Complements

The degenerate metric of rigid geometric algebra naturally divides the components of every quantity into two groups called the bulk and the weight.

The bulk of an element $$\mathbf u$$ is denoted by $$\mathbf u_\unicode{x25CF}$$, and it is defined as

$$\mathbf u_\unicode{x25CF} = \mathbf G \mathbf u$$,

where $$\mathbf G$$ is the metric exomorphism matrix. The bulk consists of the components of $$\mathbf u$$ that do not have the projective basis vector $$\mathbf e_4$$ as a factor.

The weight is denoted by $$\mathbf u_\unicode{x25CB}$$, and it is defined as

$$\mathbf u_\unicode{x25CB} = \mathbb G \mathbf u$$,

where $$\mathbb G$$ is the metric anti-exomorphism matrix. The weight consists of the components of $$\mathbf u$$ that do have the projective basis vector $$\mathbf e_4$$ as a factor.

The bulk generally contains information about the position of an element relative to the origin, and the weight generally contains information about the attitude and orientation of an element. An object with zero bulk contains the origin. An object with zero weight is contained by the horizon.

An element is unitized when the magnitude of its weight is one.

The following table lists the bulk and weight for the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.

Type	Definition	Bulk	Weight
Magnitude	$$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$	$$\mathbf z_\unicode{x25CF} = x \mathbf 1$$	$$\mathbf z_\unicode{x25CB} = y {\large\unicode{x1d7d9}}$$
Point	$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$	$$\mathbf p_\unicode{x25CF} = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3$$	$$\mathbf p_\unicode{x25CB} = p_w \mathbf e_4$$
Line	$$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$	$$\boldsymbol l_\unicode{x25CF} = l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$	$$\boldsymbol l_\unicode{x25CB} = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43}$$
Plane	$$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$	$$\mathbf g_\unicode{x25CF} = g_w \mathbf e_{321}$$	$$\mathbf g_\unicode{x25CB} = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412}$$
Motor	$$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$	$$\mathbf Q_\unicode{x25CF} = Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$	$$\mathbf Q_\unicode{x25CB} = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}}$$
Flector	$$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$	$$\mathbf F_\unicode{x25CF} = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{gw} \mathbf e_{321}$$	$$\mathbf F_\unicode{x25CB} = F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412}$$

See Also

• Attitude
• Geometric norm
• Unitization
• Complements
• Duals