Join and meet

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 flat points, lines, planes, round points, dipoles, circles, and spheres appearing in the following tables are defined as follows:


 * $$\mathbf p = p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + p_w \mathbf e_{45}$$
 * $$\boldsymbol l = l_{vx} \mathbf e_{415} + l_{vy} \mathbf e_{425} + l_{vz} \mathbf e_{435} + l_{mx} \mathbf e_{235} + l_{my} \mathbf e_{315} + l_{mz} \mathbf e_{125}$$
 * $$\mathbf g = g_x \mathbf e_{4235} + g_y \mathbf e_{4315} + g_z \mathbf e_{4125} + g_w \mathbf e_{3215}$$
 * $$\mathbf h = h_x \mathbf e_{4235} + h_y \mathbf e_{4315} + h_z \mathbf e_{4125} + h_w \mathbf e_{3215}$$


 * $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$
 * $$\mathbf b = b_x \mathbf e_1 + b_y \mathbf e_2 + b_z \mathbf e_3 + b_w \mathbf e_4 + b_u \mathbf e_5$$
 * $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$
 * $$\mathbf f = f_{vx} \mathbf e_{41} + f_{vy} \mathbf e_{42} + f_{vz} \mathbf e_{43} + f_{mx} \mathbf e_{23} + f_{my} \mathbf e_{31} + f_{mz} \mathbf e_{12} + f_{px} \mathbf e_{15} + f_{py} \mathbf e_{25} + f_{pz} \mathbf e_{35} + f_{pw} \mathbf e_{45}$$
 * $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$
 * $$\mathbf o = o_{gx} \mathbf e_{423} + o_{gy} \mathbf e_{431} + o_{gz} \mathbf e_{412} + o_{gw} \mathbf e_{321} + o_{vx} \mathbf e_{415} + o_{vy} \mathbf e_{425} + o_{vz} \mathbf e_{435} + o_{mx} \mathbf e_{235} + o_{my} \mathbf e_{315} + o_{mz} \mathbf e_{125}$$
 * $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$
 * $$\mathbf t = t_u \mathbf e_{1234} + t_x \mathbf e_{4235} + t_y \mathbf e_{4315} + t_z \mathbf e_{4125} + t_w \mathbf e_{3215}$$

The Join Operation
The join operation is performed by taking the wedge product between two geometric objects.

The Meet Operation
The meet operation is performed by taking the antiwedge product between two geometric objects.