Magnitude and Reverses: Difference between pages

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A ''magnitude'' is a quantity that represents a concrete measurement of some kind. In rigid geometric algebra, a magnitude $$\mathbf z$$ is composed of two components, a scalar and an antiscalar, as follows:
''Reverses'' are unary operations in geometric algebra that are analogs of conjugate or transpose operations.


:$$\mathbf z = x\mathbf 1 + y {\large\unicode{x1d7d9}}$$
For any element $$\mathbf x$$ that is the [[wedge product]] of $$k$$ vectors, the ''reverse'' of $$\mathbf x$$, which we denote by $$\mathbf{\tilde x}$$, is the result of multiplying those same $$k$$ vectors in reverse order. For example, the reverse of $$\mathbf e_{234}$$ is $$\mathbf e_4 \wedge \mathbf e_3 \wedge \mathbf e_2$$, which we would write as $$-\mathbf e_{234}$$since 432 is an odd permutation of 234. In general, the reverse of an element $$\mathbf x$$ is given by


Magnitudes are homogeneous just like everything else in a projective geometric algebra. This means it has both a [[bulk]] and a [[weight]], and it is [[unitized]] by making the magnitude of its weight one.
:$$\mathbf{\tilde x} = (-1)^{\operatorname{gr}(\mathbf x)(\operatorname{gr}(\mathbf x) - 1)/2}\,\mathbf x$$ .


=== Examples ===
Symmetrically, for any element $$\mathbf x$$ that is the [[antiwedge product]] of $$m$$ antivectors, the ''antireverse'' of $$\mathbf x$$, which we denote by $$\smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}}$$, 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 $$\mathbf x$$ is given by


* The [[geometric norm]] produces a magnitude that gives the perpendicular distance between an object and the origin. This is also half the distance that the origin is moved by an object used as an operator.
:$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}} = (-1)^{\operatorname{ag}(\mathbf x)(\operatorname{ag}(\mathbf x) - 1)/2}\,\mathbf x$$ .
* [[Euclidean distances]] between objects are expressed as magnitudes given by the sum of the [[bulk norm]] and [[weight norm]] of expressions involving [[attitudes]].
 
* Exponentiating the magnitude $$\delta\mathbf 1 + \phi {\large\unicode{x1d7d9}}$$ produces a [[motor]] for which $$\delta/\phi$$ is the pitch of the screw transformation.
The reverse and antireverse of any element $$\mathbf x$$ are related by
 
:$$\smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}} = (-1)^{\operatorname{gr}(\mathbf x)\operatorname{ag}(\mathbf x)}(-1)^{n(n-1)/2}\,\mathbf{\tilde x}$$ ,
 
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 $$\mathcal G_{3,0,1}$$.
 
[[Image:Reverses.svg|720px]]


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


* [[Geometric norm]]
* [[Complements]]
* [[Unitization]]

Revision as of 08:01, 22 January 2024

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

For any element $$\mathbf x$$ that is the wedge product of $$k$$ vectors, the reverse of $$\mathbf x$$, which we denote by $$\mathbf{\tilde x}$$, is the result of multiplying those same $$k$$ vectors in reverse order. For example, the reverse of $$\mathbf e_{234}$$ is $$\mathbf e_4 \wedge \mathbf e_3 \wedge \mathbf e_2$$, which we would write as $$-\mathbf e_{234}$$since 432 is an odd permutation of 234. In general, the reverse of an element $$\mathbf x$$ is given by

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

Symmetrically, for any element $$\mathbf x$$ that is the antiwedge product of $$m$$ antivectors, the antireverse of $$\mathbf x$$, which we denote by $$\smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}}$$, 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 $$\mathbf x$$ is given by

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

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

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

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 $$\mathcal G_{3,0,1}$$.

See Also