh_z + s_wh_w = 0$$ |