Geometric norm and Geometric constraint: Difference between pages

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The ''geometric norm'' is a measure of the magnitude of an element. It has two components called the bulk norm and the weight norm.
An element $$\mathbf x$$ of a geometric algebra possesses the ''geometric property'' if and only if the [[geometric product]] between $$\mathbf x$$ and its own reverse is a scalar, which is given by the [[dot product]], and the [[geometric antiproduct]] between $$\mathbf x$$ and its own antireverse is an antiscalar, which is given by the [[antidot product]]. That is,


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.
:$$\mathbf x \mathbin{\unicode{x27D1}} \mathbf{\tilde x} = \mathbf x \mathbin{\unicode{x25CF}} \mathbf x$$


== Bulk Norm ==
and


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:
:$$\mathbf x \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}} = \mathbf x \mathbin{\unicode{x25CB}} \mathbf x$$ .


:$$\left\Vert\mathbf x\right\Vert_\unicode{x25CF} = \sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf x}$$ .
The set of all elements possessing the geometric property is closed under both the [[geometric product]] and [[geometric antiproduct]].


An element that has a bulk norm of '''1''' is said to be ''bulk normalized''.
The following table lists the requirements that must be satisfied for the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ to possess the geometric property. Points and planes do not have any requirements—they all possess the geometric property.
 
The following table lists the bulk norms of the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.


{| class="wikitable"
{| class="wikitable"
! Type !! Definition !! Bulk Norm
! Type !! Definition !! Requirement
|-
|-
| 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;" | $$xy = 0$$
|-
|-
| 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;" | —
|-
|-
| 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;" | $$l_{vx} l_{mx} + l_{vy} l_{my} + l_{vz} l_{mz} = 0$$
|-
|-
| 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;" | —
|-
| 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 itself:
 
:$$\left\Vert\mathbf x\right\Vert_\unicode{x25CB} = \sqrt{\mathbf x \mathbin{\unicode{x25CB}} \mathbf x}$$ .
 
(Note that the square root in this case is taken with respect to the geometric antiproduct.)
 
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 the ''geometric norm'' given by
 
:$$\left\Vert\mathbf x\right\Vert = \left\Vert\mathbf x\right\Vert_\unicode{x25CF} + \left\Vert\mathbf x\right\Vert_\unicode{x25CB} = \sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf x} + \sqrt{\mathbf x \mathbin{\unicode{x25CB}} \mathbf x}$$ .
 
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
 
:$$\widehat{\left\Vert\mathbf x\right\Vert} = \dfrac{\left\Vert\mathbf x\right\Vert}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} = \dfrac{\left\Vert\mathbf x\right\Vert_\unicode{x25CF}}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} + {\large\unicode{x1D7D9}} = \dfrac{\sqrt{\mathbf x \mathbin{\unicode{x25CF}} \mathbf x}}{\sqrt{\mathbf x \mathbin{\unicode{x25CB}} \mathbf x}} + {\large\unicode{x1D7D9}}$$ .
 
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.
 
{| 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;" | [[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;" | $$\widehat{\left\Vert\mathbf Q\right\Vert} = \sqrt{\dfrac{Q_{mx}^2 + Q_{my}^2 + Q_{mz}^2 + Q_{mw}^2}{Q_{vx}^2 + Q_{vy}^2 + Q_{vz}^2 + Q_{vw}^2}}$$
| style="padding: 12px;" | $$Q_{vx} Q_{mx} + Q_{vy} Q_{my} + Q_{vz} Q_{mz} + Q_{vw} Q_{mw} = 0$$
| style="padding: 12px;" | Half the distance that the origin is moved by the [[motor]] $$\mathbf Q$$.
|-
|-
| 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;" | $$\widehat{\left\Vert\mathbf F\right\Vert} = \sqrt{\dfrac{F_{px}^2 + F_{py}^2 + F_{pz}^2 + F_{gw}^2}{F_{gx}^2 + F_{gy}^2 + F_{gz}^2 + F_{pw}^2}}$$
| style="padding: 12px;" | $$F_{px} F_{gx} + F_{py} F_{gy} + F_{pz} F_{gz} + F_{pw} F_{gw} = 0$$
| style="padding: 12px;" | Half the distance that the origin is moved by the [[flector]] $$\mathbf F$$.
|}
|}


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


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

Revision as of 19:22, 25 August 2023

An element $$\mathbf x$$ of a geometric algebra possesses the geometric property if and only if the geometric product between $$\mathbf x$$ and its own reverse is a scalar, which is given by the dot product, and the geometric antiproduct between $$\mathbf x$$ and its own antireverse is an antiscalar, which is given by the antidot product. That is,

$$\mathbf x \mathbin{\unicode{x27D1}} \mathbf{\tilde x} = \mathbf x \mathbin{\unicode{x25CF}} \mathbf x$$

and

$$\mathbf x \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{x}}} = \mathbf x \mathbin{\unicode{x25CB}} \mathbf x$$ .

The set of all elements possessing the geometric property is closed under both the geometric product and geometric antiproduct.

The following table lists the requirements that must be satisfied for the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$ to possess the geometric property. Points and planes do not have any requirements—they all possess the geometric property.

Type Definition Requirement
Magnitude $$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$ $$xy = 0$$
Point $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ —
Line $$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$ $$l_{vx} l_{mx} + l_{vy} l_{my} + l_{vz} l_{mz} = 0$$
Plane $$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$ —
Motor $$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$ $$Q_{vx} Q_{mx} + Q_{vy} Q_{my} + Q_{vz} Q_{mz} + Q_{vw} Q_{mw} = 0$$
Flector $$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$ $$F_{px} F_{gx} + F_{py} F_{gy} + F_{pz} F_{gz} + F_{pw} F_{gw} = 0$$

See Also

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