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The dual of an object $$\mathbf u$$, denoted by $$\mathbf u^\unicode["segoe ui symbol"]{x2605}$$, is given by

$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf G \mathbf u}$$ ,

where $$\mathbf G$$ is the metric exomorphism matrix, and the overline is the complement operation.

The antidual of an object $$\mathbf u$$, denoted by $$\mathbf u^\unicode["segoe ui symbol"]{x2606}$$, is given by

$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbb G \mathbf u}$$ ,

where $$\mathbb G$$ is the metric anti-exomorphism matrix. In conformal geometric algebra, it is always the case that $$\mathbf u^\unicode["segoe ui symbol"]{x2606} = -\mathbf u^\unicode["segoe ui symbol"]{x2605}$$, so the two duals differ only in sign.

The following table lists the duals for the geometric objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.

Type Definition Dual
Flat point $$\mathbf p = p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + p_w \mathbf e_{45}$$ $$\mathbf p^\unicode["segoe ui symbol"]{x2605} = -p_w \mathbf e_{321} + p_x \mathbf e_{235} + p_y \mathbf e_{315} + p_z \mathbf e_{125}$$
Line $$\boldsymbol l = l_{vx} \mathbf e_{415} + l_{vy} \mathbf e_{425} + l_{vz} \mathbf e_{435} + l_{mx} \mathbf e_{235} + l_{my} \mathbf e_{315} + l_{mz} \mathbf e_{125}$$ $$\boldsymbol l^\unicode["segoe ui symbol"]{x2605} = -l_{vx} \mathbf e_{23} - l_{vy} \mathbf e_{31} - l_{vz} \mathbf e_{12} - l_{mx} \mathbf e_{15} - l_{my} \mathbf e_{25} - l_{mz} \mathbf e_{35}$$
Plane $$\mathbf g = g_x \mathbf e_{4235} + g_y \mathbf e_{4315} + g_z \mathbf e_{4125} + g_w \mathbf e_{3215}$$ $$\mathbf g^\unicode["segoe ui symbol"]{x2605} = g_x \mathbf e_1 + g_y \mathbf e_2 + g_z \mathbf e_3 - g_w \mathbf e_5$$
Round point $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$ $$\mathbf a^\unicode["segoe ui symbol"]{x2605} = a_w \mathbf e_{1234} - a_x \mathbf e_{4235} - a_y \mathbf e_{4315} - a_z \mathbf e_{4125} + a_u \mathbf e_{3215}$$
Dipole $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$ $$\mathbf d^\unicode["segoe ui symbol"]{x2605} = d_{vx} \mathbf e_{423} + d_{vy} \mathbf e_{431} + d_{vz} \mathbf e_{412} - d_{pw} \mathbf e_{321} + d_{mx} \mathbf e_{415} + d_{my} \mathbf e_{425} + d_{mz} \mathbf e_{435} + d_{px} \mathbf e_{235} + d_{py} \mathbf e_{315} + d_{pz} \mathbf e_{125}$$
Circle $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$ $$\mathbf c^\unicode["segoe ui symbol"]{x2605} = -c_{gx} \mathbf e_{41} - c_{gy} \mathbf e_{42} - c_{gz} \mathbf e_{43} - c_{vx} \mathbf e_{23} - c_{vy} \mathbf e_{31} - c_{vz} \mathbf e_{12} - c_{mx} \mathbf e_{15} - c_{my} \mathbf e_{25} - c_{mz} \mathbf e_{35} + c_{gw} \mathbf e_{45}$$
Sphere $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$ $$\mathbf s^\unicode["segoe ui symbol"]{x2605} = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 - s_u \mathbf e_4 - s_w \mathbf e_5$$

See Also