Unitization - Revision history

Eric Lengyel at 01:25, 8 July 2024

2024-07-08T01:25:57Z

← Older revision		Revision as of 01:25, 8 July 2024
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	An element $$\mathbf u$$ is unitized by calculating		An element $$\mathbf u$$ is unitized by calculating

	:$$\mathbf{\hat u} = \dfrac{\mathbf u}{\left\Vert\mathbf u\right\Vert_\unicode{x25CB}} = \dfrac{\mathbf u}{\sqrt{\mathbf u \mathbin{\unicode{~~x25CB~~}} \mathbf u}}$$ .		:$$\mathbf{\hat u} = \dfrac{\mathbf u}{\left\Vert\mathbf u\right\Vert_\unicode["segoe ui symbol"]{x25CB}} = \dfrac{\mathbf u}{\sqrt{\mathbf u \mathbin{\unicode["segoe ui symbol"]{x2218}} \mathbf u}}$$ .

	In general, an element is unitized when the combined magnitude of all of its components having a factor of $$\mathbf e_4$$ is unity. That is, the components of the element that extend into the projective fourth dimension collectively have a size of one.		In general, an element is unitized when the combined magnitude of all of its components having a factor of $$\mathbf e_4$$ is unity. That is, the components of the element that extend into the projective fourth dimension collectively have a size of one.

Eric Lengyel at 23:33, 13 April 2024

2024-04-13T23:33:26Z

← Older revision		Revision as of 23:33, 13 April 2024
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	\| style="padding: 12px;" \| $$F_{pw}^2 + F_{gx}^2 + F_{gy}^2 + F_{gz}^2 = 1$$		\| style="padding: 12px;" \| $$F_{pw}^2 + F_{gx}^2 + F_{gy}^2 + F_{gz}^2 = 1$$
	\|}		\|}

			== In the Book ==

			* Unitization is discussed in Section 2.10.2.

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

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

	~~== In the Book ==~~

	* Unitization is discussed in Section 2.10.2.

Eric Lengyel at 05:15, 13 April 2024

2024-04-13T05:15:32Z

← Older revision		Revision as of 05:15, 13 April 2024
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	* [[Geometric norm]]		* [[Geometric norm]]

			== In the Book ==

			* Unitization is discussed in Section 2.10.2.

Eric Lengyel at 07:19, 23 October 2023

2023-10-23T07:19:48Z

← Older revision		Revision as of 07:19, 23 October 2023
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	''Unitization'' is the process of scaling an element of a projective geometric algebra so that its [[weight norm]] becomes the [[antiscalar]] $$\large\unicode{x1D7D9}$$. An element that has a weight norm of $$\large\unicode{x1D7D9}$$ is said to be ''unitized''.		''Unitization'' is the process of scaling an element of a projective geometric algebra so that its [[weight norm]] becomes the [[antiscalar]] $$\large\unicode{x1D7D9}$$. An element that has a weight norm of $$\large\unicode{x1D7D9}$$ is said to be ''unitized''.

	An element $$\mathbf x$$ is unitized by calculating		An element $$\mathbf u$$ is unitized by calculating

	:$$\mathbf{\hat x} = \dfrac{\mathbf x}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} = \dfrac{\mathbf x}{\sqrt{\mathbf x \mathbin{\unicode{x25CB}} \mathbf x}}$$ .		:$$\mathbf{\hat u} = \dfrac{\mathbf u}{\left\Vert\mathbf u\right\Vert_\unicode{x25CB}} = \dfrac{\mathbf u}{\sqrt{\mathbf u \mathbin{\unicode{x25CB}} \mathbf u}}$$ .

	In general, an element is unitized when the combined magnitude of all of its components having a factor of $$\mathbf e_4$$ is unity. That is, the components of the element that extend into the projective fourth dimension collectively have a size of one.		In general, an element is unitized when the combined magnitude of all of its components having a factor of $$\mathbf e_4$$ is unity. That is, the components of the element that extend into the projective fourth dimension collectively have a size of one.

Eric Lengyel at 20:18, 26 August 2023

2023-08-26T20:18:34Z

← Older revision		Revision as of 20:18, 26 August 2023
Line 3:		Line 3:
	An element $$\mathbf x$$ is unitized by calculating		An element $$\mathbf x$$ is unitized by calculating

	:$$\mathbf{\hat x} = \dfrac{\mathbf x}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} = \dfrac{\mathbf x}{\sqrt{\mathbf x \mathbin{\unicode{x25CB}} ~~\smash{~~\mathbf~~{\underset{\Large\unicode{x7E}}{~~x~~}}}~~}}$$ .		:$$\mathbf{\hat x} = \dfrac{\mathbf x}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} = \dfrac{\mathbf x}{\sqrt{\mathbf x \mathbin{\unicode{x25CB}} \mathbf x}}$$ .

	In general, an element is unitized when the combined magnitude of all of its components having a factor of $$\mathbf e_4$$ is unity. That is, the components of the element that extend into the projective fourth dimension collectively have a size of one.		In general, an element is unitized when the combined magnitude of all of its components having a factor of $$\mathbf e_4$$ is unity. That is, the components of the element that extend into the projective fourth dimension collectively have a size of one.

Eric Lengyel: Created page with "''Unitization'' is the process of scaling an element of a projective geometric algebra so that its weight norm becomes the antiscalar $$\large\unicode{x1D7D9}$$. An element that has a weight norm of $$\large\unicode{x1D7D9}$$ is said to be ''unitized''. An element $$\mathbf x$$ is unitized by calculating :$$\mathbf{\hat x} = \dfrac{\mathbf x}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} = \dfrac{\mathbf x}{\sqrt{\mathbf x \mathbin{\unicode{x25CB}} \smash{\ma..."

2023-07-15T06:03:19Z

Created page with "''Unitization'' is the process of scaling an element of a projective geometric algebra so that its weight norm becomes the antiscalar $$\large\unicode{x1D7D9}$$. An element that has a weight norm of $$\large\unicode{x1D7D9}$$ is said to be ''unitized''. An element $$\mathbf x$$ is unitized by calculating :$$\mathbf{\hat x} = \dfrac{\mathbf x}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} = \dfrac{\mathbf x}{\sqrt{\mathbf x \mathbin{\unicode{x25CB}} \smash{\ma..."

New page

''Unitization'' is the process of scaling an element of a projective geometric algebra so that its [[weight norm]] becomes the [[antiscalar]] $$\large\unicode{x1D7D9}$$. An element that has a weight norm of $$\large\unicode{x1D7D9}$$ is said to be ''unitized''.

An element $$\mathbf x$$ is unitized by calculating

:$$\mathbf{\hat x} = \dfrac{\mathbf x}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} = \dfrac{\mathbf x}{\sqrt{\mathbf x \mathbin{\unicode{x25CB}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}}}}$$ .

In general, an element is unitized when the combined magnitude of all of its components having a factor of $$\mathbf e_4$$ is unity. That is, the components of the element that extend into the projective fourth dimension collectively have a size of one.

The following table lists the unitization conditions for the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.

{| class="wikitable"
! Type !! Definition !! Unitization
|-
| style="padding: 12px;" | [[Magnitude]]
| style="padding: 12px;" | $$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$
| style="padding: 12px;" | $$y^2 = 1$$
|-
| 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;" | $$p_w^2 = 1$$
|-
| 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;" | $$l_{vx}^2 + l_{vy}^2 + l_{vz}^2 = 1$$
|-
| 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;" | $$g_x^2 + g_y^2 + g_z^2 = 1$$
|-
| 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;" | $$Q_{vx}^2 + Q_{vy}^2 + Q_{vz}^2 + Q_{vw}^2 = 1$$
|-
| 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;" | $$F_{pw}^2 + F_{gx}^2 + F_{gy}^2 + F_{gz}^2 = 1$$
|}

== See Also ==

* [[Geometric norm]]