From Rigid Geometric Algebra
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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

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

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.

See Also