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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 $$\mathbf g$$ is a trivector having the general form

$$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$ .

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 $$g_x^2 + g_y^2 + g_z^2 = 1$$. The attitude of a plane is the bivector $$g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}$$ corresponding to its normal.

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 $$\mathbf g$$ to the horizon is given by

$$\mathbf T = \underline{\mathbf g} \wedge \mathbf e_{4} + 2\mathbf g \vee \mathbf e_4 = g_{x\vphantom{y}} \mathbf e_{41} + g_y \mathbf e_{42} + g_{z\vphantom{y}} \mathbf e_{43} + 2g_w$$ .

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 $$g_w = \pm 1$$. This is the horizon of three-dimensional space.

See Also