Duals

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 $$\mathbf u_\unicode["segoe ui symbol"]{x25CF}$$, and it is defined as

$$\mathbf u_\unicode["segoe ui symbol"]{x25CF} = \mathbf G \mathbf u$$,

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 $$\mathbf u_\unicode["segoe ui symbol"]{x25CB}$$, and it is defined as

$$\mathbf u_\unicode["segoe ui symbol"]{x25CB} = \mathbb G \mathbf u$$,

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

In the Book

• Bulk and weight are introduced in Section 2.8.3.

See Also

• Attitude
• Geometric norm
• Unitization
• Complements
• Duals