Geometric constraint and Duals: Difference between pages

Revision as of 01:31, 12 May 2024

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:

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

The following table lists the bulk and weight duals for all of the basis elements in the 4D geometric algebra $$\mathcal G_{3,0,1}$$.

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	$$\mathbf g^\unicode["segoe ui symbol"]{x2605} = -g_w \mathbf e_4$$	$$\mathbf g^\unicode["segoe ui symbol"]{x2606} = -g_x \mathbf e_1 - g_y \mathbf e_2 - g_z \mathbf e_3$$

In the Book

Duals are introduced in Section 2.12.

@@ Line 1: / Line 1: @@
-An element $$\mathbf x$$ of a geometric algebra possesses the ''geometric constraint'' if and only if the [[geometric product]] between $$\mathbf u$$ and its own reverse is a scalar, which is given by the [[dot product]], and the [[geometric antiproduct]] between $$\mathbf u$$ and its own antireverse is an antiscalar, which is given by the [[antidot product]]. That is,
+Every object in projective geometric algebra has two duals derived from the metric tensor, called the ''metric dual'' and ''metric antidual''.
-:$$\mathbf u \mathbin{\unicode{x27D1}} \mathbf{\tilde u} = \mathbf u \mathbin{\unicode{x25CF}} \mathbf u$$
+== Dual ==
-and
+The ''metric dual'' or just "dual" of an object $$\mathbf u$$ is denoted by $$\mathbf u^\unicode["segoe ui symbol"]{x2605}$$ and defined as
-:$$\mathbf u \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} = \mathbf u \mathbin{\unicode{x25CB}} \mathbf u$$ .
+:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf{Gu}}$$ ,
-The set of all elements satisfying the geometric constraint is closed under both the [[geometric product]] and [[geometric antiproduct]].
+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
-The following table lists the requirements that must be satisfied for the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ due to the geometric constraint. Points and planes do not have any requirements&mdash;they have no constraints.
+:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf u_\unicode{x25CF}}$$ .
+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]]:
+:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} \mathbin{\unicode{x27C7}} \mathbf 1$$ .
+The following table lists the bulk and weight duals for all of the basis elements in the 4D geometric algebra $$\mathcal G_{3,0,1}$$.
+[[Image:Duals.svg|720px]]
+== Geometries ==
+The bulk duals and weight duals of geometries in the 4D rigid geometric algebra are listed in the following table.
 {| class="wikitable"
-! Type !! Definition !! Requirement
+! Type !! Bulk Dual !! Weight Dual
-|-
-| style="padding: 12px;" | [[Magnitude]]
-| style="padding: 12px;" | $$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$
-| style="padding: 12px;" | $$xy = 0$$
 |-
 | style="padding: 12px;" | [[Point]]
-| 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;" | $$\mathbf p^\unicode["segoe ui symbol"]{x2605} = p_x \mathbf e_{423} + p_y \mathbf e_{431} + p_z \mathbf e_{412}$$
-| style="padding: 12px;" | &mdash;
+| style="padding: 12px;" | $$\mathbf p^\unicode["segoe ui symbol"]{x2606} = p_w \mathbf e_{321}$$
 |-
 | style="padding: 12px;" | [[Line]]
-| style="padding: 12px;" | $$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$
+| style="padding: 12px;" | $$\boldsymbol l^\unicode["segoe ui symbol"]{x2605} = -l_{mx} \mathbf e_{41} - l_{my} \mathbf e_{42} - l_{mz} \mathbf e_{43}$$
-| style="padding: 12px;" | $$l_{vx} l_{mx} + l_{vy} l_{my} + l_{vz} l_{mz} = 0$$
+| style="padding: 12px;" | $$\boldsymbol l^\unicode["segoe ui symbol"]{x2606} = -l_{vx} \mathbf e_{23} - l_{vy} \mathbf e_{31} - l_{vz} \mathbf e_{12}$$
 |-
 | style="padding: 12px;" | [[Plane]]
-| style="padding: 12px;" | $$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$
+| style="padding: 12px;" | $$\mathbf g^\unicode["segoe ui symbol"]{x2605} = -g_w \mathbf e_4$$
-| style="padding: 12px;" | &mdash;
+| style="padding: 12px;" | $$\mathbf g^\unicode["segoe ui symbol"]{x2606} = -g_x \mathbf e_1 - g_y \mathbf e_2 - g_z \mathbf e_3$$
-|-
-| style="padding: 12px;" | [[Motor]]
-| style="padding: 12px;" | $$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$
-| style="padding: 12px;" | $$Q_{vx} Q_{mx} + Q_{vy} Q_{my} + Q_{vz} Q_{mz} + Q_{vw} Q_{mw} = 0$$
-|-
-| style="padding: 12px;" | [[Flector]]
-| style="padding: 12px;" | $$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$
-| style="padding: 12px;" | $$F_{px} F_{gx} + F_{py} F_{gy} + F_{pz} F_{gz} + F_{pw} F_{gw} = 0$$
 |}
+== In the Book ==
+* Duals are introduced in Section 2.12.
 == See Also ==
-* [[Geometric norm]]
+* [[Complements]]
+* [[Bulk and weight]]

Geometric constraint and Duals: Difference between pages

Revision as of 01:31, 12 May 2024

Contents

Dual

Antidual

Geometries

In the Book

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