Duals and Reverses: Difference between pages

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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''.
''Reverses'' are unary operations in geometric algebra that are analogs of conjugate or transpose operations.


== Dual ==
For any element $$\mathbf u$$ that is the [[wedge product]] of $$k$$ vectors, the ''reverse'' of $$\mathbf u$$, which we denote by $$\mathbf{\tilde u}$$, is the result of multiplying those same $$k$$ vectors in reverse order. For example, the reverse of $$\mathbf e_{423}$$ is $$\mathbf e_3 \wedge \mathbf e_2 \wedge \mathbf e_4$$, which we would write as $$-\mathbf e_{423}$$since 324 is an odd permutation of 423. In general, the reverse of an element $$\mathbf u$$ is given by


The ''metric dual'' or just "dual" of an object $$\mathbf u$$ is denoted by $$\mathbf u^\unicode["segoe ui symbol"]{x2605}$$ and defined as
:$$\mathbf{\tilde u} = (-1)^{\operatorname{gr}(\mathbf u)(\operatorname{gr}(\mathbf u) - 1)/2}\,\mathbf u$$ .


:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf{Gu}}$$ ,
Symmetrically, for any element $$\mathbf u$$ that is the [[antiwedge product]] of $$m$$ antivectors, the ''antireverse'' of $$\mathbf u$$, which we denote by $$\smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}}$$, is the result of multiplying those same $$m$$ antivectors in reverse order (but this time under the [[antiwedge product]]). In general, the antireverse of an element $$\mathbf u$$ is given by


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
:$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} = (-1)^{\operatorname{ag}(\mathbf u)(\operatorname{ag}(\mathbf u) - 1)/2}\,\mathbf u$$ .


:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf u_\unicode{x25CF}}$$ .
The reverse and antireverse of any element $$\mathbf u$$ are related by


The bulk dual satisfies the following identity based on the [[geometric product]]:
:$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} = (-1)^{\operatorname{gr}(\mathbf u)\operatorname{ag}(\mathbf u)}(-1)^{n(n-1)/2}\,\mathbf{\tilde u}$$ ,


:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \tilde{\mathbf u} \mathbin{\unicode{x27D1}} {\large\unicode{x1D7D9}}$$ .
where $$n$$ is the number of dimensions in the algebra. To extend the reversals to all elements of an algebra, we simply require that it is a linear operation. For any basis elements $$\mathbf x$$ and $$\mathbf y$$, and for any scalars $$a$$ and $$b$$, we must have, for the reverse


== Antidual ==
:$$\widetilde{(a\mathbf x + b\mathbf y)} = a\mathbf{\tilde x} + b\mathbf{\tilde y}$$ ,


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


:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbb G \mathbf u}$$ ,
The following table lists the reverse and antireverse for all of the basis elements in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.


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
[[Image:Reverses.svg|720px]]


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


{| class="wikitable"
* Reverses and antireverses are introduced in Section 3.4.
! Type !! Bulk Dual !! Weight Dual
|-
| style="padding: 12px;" | [[Point]]
| 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;" | $$\mathbf p^\unicode["segoe ui symbol"]{x2606} = p_w \mathbf e_{321}$$
|-
| style="padding: 12px;" | [[Line]]
| 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;" | $$\boldsymbol l^\unicode["segoe ui symbol"]{x2606} = -l_{vx} \mathbf e_{23} - l_{vy} \mathbf e_{31} - l_{vz} \mathbf e_{12}$$
|-
| style="padding: 12px;" | [[Plane]]
| style="padding: 12px;" | $$\mathbf g^\unicode["segoe ui symbol"]{x2605} = -g_w \mathbf e_4$$
| style="padding: 12px;" | $$\mathbf g^\unicode["segoe ui symbol"]{x2606} = -g_x \mathbf e_1 - g_y \mathbf e_2 - g_z \mathbf e_3$$
|}


== See Also ==
== See Also ==


* [[Complements]]
* [[Complements]]
* [[Bulk and weight]]
== In the Book ==
* Duals are introduced in Section 2.12.

Latest revision as of 23:32, 13 April 2024

Reverses are unary operations in geometric algebra that are analogs of conjugate or transpose operations.

For any element $$\mathbf u$$ that is the wedge product of $$k$$ vectors, the reverse of $$\mathbf u$$, which we denote by $$\mathbf{\tilde u}$$, is the result of multiplying those same $$k$$ vectors in reverse order. For example, the reverse of $$\mathbf e_{423}$$ is $$\mathbf e_3 \wedge \mathbf e_2 \wedge \mathbf e_4$$, which we would write as $$-\mathbf e_{423}$$since 324 is an odd permutation of 423. In general, the reverse of an element $$\mathbf u$$ is given by

$$\mathbf{\tilde u} = (-1)^{\operatorname{gr}(\mathbf u)(\operatorname{gr}(\mathbf u) - 1)/2}\,\mathbf u$$ .

Symmetrically, for any element $$\mathbf u$$ that is the antiwedge product of $$m$$ antivectors, the antireverse of $$\mathbf u$$, which we denote by $$\smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}}$$, is the result of multiplying those same $$m$$ antivectors in reverse order (but this time under the antiwedge product). In general, the antireverse of an element $$\mathbf u$$ is given by

$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} = (-1)^{\operatorname{ag}(\mathbf u)(\operatorname{ag}(\mathbf u) - 1)/2}\,\mathbf u$$ .

The reverse and antireverse of any element $$\mathbf u$$ are related by

$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{u}}} = (-1)^{\operatorname{gr}(\mathbf u)\operatorname{ag}(\mathbf u)}(-1)^{n(n-1)/2}\,\mathbf{\tilde u}$$ ,

where $$n$$ is the number of dimensions in the algebra. To extend the reversals to all elements of an algebra, we simply require that it is a linear operation. For any basis elements $$\mathbf x$$ and $$\mathbf y$$, and for any scalars $$a$$ and $$b$$, we must have, for the reverse

$$\widetilde{(a\mathbf x + b\mathbf y)} = a\mathbf{\tilde x} + b\mathbf{\tilde y}$$ ,

and similarly for the antireverse.

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

In the Book

  • Reverses and antireverses are introduced in Section 3.4.

See Also

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