# Unitization

Revision as of 06:31, 9 November 2021 by Eric Lengyel (talk | contribs)

*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 projective geometric algebra $$\mathcal G_{3,0,1}$$.

Type | Definition | Unitization |
---|---|---|

Magnitude | $$\mathbf z = x + y {\large\unicode{x1d7d9}}$$ | $$y^2 = 1$$ |

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 | $$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$ | $$f_x^2 + f_y^2 + f_z^2 = 1$$ |

Motor | $$\mathbf Q = r_x \mathbf e_{41} + r_y \mathbf e_{42} + r_z \mathbf e_{43} + r_w {\large\unicode{x1d7d9}} + u_x \mathbf e_{23} + u_y \mathbf e_{31} + u_z \mathbf e_{12} + u_w$$ | $$r_x^2 + r_y^2 + r_z^2 + r_w^2 = 1$$ |

Flector | $$\mathbf G = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 + s_w \mathbf e_4 + h_x \mathbf e_{234} + h_y \mathbf e_{314} + h_z \mathbf e_{124} + h_w \mathbf e_{321}$$ | $$s_w^2 + h_x^2 + h_y^2 + h_z^2 = 1$$ |