Unitization

From Projective Geometric Algebra
Revision as of 07:36, 14 June 2021 by Eric Lengyel (talk | contribs)
Jump to navigation Jump to search

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 $$\mathbf a$$ that possesses the geometric property is unitized by calculating

$$\mathbf{\hat a} = \dfrac{\mathbf a}{\left\Vert\mathbf a\right\Vert_\unicode{x25CB}} = \dfrac{\mathbf a}{\sqrt{\mathbf a \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{a}}}}}$$ .

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

See Also