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| Every object in projective geometric algebra has two duals derived from the metric tensor, called the ''metric dual'' and ''metric antidual''.
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| == Dual ==
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| The ''metric dual'' or just "dual" of an object $$\mathbf u$$ is denoted by $$\mathbf u^\unicode["segoe ui symbol"]{x2605}$$ and defined as
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| :$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf{Gu}}$$ ,
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| 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
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| :$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf u_\unicode{x25CF}}$$ .
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| The bulk dual satisfies the following identity based on the [[geometric product]]:
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| :$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \tilde{\mathbf u} \mathbin{\unicode{x27D1}} {\large\unicode{x1D7D9}}$$ .
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| == Antidual ==
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| The ''metric antidual'' or just "antidual" of an object $$\mathbf u$$ is denoted by $$\mathbf u^\unicode["segoe ui symbol"]{x2606}$$ and defined as
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| :$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbb G \mathbf u}$$ ,
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| 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
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| :$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbf u_\unicode{x25CB}}$$ .
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| The weight dual satisfies the following identity based on the [[geometric antiproduct]]:
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| :$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} \mathbin{\unicode{x27C7}} \mathbf 1$$ .
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| The following table lists the bulk and weight duals for all of the basis elements in the 4D geometric algebra $$\mathcal G_{3,0,1}$$.
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| [[Image:Duals.svg|720px]]
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| == Geometries ==
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| The bulk duals and weight duals of geometries in the 4D rigid geometric algebra are listed in the following table.
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| {| class="wikitable"
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| ! Type !! Bulk Dual !! Weight Dual
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| | style="padding: 12px;" | [[Point]]
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| | style="padding: 12px;" | $$\mathbf p^\unicode["segoe ui symbol"]{x2605} = p_x \mathbf e_{423} + p_y \mathbf e_{431} + p_z \mathbf e_{412}$$
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| | style="padding: 12px;" | $$\mathbf p^\unicode["segoe ui symbol"]{x2606} = p_w \mathbf e_{321}$$
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| |-
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| | style="padding: 12px;" | [[Line]]
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| | style="padding: 12px;" | $$\boldsymbol l^\unicode["segoe ui symbol"]{x2605} = -l_{mx} \mathbf e_{41} - l_{my} \mathbf e_{42} - l_{mz} \mathbf e_{43}$$
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| | style="padding: 12px;" | $$\boldsymbol l^\unicode["segoe ui symbol"]{x2606} = -l_{vx} \mathbf e_{23} - l_{vy} \mathbf e_{31} - l_{vz} \mathbf e_{12}$$
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| | style="padding: 12px;" | [[Plane]]
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| | style="padding: 12px;" | $$\mathbf g^\unicode["segoe ui symbol"]{x2605} = -g_w \mathbf e_4$$
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| | style="padding: 12px;" | $$\mathbf g^\unicode["segoe ui symbol"]{x2606} = -g_x \mathbf e_1 - g_y \mathbf e_2 - g_z \mathbf e_3$$
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| |}
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| == In the Book ==
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| * Duals are introduced in Section 2.12.
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| == See Also ==
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| * [[Complements]]
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| * [[Bulk and weight]]
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