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 $$\mathbf u$$ is unitized by calculating
- $$\mathbf{\hat u} = \dfrac{\mathbf u}{\left\Vert\mathbf u\right\Vert_\unicode{x25CB}} = \dfrac{\mathbf u}{\sqrt{\mathbf u \mathbin{\unicode{x25CB}} \mathbf u}}$$ .
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 | $$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$ | $$l_{vx}^2 + l_{vy}^2 + l_{vz}^2 = 1$$ |
| Plane | $$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$ | $$g_x^2 + g_y^2 + g_z^2 = 1$$ |
| Motor | $$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$ | $$Q_{vx}^2 + Q_{vy}^2 + Q_{vz}^2 + Q_{vw}^2 = 1$$ |
| Flector | $$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$ | $$F_{pw}^2 + F_{gx}^2 + F_{gy}^2 + F_{gz}^2 = 1$$ |