Duals

Every object in projective geometric algebra has two duals derived from the metric tensor, called the metric dual and metric antidual.

Dual

The metric dual or just "dual" of an object $$\mathbf u$$ is denoted by $$\mathbf u^\unicode["segoe ui symbol"]{x2605}$$ and defined as

$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf{Gu}}$$ ,

where $$\mathbf G$$ is the $$16 \times 16$$ metric exomorphism matrix. In projective geometric algebra, this dual is also called the bulk dual because it is the complement of the bulk components, as expressed by

$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf u_\unicode{x25CF}}$$ .

The bulk dual satisfies the following identity based on the geometric product:

$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \tilde{\mathbf u} \mathbin{\unicode{x27D1}} {\large\unicode{x1D7D9}}$$ .

Antidual

The metric antidual or just "antidual" of an object $$\mathbf u$$ is denoted by $$\mathbf u^\unicode["segoe ui symbol"]{x2606}$$ and defined as

$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbb G \mathbf u}$$ ,

where $$\mathbb G$$ is the $$16 \times 16$$ metric antiexomorphism matrix. In projective geometric algebra, this dual is also called the weight dual because it is the complement of the weight components, as expressed by

$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbf u_\unicode{x25CB}}$$ .

The weight dual satisfies the following identity based on the geometric antiproduct:

$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} \mathbin{\unicode{x27C7}} \mathbf 1$$ .

Geometries

The bulk duals and weight duals of geometries in the 4D rigid geometric algebra are listed in the following table.

Type	Bulk Dual	Weight Dual
Point	$$\mathbf p^\unicode["segoe ui symbol"]{x2605} = p_x \mathbf e_{423} + p_y \mathbf e_{431} + p_z \mathbf e_{412}$$	$$\mathbf p^\unicode["segoe ui symbol"]{x2606} = p_w \mathbf e_{321}$$
Line	$$\boldsymbol l^\unicode["segoe ui symbol"]{x2605} = -l_{mx} \mathbf e_{41} - l_{my} \mathbf e_{42} - l_{mz} \mathbf e_{43}$$	$$\boldsymbol l^\unicode["segoe ui symbol"]{x2606} = -l_{vx} \mathbf e_{23} - l_{vy} \mathbf e_{31} - l_{vz} \mathbf e_{12}$$
Plane	$$\mathbf g^\unicode["segoe ui symbol"]{x2605} = -g_w \mathbf e_4$$	$$\mathbf g^\unicode["segoe ui symbol"]{x2606} = -g_x \mathbf e_1 - g_y \mathbf e_2 - g_z \mathbf e_3$$

See Also

• Complements
• Bulk and weight

In the Book

• Duals are introduced in Section 2.12.