Bulk and weight

The components of an element of a rigid geometric algebra can be divided into two groups called the bulk and the weight of the element. The bulk of an element $$\mathbf a$$ is denoted by $$\mathbf a_\unicode{x25CF}$$, and it consists of the components of $$\mathbf a$$ that do not have the projective basis vector as a factor. The weight is denoted by $$\mathbf a_\unicode{x25CB}$$, and it consists of the components that do have the projective basis vector as a factor. In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, the bulk is thus all components that do not contain the factor $$\mathbf e_4$$, and the weight is all components that do contain the factor $$\mathbf e_4$$.

The bulk generally contains information about the position of an element relative to the origin, and the weight generally contains information about the attitude and orientation of an element. An object with zero bulk contains the origin. An object with zero weight is contained by the horizon.

An element is unitized when the magnitude of its weight is one.

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

Type Definition Bulk Weight
Magnitude $$\mathbf z = x + y {\large\unicode{x1d7d9}}$$ $$\mathbf z_\unicode{x25CF} = x$$ $$\mathbf z_\unicode{x25CB} = y {\large\unicode{x1d7d9}}$$
Point $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ $$\mathbf p_\unicode{x25CF} = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3$$ $$\mathbf p_\unicode{x25CB} = p_w \mathbf e_4$$
Line $$\mathbf L = v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} + m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$ $$\mathbf L_\unicode{x25CF} = m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$ $$\mathbf L_\unicode{x25CB} = v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43}$$
Plane $$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$ $$\mathbf f_\unicode{x25CF} = f_w \mathbf e_{321}$$ $$\mathbf f_\unicode{x25CB} = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124}$$
Motor $$\mathbf Q = r_x \mathbf e_{41} + r_y \mathbf e_{42} + r_z \mathbf e_{43} + r_w {\large\unicode{x1d7d9}} + u_x \mathbf e_{23} + u_y \mathbf e_{31} + u_z \mathbf e_{12} + u_w$$ $$\mathbf Q_\unicode{x25CF} = u_x \mathbf e_{23} + u_y \mathbf e_{31} + u_z \mathbf e_{12} + u_w$$ $$\mathbf Q_\unicode{x25CB} = r_x \mathbf e_{41} + r_y \mathbf e_{42} + r_z \mathbf e_{43} + r_w {\large\unicode{x1d7d9}}$$
Flector $$\mathbf G = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 + s_w \mathbf e_4 + h_x \mathbf e_{234} + h_y \mathbf e_{314} + h_z \mathbf e_{124} + h_w \mathbf e_{321}$$ $$\mathbf G_\unicode{x25CF} = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 + h_w \mathbf e_{321}$$ $$\mathbf G_\unicode{x25CB} = s_w \mathbf e_4 + h_x \mathbf e_{234} + h_y \mathbf e_{314} + h_z \mathbf e_{124}$$