# Plane

Revision as of 20:27, 21 October 2021 by Eric Lengyel (talk | contribs)

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 $$f_w = \pm 1$$.