Unitization

Unitization is the process of scaling an element of a projective geometric algebra so that its weight norm becomes the antiscalar $$\large\unicode{x1D7D9}$$. An element that has a weight norm of $$\large\unicode{x1D7D9}$$ is said to be unitized.

An element $$\mathbf a$$ is unitized by calculating

$$\mathbf{\hat a} = \dfrac{\mathbf a}{\left\Vert\mathbf a\right\Vert_\unicode{x25CB}} = \dfrac{\mathbf a}{\sqrt{\mathbf a \mathbin{\unicode{x25CB}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{a}}}}}$$ .

In general, an element is unitized when the combined magnitude of all of its components having a factor of $$\mathbf e_4$$ is unity. That is, the components of the element that extend into the projective fourth dimension collectively have a size of one.

The following table lists the unitization conditions for the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.

Type	Definition	Unitization
Magnitude	$$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$	$$y^2 = 1$$
Point	$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$	$$p_w^2 = 1$$
Line	$$\mathbf L = v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} + m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$	$$v_x^2 + v_y^2 + v_z^2 = 1$$
Plane	$$\mathbf f = f_x \mathbf e_{423} + f_y \mathbf e_{431} + f_z \mathbf e_{412} + f_w \mathbf e_{321}$$	$$f_x^2 + f_y^2 + f_z^2 = 1$$
Motor	$$\mathbf Q = r_x \mathbf e_{41} + r_y \mathbf e_{42} + r_z \mathbf e_{43} + r_w {\large\unicode{x1d7d9}} + u_x \mathbf e_{23} + u_y \mathbf e_{31} + u_z \mathbf e_{12} + u_w \mathbf 1$$	$$r_x^2 + r_y^2 + r_z^2 + r_w^2 = 1$$
Flector	$$\mathbf G = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 + s_w \mathbf e_4 + h_x \mathbf e_{423} + h_y \mathbf e_{431} + h_z \mathbf e_{412} + h_w \mathbf e_{321}$$	$$s_w^2 + h_x^2 + h_y^2 + h_z^2 = 1$$

See Also

• Geometric norm