Join and meet
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The join is a binary operation that calculates the higher-dimensional geometry containing its two operands, similar to a union. The meet is another binary operation that calculates the lower-dimensional geometry shared by its two operands, similar to an intersection.
The points, lines, and planes appearing in the following tables are defined as follows:
- $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$
- $$\mathbf q = q_x \mathbf e_1 + q_y \mathbf e_2 + q_z \mathbf e_3 + q_w \mathbf e_4$$
- $$\mathbf K = w_x \mathbf e_{41} + w_y \mathbf e_{42} + w_z \mathbf e_{43} + n_x \mathbf e_{23} + n_y \mathbf e_{31} + n_z \mathbf e_{12}$$
- $$\mathbf L = v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} + m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$
- $$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$
- $$\mathbf g = g_x \mathbf e_{234} + g_y \mathbf e_{314} + g_z \mathbf e_{124} + g_w \mathbf e_{321}$$
The join operation is performed by taking the wedge product between two geometric objects. The meet operation is performed by taking the antiwedge product between two geometric objects.