# Difference between pages "Quaternion" and "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 rigid geometric algebra $$\mathcal G_{3,0,1}$$, a ''plane'' $$\mathbf f$$ is a trivector having the general form | |||

:$$\mathbf | :$$\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 with the [[geometric antiproduct]], a plane is a specific kind of [[flector]] that performs a [[reflection]] through itself. | |||

A [[dual translation]] operator $$\mathbf T$$ that moves a plane $$\mathbf f$$ to the horizon is given by | |||

:$$\mathbf T = \underline{\mathbf f} \wedge \mathbf e_{4} + 2\mathbf f \vee \mathbf e_4 = f_{x\vphantom{y}} \mathbf e_{41} + f_y \mathbf e_{42} + f_{z\vphantom{y}} \mathbf e_{43} + 2f_w$$ . | |||

== | == 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 $$f_w = \pm 1$$. This is the ''horizon'' of three-dimensional space. | |||

<br clear="right" /> | |||

== See Also == | == See Also == | ||

* [[ | * [[Point]] | ||

* [[ | * [[Line]] |

## Revision as of 06:57, 19 June 2022

In the 4D rigid 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 with the geometric antiproduct, a plane is a specific kind of flector that performs a reflection through itself.

A dual translation operator $$\mathbf T$$ that moves a plane $$\mathbf f$$ to the horizon is given by

- $$\mathbf T = \underline{\mathbf f} \wedge \mathbf e_{4} + 2\mathbf f \vee \mathbf e_4 = f_{x\vphantom{y}} \mathbf e_{41} + f_y \mathbf e_{42} + f_{z\vphantom{y}} \mathbf e_{43} + 2f_w$$ .

## 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 $$f_w = \pm 1$$. This is the *horizon* of three-dimensional space.