Difference between revisions of "Geometric property"

From Projective Geometric Algebra
Jump to navigation Jump to search
Line 1: Line 1:
 
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]].
  
Every point $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ possesses the geometric property.
+
{| class="wikitable"
 
+
! Type !! Definition !! Requirement
A [[line]] $$\mathbf L = v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} + m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$ possesses the geometric property when $$v_xm_x + v_ym_y + v_zm_z = 0$$.
+
|-
 
+
| style="padding: 12px;" | [[Point]]
Every [[plane]] $$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$ possesses the geometric property.
+
| 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;" | —
A [[motor]] $$\mathbf Q = r_x \mathbf e_{41} + r_y \mathbf e_{42} + r_z \mathbf e_{43} + r_w {\large\unicode{x1d7d9}} + u_x \mathbf e_{23} + u_y \mathbf e_{31} + u_z \mathbf e_{12} + u_w$$ possesses the geometric property when $$r_xy_x + r_yu_y + r_zu_z + r_wu_w = 0$$.
+
|-
 
+
| style="padding: 12px;" | [[Line]]
A [[flector]] $$\mathbf F = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 + s_w \mathbf e_4 + h_x \mathbf e_{234} + h_y \mathbf e_{314} + h_z \mathbf e_{124} + h_w \mathbf e_{321}$$ possesses the geometric property when $$s_xh_x + s_yh_y + s_zh_z + s_wh_w = 0$$.
+
| 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 $$\mathbf F = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 + s_w \mathbf e_4 + h_x \mathbf e_{234} + h_y \mathbf e_{314} + h_z \mathbf e_{124} + h_w \mathbf e_{321}$$ $$s_xh_x + s_yh_y + s_zh_z + s_wh_w = 0$$

See Also