# Duals

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The dual of an object $$\mathbf x$$, denoted by $$\mathbf x^*$$, is given by

$$\mathbf x^* = \mathbf{\tilde x} \mathbin{\unicode{x27D1}} {\large\unicode{x1d7d9}}$$ .

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