Magnitude - Revision history

Eric Lengyel at 08:12, 25 November 2023

2023-11-25T08:12:17Z

← Older revision		Revision as of 08:12, 25 November 2023
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	A ''magnitude'' is a quantity that represents a concrete ~~distance~~ of some kind. In rigid geometric algebra, a magnitude $$\mathbf z$$ is composed of two components, a scalar and an antiscalar, as follows:		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:

	:$$\mathbf z = x\mathbf 1 + y {\large\unicode{x1d7d9}}$$		:$$\mathbf z = x\mathbf 1 + y {\large\unicode{x1d7d9}}$$

Eric Lengyel: /* Examples */

2023-08-01T05:07:23Z

Examples

← Older revision		Revision as of 05:07, 1 August 2023
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	* 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.		* 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.
	* [[Euclidean distances]] between objects are expressed as magnitudes given by the sum of the [[bulk ~~norms~~]] and [[weight ~~norms~~]] of [[~~commutators~~]].		* [[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 $$d\mathbf 1 + \phi {\large\unicode{x1d7d9}}$$ produces a [[motor]] for which $$d/\phi$$ is the pitch of the screw transformation.		* Exponentiating the magnitude $$\delta\mathbf 1 + \phi {\large\unicode{x1d7d9}}$$ produces a [[motor]] for which $$\delta/\phi$$ is the pitch of the screw transformation.

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

Eric Lengyel: Created page with "A ''magnitude'' is a quantity that represents a concrete distance of some kind. In rigid geometric algebra, a magnitude $$\mathbf z$$ is composed of two components, a scalar and an antiscalar, as follows: :$$\mathbf z = x\mathbf 1 + y {\large\unicode{x1d7d9}}$$ 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. ===..."

2023-07-15T05:50:33Z

Created page with "A ''magnitude'' is a quantity that represents a concrete distance of some kind. In rigid geometric algebra, a magnitude $$\mathbf z$$ is composed of two components, a scalar and an antiscalar, as follows: :$$\mathbf z = x\mathbf 1 + y {\large\unicode{x1d7d9}}$$ 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. ===..."

New page

A ''magnitude'' is a quantity that represents a concrete distance of some kind. In rigid geometric algebra, a magnitude $$\mathbf z$$ is composed of two components, a scalar and an antiscalar, as follows:

:$$\mathbf z = x\mathbf 1 + y {\large\unicode{x1d7d9}}$$

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.

=== Examples ===

* 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.
* [[Euclidean distances]] between objects are expressed as magnitudes given by the sum of the [[bulk norms]] and [[weight norms]] of [[commutators]].
* Exponentiating the magnitude $$d\mathbf 1 + \phi {\large\unicode{x1d7d9}}$$ produces a [[motor]] for which $$d/\phi$$ is the pitch of the screw transformation.

== See Also ==

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