Difference between revisions of "Plane"

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[[Image:plane.svg|400px|thumb|right|'''Figure 1.''' A plane is the intersection of a 4D trivector with the 3D subspace where $$w = 1$$.]]
 
In the 4D projective geometric algebra $$\mathcal G_{3,0,1}$$, a ''plane'' $$\mathbf f$$ is a trivector having the general form
 
In the 4D projective geometric algebra $$\mathcal G_{3,0,1}$$, a ''plane'' $$\mathbf f$$ is a trivector having the general form
  
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All planes possess the [[geometric property]].
 
All planes possess the [[geometric property]].
  
A plane is [[unitized]] when $$f_x^2 + f_y^2 + f_z^2 = 1$$.
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The [[bulk]] of a plane is given by its $$w$$ coordinate, and the [[weight]] of a plane is given by its $$x$$, $$y$$, and $$z$$ coordinates. A plane is [[unitized]] when $$f_x^2 + f_y^2 + f_z^2 = 1$$.
  
When used as an operator in the sandwich product, a [[unitized]] plane is a specific kind of [[flector]] that performs a [[reflection]].
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When used as an operator in the sandwich product, a unitized plane is a specific kind of [[flector]] that performs a [[reflection]] through itself.
  
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== Plane at Infinity ==
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If the weight of a plane is zero (i.e., its $$x$$, $$y$$, and $$z$$ coordinates are all zero), then the plane lies at infinity in all directions. Such a plane is normalized when $$f_w = \pm 1$$.
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== See Also ==
 
== See Also ==
  
 
* [[Point]]
 
* [[Point]]
 
* [[Line]]
 
* [[Line]]

Latest revision as of 20:27, 21 October 2021

Figure 1. A plane is the intersection of a 4D trivector with the 3D subspace where $$w = 1$$.

In the 4D projective geometric algebra $$\mathcal G_{3,0,1}$$, a plane $$\mathbf f$$ is a trivector having the general form

$$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$ .

All planes possess the geometric property.

The bulk of a plane is given by its $$w$$ coordinate, and the weight of a plane is given by its $$x$$, $$y$$, and $$z$$ coordinates. A plane is unitized when $$f_x^2 + f_y^2 + f_z^2 = 1$$.

When used as an operator in the sandwich product, a unitized plane is a specific kind of flector that performs a reflection through itself.

Plane at Infinity

If the weight of a plane is zero (i.e., its $$x$$, $$y$$, and $$z$$ coordinates are all zero), then the plane lies at infinity in all directions. Such a plane is normalized when $$f_w = \pm 1$$.


See Also