Geometric norm - Revision history

Eric Lengyel at 01:24, 8 July 2024

2024-07-08T01:24:58Z

Eric Lengyel at 23:32, 13 April 2024

2024-04-13T23:32:58Z

← Older revision		Revision as of 23:32, 13 April 2024
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	\| style="padding: 12px;" \| Half the distance that the origin is moved by the [[flector]] $$\mathbf F$$.		\| style="padding: 12px;" \| Half the distance that the origin is moved by the [[flector]] $$\mathbf F$$.
	\|}		\|}

			== In the Book ==

			* The geometric norm is discussed in Section 2.10.

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

	* [[Geometric constraint]]		* [[Geometric constraint]]

	~~== In the Book ==~~

	* The geometric norm is discussed in Section 2.10.

Eric Lengyel at 05:14, 13 April 2024

2024-04-13T05:14:11Z

← Older revision		Revision as of 05:14, 13 April 2024
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	* [[Geometric constraint]]		* [[Geometric constraint]]

			== In the Book ==

			* The geometric norm is discussed in Section 2.10.

Eric Lengyel: /* See Also */

2024-02-09T01:00:58Z

Eric Lengyel at 07:23, 23 October 2023

2023-10-23T07:23:09Z

← Older revision		Revision as of 07:23, 23 October 2023
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	== Bulk Norm ==		== Bulk Norm ==

	The ''bulk norm'' of an element $$\mathbf x$$, denoted $$\left\Vert\mathbf x\right\Vert_\unicode{x25CF}$$, is the magnitude of its [[bulk]] components. It can be calculated by taking the square root of the [[dot product]] of $$\mathbf x$$ with itself:		The ''bulk norm'' of an element $$\mathbf u$$, denoted $$\left\Vert\mathbf u\right\Vert_\unicode{x25CF}$$, is the magnitude of its [[bulk]] components. It can be calculated by taking the square root of the [[dot product]] of $$\mathbf u$$ with itself:

	:$$\left\Vert\mathbf x\right\Vert_\unicode{x25CF} = \sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf x}$$ .		:$$\left\Vert\mathbf u\right\Vert_\unicode{x25CF} = \sqrt{\mathbf u \mathbin{\unicode{x25CF}} \mathbf u}$$ .

	An element that has a bulk norm of '''1''' is said to be ''bulk normalized''.		An element that has a bulk norm of '''1''' is said to be ''bulk normalized''.
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	== Weight Norm ==		== Weight Norm ==

	The ''weight norm'' of an element $$\mathbf x$$, denoted $$\left\Vert\mathbf x\right\Vert_\unicode{x25CB}$$, is the magnitude of its [[weight]] components. It can be calculated by taking the square root of the [[antidot product]] of $$\mathbf x$$ with itself:		The ''weight norm'' of an element $$\mathbf u$$, denoted $$\left\Vert\mathbf u\right\Vert_\unicode{x25CB}$$, is the magnitude of its [[weight]] components. It can be calculated by taking the square root of the [[antidot product]] of $$\mathbf u$$ with itself:

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

	(Note that the square root in this case is taken with respect to the geometric antiproduct.)		(Note that the square root in this case is taken with respect to the geometric antiproduct.)
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	The bulk norm and weight norm are summed to construct the ''geometric norm'' given by		The bulk norm and weight norm are summed to construct the ''geometric norm'' given by

	:$$\left\Vert\mathbf x\right\Vert = \left\Vert\mathbf x\right\Vert_\unicode{x25CF} + \left\Vert\mathbf x\right\Vert_\unicode{x25CB} = \sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf x} + \sqrt{\mathbf x \mathbin{\unicode{x25CB}} \mathbf x}$$ .		:$$\left\Vert\mathbf u\right\Vert = \left\Vert\mathbf u\right\Vert_\unicode{x25CF} + \left\Vert\mathbf u\right\Vert_\unicode{x25CB} = \sqrt{\mathbf u \mathbin{\unicode{x25CF}} \mathbf u} + \sqrt{\mathbf u \mathbin{\unicode{x25CB}} \mathbf u}$$ .

	This quantity is the sum of a scalar $$a\mathbf 1$$ and antiscalar $$b{\large\unicode{x1D7D9}}$$ representing a ''homogeneous magnitude'' that itself has a bulk and a weight. Its bulk norm is simply the magnitude of its scalar part, and its weight norm is simply the magnitude of its antiscalar part. The geometric norm is idempotent because		This quantity is the sum of a scalar $$a\mathbf 1$$ and antiscalar $$b{\large\unicode{x1D7D9}}$$ representing a ''homogeneous magnitude'' that itself has a bulk and a weight. Its bulk norm is simply the magnitude of its scalar part, and its weight norm is simply the magnitude of its antiscalar part. The geometric norm is idempotent because
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	:$$\left\Vert a\mathbf 1 + b{\large\unicode{x1D7D9}}\right\Vert = \|a\|\mathbf 1 + \|b\|{\large\unicode{x1D7D9}}$$ .		:$$\left\Vert a\mathbf 1 + b{\large\unicode{x1D7D9}}\right\Vert = \|a\|\mathbf 1 + \|b\|{\large\unicode{x1D7D9}}$$ .

	Like all other homogeneous quantities, the magnitude given by the geometric norm is [[unitized]] by dividing by its weight norm. The unitized magnitude of an element $$\mathbf x$$ is given by		Like all other homogeneous quantities, the magnitude given by the geometric norm is [[unitized]] by dividing by its weight norm. The unitized magnitude of an element $$\mathbf u$$ is given by

	:$$\widehat{\left\Vert\mathbf x\right\Vert} = \dfrac{\left\Vert\mathbf x\right\Vert}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} = \dfrac{\left\Vert\mathbf x\right\Vert_\unicode{x25CF}}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} + {\large\unicode{x1D7D9}} = \dfrac{\sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf x}}{\sqrt{\mathbf x \mathbin{\unicode{x25CB}} \mathbf x}} + {\large\unicode{x1D7D9}}$$ .		:$$\widehat{\left\Vert\mathbf u\right\Vert} = \dfrac{\left\Vert\mathbf u\right\Vert}{\left\Vert\mathbf u\right\Vert_\unicode{x25CB}} = \dfrac{\left\Vert\mathbf u\right\Vert_\unicode{x25CF}}{\left\Vert\mathbf u\right\Vert_\unicode{x25CB}} + {\large\unicode{x1D7D9}} = \dfrac{\sqrt{\mathbf u \mathbin{\unicode{x25CF}} \mathbf u}}{\sqrt{\mathbf u \mathbin{\unicode{x25CB}} \mathbf u}} + {\large\unicode{x1D7D9}}$$ .

	The following table lists the unitized geometric norms of the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ after dropping the constant $${\large\unicode{x1D7D9}}$$ term.		The following table lists the unitized geometric norms of the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ after dropping the constant $${\large\unicode{x1D7D9}}$$ term.

Eric Lengyel at 19:20, 25 August 2023

2023-08-25T19:20:26Z

Eric Lengyel: Created page with "The ''geometric norm'' is a measure of the magnitude of an element. It has two components called the bulk norm and the weight norm. For points, lines, and planes, the geometric norm is equal to the shortest Euclidean distance between the geometry and the origin. For motors and flectors, the geometric norm is equal to half the distance that the origin is moved by the isometry operator. == Bulk Norm == The ''bulk norm'' of an element $$\mathbf x$$, d..."

2023-07-15T06:02:43Z

Created page with "The ''geometric norm'' is a measure of the magnitude of an element. It has two components called the bulk norm and the weight norm. For points, lines, and planes, the geometric norm is equal to the shortest Euclidean distance between the geometry and the origin. For motors and flectors, the geometric norm is equal to half the distance that the origin is moved by the isometry operator. == Bulk Norm == The ''bulk norm'' of an element $$\mathbf x$$, d..."

New page

The ''geometric norm'' is a measure of the magnitude of an element. It has two components called the bulk norm and the weight norm.

For [[points]], [[lines]], and [[planes]], the geometric norm is equal to the shortest Euclidean distance between the geometry and the origin. For [[motors]] and [[flectors]], the geometric norm is equal to half the distance that the origin is moved by the isometry operator.

== Bulk Norm ==

The ''bulk norm'' of an element $$\mathbf x$$, denoted $$\left\Vert\mathbf x\right\Vert_\unicode{x25CF}$$, is the magnitude of its [[bulk]] components. It can be calculated by taking the square root of the [[dot product]] of $$\mathbf x$$ with its own [[reverse]]:

:$$\left\Vert\mathbf x\right\Vert_\unicode{x25CF} = \sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf{\tilde x}}$$ .

An element that has a bulk norm of '''1''' is said to be ''bulk normalized''.

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

{| class="wikitable"
! Type !! Definition !! Bulk Norm
|-
| style="padding: 12px;" | [[Magnitude]]
| style="padding: 12px;" | $$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$
| style="padding: 12px;" | $$\left\Vert\mathbf z\right\Vert_\unicode{x25CF} = |x|$$
|-
| 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;" | $$\left\Vert\mathbf p\right\Vert_\unicode{x25CF} = \sqrt{p_x^2 + p_y^2 + p_z^2}$$
|-
| 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;" | $$\left\Vert\boldsymbol l\right\Vert_\unicode{x25CF} = \sqrt{l_{mx}^2 + l_{my}^2 + l_{mz}^2}$$
|-
| 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;" | $$\left\Vert\mathbf g\right\Vert_\unicode{x25CF} = |g_w|$$
|-
| 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;" | $$\left\Vert\mathbf Q\right\Vert_\unicode{x25CF} = \sqrt{Q_{mx}^2 + Q_{my}^2 + Q_{mz}^2 + Q_{mw}^2}$$
|-
| 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;" | $$\left\Vert\mathbf F\right\Vert_\unicode{x25CF} = \sqrt{F_{px}^2 + F_{py}^2 + F_{pz}^2 + F_{gw}^2}$$
|}

== Weight Norm ==

The ''weight norm'' of an element $$\mathbf x$$, denoted $$\left\Vert\mathbf x\right\Vert_\unicode{x25CB}$$, is the magnitude of its [[weight]] components. It can be calculated by taking the square root of the [[antidot product]] of $$\mathbf x$$ with its own [[antireverse]]:

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

(Note that the square root in this case is taken with respect to the geometric antiproduct.)

An element that has a weight norm of $$\large\unicode{x1D7D9}$$ is said to be ''weight normalized'' or ''[[unitized]]''.

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

{| class="wikitable"
! Type !! Definition !! Weight Norm
|-
| style="padding: 12px;" | [[Magnitude]]
| style="padding: 12px;" | $$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$
| style="padding: 12px;" | $$\left\Vert\mathbf z\right\Vert_\unicode{x25CB} = |y|{\large\unicode{x1D7D9}}$$
|-
| 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;" | $$\left\Vert\mathbf p\right\Vert_\unicode{x25CB} = |p_w|{\large\unicode{x1D7D9}}$$
|-
| 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;" | $$\left\Vert\boldsymbol l\right\Vert_\unicode{x25CB} = {\large\unicode{x1D7D9}}\sqrt{l_{vx}^2 + l_{vy}^2 + l_{vz}^2}$$
|-
| style="padding: 12px;" | [[Plane]]
| style="padding: 12px;" | $$\mathbf g = g_x \mathbf e_{234} + g_y \mathbf e_{314} + g_z \mathbf e_{124} + g_w \mathbf e_{321}$$
| style="padding: 12px;" | $$\left\Vert\mathbf g\right\Vert_\unicode{x25CB} = {\large\unicode{x1D7D9}}\sqrt{g_x^2 + g_y^2 + g_z^2}$$
|-
| 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;" | $$\left\Vert\mathbf Q\right\Vert_\unicode{x25CB} = {\large\unicode{x1D7D9}}\sqrt{Q_{vx}^2 + Q_{vy}^2 + Q_{vz}^2 + Q_{vw}^2}$$
|-
| 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;" | $$\left\Vert\mathbf F\right\Vert_\unicode{x25CB} = {\large\unicode{x1D7D9}}\sqrt{F_{gx}^2 + F_{gy}^2 + F_{gz}^2 + F_{pw}^2}$$
|}

== Geometric Norm ==

The bulk norm and weight norm are summed to construct a ''geometric norm'' given by

:$$\left\Vert\mathbf x\right\Vert = \left\Vert\mathbf x\right\Vert_\unicode{x25CF} + \left\Vert\mathbf x\right\Vert_\unicode{x25CB} = \sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf{\tilde x}} + \sqrt{\mathbf x \mathbin{\unicode{x25CB}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}}\vphantom{\mathbf{\tilde x}}}$$ .

This quantity is the sum of a scalar $$a\mathbf 1$$ and antiscalar $$b{\large\unicode{x1D7D9}}$$ representing a ''homogeneous magnitude'' that itself has a bulk and a weight. Its bulk norm is simply the magnitude of its scalar part, and its weight norm is simply the magnitude of its antiscalar part. The geometric norm is idempotent because

:$$\left\Vert a\mathbf 1 + b{\large\unicode{x1D7D9}}\right\Vert = |a|\mathbf 1 + |b|{\large\unicode{x1D7D9}}$$ .

Like all other homogeneous quantities, the magnitude given by the geometric norm is [[unitized]] by dividing by its weight norm. The unitized magnitude of an element $$\mathbf x$$ is given by

:$$\widehat{\left\Vert\mathbf x\right\Vert} = \dfrac{\left\Vert\mathbf x\right\Vert}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} = \dfrac{\left\Vert\mathbf x\right\Vert_\unicode{x25CF}}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} + {\large\unicode{x1D7D9}} = \dfrac{\sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf{\tilde x}}}{\sqrt{\mathbf x \mathbin{\unicode{x25CB}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}}\vphantom{\mathbf{\tilde x}}}} + {\large\unicode{x1D7D9}}$$ .

The following table lists the unitized geometric norms of the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ after dropping the constant $${\large\unicode{x1D7D9}}$$ term.

{| class="wikitable"
! Type !! Definition !! Geometric Norm !! Interpretation
|-
| style="padding: 12px;" | [[Magnitude]]
| style="padding: 12px;" | $$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$
| style="padding: 12px;" | $$\widehat{\left\Vert\mathbf z\right\Vert} = \dfrac{|x|}{|y|}$$
| style="padding: 12px;" | A Euclidean distance.
|-
| 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;" | $$\widehat{\left\Vert\mathbf p\right\Vert} = \dfrac{\sqrt{p_x^2 + p_y^2 + p_z^2}}{|p_w|}$$
| style="padding: 12px;" | Distance from the origin to the point $$\mathbf p$$.

Half the distance that the origin is moved by the [[flector]] $$\mathbf p$$.
|-
| 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;" | $$\widehat{\left\Vert\boldsymbol l\right\Vert} = \sqrt{\dfrac{l_{mx}^2 + l_{my}^2 + l_{mz}^2}{l_{vx}^2 + l_{vy}^2 + l_{vz}^2}}$$
| style="padding: 12px;" | Perpendicular distance from the origin to the line $$\boldsymbol l$$.

Half the distance that the origin is moved by the [[motor]] $$\boldsymbol l$$.
|-
| 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;" | $$\widehat{\left\Vert\mathbf g\right\Vert} = \dfrac{|g_w|}{\sqrt{g_x^2 + g_y^2 + g_z^2}}$$
| style="padding: 12px;" | Perpendicular distance from the origin to the plane $$\mathbf g$$.

Half the distance that the origin is moved by the [[flector]] $$\mathbf g$$.
|-
| 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;" | $$\widehat{\left\Vert\mathbf Q\right\Vert} = \sqrt{\dfrac{Q_{mx}^2 + Q_{my}^2 + Q_{mz}^2 + Q_{mw}^2}{Q_{vx}^2 + Q_{vy}^2 + Q_{vz}^2 + Q_{vw}^2}}$$
| style="padding: 12px;" | Half the distance that the origin is moved by the [[motor]] $$\mathbf Q$$.
|-
| 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;" | $$\widehat{\left\Vert\mathbf F\right\Vert} = \sqrt{\dfrac{F_{px}^2 + F_{py}^2 + F_{pz}^2 + F_{gw}^2}{F_{gx}^2 + F_{gy}^2 + F_{gz}^2 + F_{pw}^2}}$$
| style="padding: 12px;" | Half the distance that the origin is moved by the [[flector]] $$\mathbf F$$.
|}

== See Also ==

* [[Geometric property]]

← Older revision		Revision as of 01:24, 8 July 2024
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	== Bulk Norm ==		== Bulk Norm ==

	The ''bulk norm'' of an element $$\mathbf u$$, denoted $$\left\Vert\mathbf u\right\Vert_\unicode{x25CF}$$, is the magnitude of its [[bulk]] components. It can be calculated by taking the square root of the [[dot product]] of $$\mathbf u$$ with itself:		The ''bulk norm'' of an element $$\mathbf u$$, denoted $$\left\Vert\mathbf u\right\Vert_\unicode["segoe ui symbol"]{x25CF}$$, is the magnitude of its [[bulk]] components. It can be calculated by taking the square root of the [[dot product]] of $$\mathbf u$$ with itself:

	:$$\left\Vert\mathbf u\right\Vert_\unicode{x25CF} = \sqrt{\mathbf u \mathbin{\unicode{~~x25CF~~}} \mathbf u}$$ .		:$$\left\Vert\mathbf u\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \sqrt{\mathbf u \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf u}$$ .

	An element that has a bulk norm of '''1''' is said to be ''bulk normalized''.		An element that has a bulk norm of '''1''' is said to be ''bulk normalized''.
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	\| style="padding: 12px;" \| [[Magnitude]]		\| style="padding: 12px;" \| [[Magnitude]]
	\| style="padding: 12px;" \| $$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$		\| style="padding: 12px;" \| $$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$
	\| style="padding: 12px;" \| $$\left\Vert\mathbf z\right\Vert_\unicode{x25CF} = \|x\|$$		\| style="padding: 12px;" \| $$\left\Vert\mathbf z\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \|x\|$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Point]]		\| 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;" \| $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$
	\| style="padding: 12px;" \| $$\left\Vert\mathbf p\right\Vert_\unicode{x25CF} = \sqrt{p_x^2 + p_y^2 + p_z^2}$$		\| style="padding: 12px;" \| $$\left\Vert\mathbf p\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \sqrt{p_x^2 + p_y^2 + p_z^2}$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Line]]		\| 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;" \| $$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$
	\| style="padding: 12px;" \| $$\left\Vert\boldsymbol l\right\Vert_\unicode{x25CF} = \sqrt{l_{mx}^2 + l_{my}^2 + l_{mz}^2}$$		\| style="padding: 12px;" \| $$\left\Vert\boldsymbol l\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \sqrt{l_{mx}^2 + l_{my}^2 + l_{mz}^2}$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Plane]]		\| 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;" \| $$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$
	\| style="padding: 12px;" \| $$\left\Vert\mathbf g\right\Vert_\unicode{x25CF} = \|g_w\|$$		\| style="padding: 12px;" \| $$\left\Vert\mathbf g\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \|g_w\|$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Motor]]		\| 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;" \| $$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$
	\| style="padding: 12px;" \| $$\left\Vert\mathbf Q\right\Vert_\unicode{x25CF} = \sqrt{Q_{mx}^2 + Q_{my}^2 + Q_{mz}^2 + Q_{mw}^2}$$		\| style="padding: 12px;" \| $$\left\Vert\mathbf Q\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \sqrt{Q_{mx}^2 + Q_{my}^2 + Q_{mz}^2 + Q_{mw}^2}$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Flector]]		\| 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;" \| $$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$
	\| style="padding: 12px;" \| $$\left\Vert\mathbf F\right\Vert_\unicode{x25CF} = \sqrt{F_{px}^2 + F_{py}^2 + F_{pz}^2 + F_{gw}^2}$$		\| style="padding: 12px;" \| $$\left\Vert\mathbf F\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \sqrt{F_{px}^2 + F_{py}^2 + F_{pz}^2 + F_{gw}^2}$$
	\|}		\|}

	== Weight Norm ==		== Weight Norm ==

	The ''weight norm'' of an element $$\mathbf u$$, denoted $$\left\Vert\mathbf u\right\Vert_\unicode{x25CB}$$, is the magnitude of its [[weight]] components. It can be calculated by taking the square root of the [[antidot product]] of $$\mathbf u$$ with itself:		The ''weight norm'' of an element $$\mathbf u$$, denoted $$\left\Vert\mathbf u\right\Vert_\unicode["segoe ui symbol"]{x25CB}$$, is the magnitude of its [[weight]] components. It can be calculated by taking the square root of the [[antidot product]] of $$\mathbf u$$ with itself:

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

	(Note that the square root in this case is taken with respect to the geometric antiproduct.)		(Note that the square root in this case is taken with respect to the geometric antiproduct.)
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	\| style="padding: 12px;" \| [[Magnitude]]		\| style="padding: 12px;" \| [[Magnitude]]
	\| style="padding: 12px;" \| $$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$		\| style="padding: 12px;" \| $$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$
	\| style="padding: 12px;" \| $$\left\Vert\mathbf z\right\Vert_\unicode{x25CB} = \|y\|{\large\unicode{x1D7D9}}$$		\| style="padding: 12px;" \| $$\left\Vert\mathbf z\right\Vert_\unicode["segoe ui symbol"]{x25CB} = \|y\|{\large\unicode{x1D7D9}}$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Point]]		\| 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;" \| $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$
	\| style="padding: 12px;" \| $$\left\Vert\mathbf p\right\Vert_\unicode{x25CB} = \|p_w\|{\large\unicode{x1D7D9}}$$		\| style="padding: 12px;" \| $$\left\Vert\mathbf p\right\Vert_\unicode["segoe ui symbol"]{x25CB} = \|p_w\|{\large\unicode{x1D7D9}}$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Line]]		\| 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;" \| $$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$
	\| style="padding: 12px;" \| $$\left\Vert\boldsymbol l\right\Vert_\unicode{x25CB} = {\large\unicode{x1D7D9}}\sqrt{l_{vx}^2 + l_{vy}^2 + l_{vz}^2}$$		\| style="padding: 12px;" \| $$\left\Vert\boldsymbol l\right\Vert_\unicode["segoe ui symbol"]{x25CB} = {\large\unicode{x1D7D9}}\sqrt{l_{vx}^2 + l_{vy}^2 + l_{vz}^2}$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Plane]]		\| style="padding: 12px;" \| [[Plane]]
	\| style="padding: 12px;" \| $$\mathbf g = g_x \mathbf e_{234} + g_y \mathbf e_{314} + g_z \mathbf e_{124} + g_w \mathbf e_{321}$$		\| style="padding: 12px;" \| $$\mathbf g = g_x \mathbf e_{234} + g_y \mathbf e_{314} + g_z \mathbf e_{124} + g_w \mathbf e_{321}$$
	\| style="padding: 12px;" \| $$\left\Vert\mathbf g\right\Vert_\unicode{x25CB} = {\large\unicode{x1D7D9}}\sqrt{g_x^2 + g_y^2 + g_z^2}$$		\| style="padding: 12px;" \| $$\left\Vert\mathbf g\right\Vert_\unicode["segoe ui symbol"]{x25CB} = {\large\unicode{x1D7D9}}\sqrt{g_x^2 + g_y^2 + g_z^2}$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Motor]]		\| 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;" \| $$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$
	\| style="padding: 12px;" \| $$\left\Vert\mathbf Q\right\Vert_\unicode{x25CB} = {\large\unicode{x1D7D9}}\sqrt{Q_{vx}^2 + Q_{vy}^2 + Q_{vz}^2 + Q_{vw}^2}$$		\| style="padding: 12px;" \| $$\left\Vert\mathbf Q\right\Vert_\unicode["segoe ui symbol"]{x25CB} = {\large\unicode{x1D7D9}}\sqrt{Q_{vx}^2 + Q_{vy}^2 + Q_{vz}^2 + Q_{vw}^2}$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Flector]]		\| 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;" \| $$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$
	\| style="padding: 12px;" \| $$\left\Vert\mathbf F\right\Vert_\unicode{x25CB} = {\large\unicode{x1D7D9}}\sqrt{F_{gx}^2 + F_{gy}^2 + F_{gz}^2 + F_{pw}^2}$$		\| style="padding: 12px;" \| $$\left\Vert\mathbf F\right\Vert_\unicode["segoe ui symbol"]{x25CB} = {\large\unicode{x1D7D9}}\sqrt{F_{gx}^2 + F_{gy}^2 + F_{gz}^2 + F_{pw}^2}$$
	\|}		\|}

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	The bulk norm and weight norm are summed to construct the ''geometric norm'' given by		The bulk norm and weight norm are summed to construct the ''geometric norm'' given by

	:$$\left\Vert\mathbf u\right\Vert = \left\Vert\mathbf u\right\Vert_\unicode{x25CF} + \left\Vert\mathbf u\right\Vert_\unicode{x25CB} = \sqrt{\mathbf u \mathbin{\unicode{~~x25CF~~}} \mathbf u} + \sqrt{\mathbf u \mathbin{\unicode{~~x25CB~~}} \mathbf u}$$ .		:$$\left\Vert\mathbf u\right\Vert = \left\Vert\mathbf u\right\Vert_\unicode["segoe ui symbol"]{x25CF} + \left\Vert\mathbf u\right\Vert_\unicode["segoe ui symbol"]{x25CB} = \sqrt{\mathbf u \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf u} + \sqrt{\mathbf u \mathbin{\unicode["segoe ui symbol"]{x2218}} \mathbf u}$$ .

	This quantity is the sum of a scalar $$a\mathbf 1$$ and antiscalar $$b{\large\unicode{x1D7D9}}$$ representing a ''homogeneous magnitude'' that itself has a bulk and a weight. Its bulk norm is simply the magnitude of its scalar part, and its weight norm is simply the magnitude of its antiscalar part. The geometric norm is idempotent because		This quantity is the sum of a scalar $$a\mathbf 1$$ and antiscalar $$b{\large\unicode{x1D7D9}}$$ representing a ''homogeneous magnitude'' that itself has a bulk and a weight. Its bulk norm is simply the magnitude of its scalar part, and its weight norm is simply the magnitude of its antiscalar part. The geometric norm is idempotent because
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	Like all other homogeneous quantities, the magnitude given by the geometric norm is [[unitized]] by dividing by its weight norm. The unitized magnitude of an element $$\mathbf u$$ is given by		Like all other homogeneous quantities, the magnitude given by the geometric norm is [[unitized]] by dividing by its weight norm. The unitized magnitude of an element $$\mathbf u$$ is given by

	:$$\widehat{\left\Vert\mathbf u\right\Vert} = \dfrac{\left\Vert\mathbf u\right\Vert}{\left\Vert\mathbf u\right\Vert_\unicode{x25CB}} = \dfrac{\left\Vert\mathbf u\right\Vert_\unicode{x25CF}}{\left\Vert\mathbf u\right\Vert_\unicode{x25CB}} + {\large\unicode{x1D7D9}} = \dfrac{\sqrt{\mathbf u \mathbin{\unicode{~~x25CF~~}} \mathbf u}}{\sqrt{\mathbf u \mathbin{\unicode{~~x25CB~~}} \mathbf u}} + {\large\unicode{x1D7D9}}$$ .		:$$\widehat{\left\Vert\mathbf u\right\Vert} = \dfrac{\left\Vert\mathbf u\right\Vert}{\left\Vert\mathbf u\right\Vert_\unicode["segoe ui symbol"]{x25CB}} = \dfrac{\left\Vert\mathbf u\right\Vert_\unicode["segoe ui symbol"]{x25CF}}{\left\Vert\mathbf u\right\Vert_\unicode["segoe ui symbol"]{x25CB}} + {\large\unicode{x1D7D9}} = \dfrac{\sqrt{\mathbf u \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf u}}{\sqrt{\mathbf u \mathbin{\unicode["segoe ui symbol"]{x2218}} \mathbf u}} + {\large\unicode{x1D7D9}}$$ .

	The following table lists the unitized geometric norms of the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ after dropping the constant $${\large\unicode{x1D7D9}}$$ term.		The following table lists the unitized geometric norms of the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ after dropping the constant $${\large\unicode{x1D7D9}}$$ term.

← Older revision		Revision as of 01:00, 9 February 2024
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	== See Also ==		== See Also ==

	* [[Geometric ~~property~~]]		* [[Geometric constraint]]

← Older revision		Revision as of 19:20, 25 August 2023
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	== Bulk Norm ==		== Bulk Norm ==

	The ''bulk norm'' of an element $$\mathbf x$$, denoted $$\left\Vert\mathbf x\right\Vert_\unicode{x25CF}$$, is the magnitude of its [[bulk]] components. It can be calculated by taking the square root of the [[dot product]] of $$\mathbf x$$ with ~~its own [[reverse]]~~:		The ''bulk norm'' of an element $$\mathbf x$$, denoted $$\left\Vert\mathbf x\right\Vert_\unicode{x25CF}$$, is the magnitude of its [[bulk]] components. It can be calculated by taking the square root of the [[dot product]] of $$\mathbf x$$ with itself:

	:$$\left\Vert\mathbf x\right\Vert_\unicode{x25CF} = \sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf~~{\tilde~~ x}}$$ .		:$$\left\Vert\mathbf x\right\Vert_\unicode{x25CF} = \sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf x}$$ .

	An element that has a bulk norm of '''1''' is said to be ''bulk normalized''.		An element that has a bulk norm of '''1''' is said to be ''bulk normalized''.
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	== Weight Norm ==		== Weight Norm ==

	The ''weight norm'' of an element $$\mathbf x$$, denoted $$\left\Vert\mathbf x\right\Vert_\unicode{x25CB}$$, is the magnitude of its [[weight]] components. It can be calculated by taking the square root of the [[antidot product]] of $$\mathbf x$$ with ~~its own [[antireverse]]~~:		The ''weight norm'' of an element $$\mathbf x$$, denoted $$\left\Vert\mathbf x\right\Vert_\unicode{x25CB}$$, is the magnitude of its [[weight]] components. It can be calculated by taking the square root of the [[antidot product]] of $$\mathbf x$$ with itself:

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

	(Note that the square root in this case is taken with respect to the geometric antiproduct.)		(Note that the square root in this case is taken with respect to the geometric antiproduct.)
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	== Geometric Norm ==		== Geometric Norm ==

	The bulk norm and weight norm are summed to construct a ''geometric norm'' given by		The bulk norm and weight norm are summed to construct the ''geometric norm'' given by

	:$$\left\Vert\mathbf x\right\Vert = \left\Vert\mathbf x\right\Vert_\unicode{x25CF} + \left\Vert\mathbf x\right\Vert_\unicode{x25CB} = \sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf~~{\tilde~~ x}} + \sqrt{\mathbf x \mathbin{\unicode{x25CB}} ~~\smash{~~\mathbf~~{\underset{\Large\unicode{x7E}}{~~x~~}}}\vphantom{\mathbf{\tilde x}}~~}$$ .		:$$\left\Vert\mathbf x\right\Vert = \left\Vert\mathbf x\right\Vert_\unicode{x25CF} + \left\Vert\mathbf x\right\Vert_\unicode{x25CB} = \sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf x} + \sqrt{\mathbf x \mathbin{\unicode{x25CB}} \mathbf x}$$ .

	This quantity is the sum of a scalar $$a\mathbf 1$$ and antiscalar $$b{\large\unicode{x1D7D9}}$$ representing a ''homogeneous magnitude'' that itself has a bulk and a weight. Its bulk norm is simply the magnitude of its scalar part, and its weight norm is simply the magnitude of its antiscalar part. The geometric norm is idempotent because		This quantity is the sum of a scalar $$a\mathbf 1$$ and antiscalar $$b{\large\unicode{x1D7D9}}$$ representing a ''homogeneous magnitude'' that itself has a bulk and a weight. Its bulk norm is simply the magnitude of its scalar part, and its weight norm is simply the magnitude of its antiscalar part. The geometric norm is idempotent because
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	Like all other homogeneous quantities, the magnitude given by the geometric norm is [[unitized]] by dividing by its weight norm. The unitized magnitude of an element $$\mathbf x$$ is given by		Like all other homogeneous quantities, the magnitude given by the geometric norm is [[unitized]] by dividing by its weight norm. The unitized magnitude of an element $$\mathbf x$$ is given by

	:$$\widehat{\left\Vert\mathbf x\right\Vert} = \dfrac{\left\Vert\mathbf x\right\Vert}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} = \dfrac{\left\Vert\mathbf x\right\Vert_\unicode{x25CF}}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} + {\large\unicode{x1D7D9}} = \dfrac{\sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf~~{\tilde~~ x}}}{\sqrt{\mathbf x \mathbin{\unicode{x25CB}} ~~\smash{~~\mathbf~~{\underset{\Large\unicode{x7E}}{~~x~~}}}\vphantom{\mathbf{\tilde x}}~~}} + {\large\unicode{x1D7D9}}$$ .		:$$\widehat{\left\Vert\mathbf x\right\Vert} = \dfrac{\left\Vert\mathbf x\right\Vert}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} = \dfrac{\left\Vert\mathbf x\right\Vert_\unicode{x25CF}}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} + {\large\unicode{x1D7D9}} = \dfrac{\sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf x}}{\sqrt{\mathbf x \mathbin{\unicode{x25CB}} \mathbf x}} + {\large\unicode{x1D7D9}}$$ .

	The following table lists the unitized geometric norms of the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ after dropping the constant $${\large\unicode{x1D7D9}}$$ term.		The following table lists the unitized geometric norms of the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ after dropping the constant $${\large\unicode{x1D7D9}}$$ term.