Magnitude: Difference between revisions
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Eric Lengyel (talk | contribs) (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. ===...") |
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A ''magnitude'' is a quantity that represents a concrete | 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: | ||
:$$\mathbf z = x\mathbf 1 + y {\large\unicode{x1d7d9}}$$ | :$$\mathbf z = x\mathbf 1 + y {\large\unicode{x1d7d9}}$$ | ||
| Line 8: | Line 8: | ||
* 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 | * [[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 $$ | * Exponentiating the magnitude $$\delta\mathbf 1 + \phi {\large\unicode{x1d7d9}}$$ produces a [[motor]] for which $$\delta/\phi$$ is the pitch of the screw transformation. | ||
== See Also == | == See Also == | ||
Latest revision as of 08:12, 25 November 2023
A magnitude is a quantity that represents a concrete measurement of some kind. In rigid geometric algebra, a magnitude $$\mathbf z$$ is composed of two components, a scalar and an antiscalar, as follows:
- $$\mathbf z = x\mathbf 1 + y {\large\unicode{x1d7d9}}$$
Magnitudes are homogeneous just like everything else in a projective geometric algebra. This means it has both a bulk and a weight, and it is unitized by making the magnitude of its weight one.
Examples
- The geometric norm produces a magnitude that gives the perpendicular distance between an object and the origin. This is also half the distance that the origin is moved by an object used as an operator.
- Euclidean distances between objects are expressed as magnitudes given by the sum of the bulk norm and weight norm of expressions involving attitudes.
- Exponentiating the magnitude $$\delta\mathbf 1 + \phi {\large\unicode{x1d7d9}}$$ produces a motor for which $$\delta/\phi$$ is the pitch of the screw transformation.