Line

From Projective Geometric Algebra
Jump to navigation Jump to search
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 $$v_x^2 + v_y^2 + v_z^2 = 1$$.

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 $$(m_x, m_y, m_z)$$, and it cannot be unitized. Such a line can be normalized by dividing by its bulk norm, which is $$\sqrt{m_x^2 + m_y^2 + m_z^2}$$.

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.

See Also