Line

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

In the 4D projective geometric algebra $$\mathcal G_{3,0,1}$$, a line $$\mathbf L$$ is a bivector having the general form

$$\mathbf L = v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} + m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$ .

The components $$(v_x, v_y, v_z)$$ correspond to the line's direction, and the components $$(m_x, m_y, m_z)$$ correspond to the line's moment. To possess the geometric property, the components of $$\mathbf L$$ must satisfy the equation

$$v_x m_x + v_y m_y + v_z m_z = 0$$ ,

which means that, when regarded as vectors, the direction and moment of a line are perpendicular.

The bulk of a line is given by its $$m_x$$, $$m_y$$, and $$m_z$$ coordinates, and the weight of a line is given by its $$v_x$$, $$v_y$$, and $$v_z$$ coordinates. A line is unitized when $$v_x^2 + v_y^2 + v_z^2 = 1$$.

When used as an operator in the sandwich product, a unitized line is a specific kind of motor that performs a 180-degree rotation about itself.

Lines at Infinity

Figure 2. A line at infinity consists of all points at infinity in directions perpendicular to the moment $$\mathbf m$$.

If the weight of a line is zero (i.e., its $$v_x$$, $$v_y$$, and $$v_z$$ coordinates are all zero), then the line lies at infinity in all directions perpendicular to its moment $$(m_x, m_y, m_z)$$, regarded as a vector, as shown in Figure 2. Such a line cannot be unitized, but it can be normalized by dividing by its bulk norm.

When the moment $$\mathbf m$$ is regarded as a bivector, a line at infinity can be thought of as all directions $$\mathbf v$$ parallel to the moment, which satisfy $$\mathbf m \wedge \mathbf v = 0$$.

Skew Lines

Figure 3. The line $$\mathbf J$$ connecting skew lines is given by a commutator.

Given two skew $$\mathbf K$$ and $$\mathbf L$$, as shown in Figure 3, a third line $$\mathbf J$$ that contains a point on each of the lines $$\mathbf K$$ and $$\mathbf L$$ is given by the commutator

$$\mathbf J = [\mathbf K, \mathbf L]^{\Large\unicode{x27C7}}_- = (v_yw_z - v_zw_y)\mathbf e_{41} + (v_zw_x - v_xw_z)\mathbf e_{42} + (v_xw_y - v_yw_x)\mathbf e_{43} + (v_yn_z - v_zn_y + m_yw_z - m_zw_y)\mathbf e_{23} + (v_zn_x - v_xn_z + m_zw_x - m_xw_z)\mathbf e_{31} + (v_xn_y - v_yn_x + m_xw_y - m_yw_x)\mathbf e_{12}$$ .

The direction of $$\mathbf J$$ is perpendicular to the directions of $$\mathbf K$$ and $$\mathbf L$$, and it contains the closest points of approach between $$\mathbf K$$ and $$\mathbf L$$.