Dipole
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a dipole $$\mathbf d$$ is a bivector with ten components having the general form
- $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$ .
If the $$vx$$, $$vy$$, $$vz$$, $$mx$$, $$my$$, and $$mz$$ components are all zero, then the dipole contains the point at infinity, and it is thus a flat point.
Given a center $$\mathbf p = (p_x, p_y, p_z)$$, a line direction $$\mathbf n = (n_x, n_y, n_z)$$, and a radius $$r$$, a dipole can be formulated as
- $$\mathbf d = n_x \mathbf e_{41} + n_y \mathbf e_{42} + n_z \mathbf e_{43} + (p_yn_z - p_zn_y) \mathbf e_{23} + (p_zn_x - p_xn_z) \mathbf e_{31} + (p_xn_y - p_yn_x) \mathbf e_{12} + (\mathbf p \cdot \mathbf n)(p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + \mathbf e_{45}) - \dfrac{p^2 + r^2}{2}(n_x \mathbf e_{15} + n_y \mathbf e_{25} + n_z \mathbf e_{35})$$ .
The various properties of a dipole are summarized in the following table.
Constraints
A valid dipole $$\mathbf d$$ must satisfy the following constraints, where $$\mathbf v = (d_{vx}, d_{vy}, d_{vz})$$, $$\mathbf m = (d_{mx}, d_{my}, d_{mz})$$, and $$\mathbf p = (d_{px}, d_{py}, d_{pz})$$ are treated as ordinary 3D vectors.
- $$\mathbf p \times \mathbf v - d_{pw}\mathbf m = \mathbf 0$$
- $$\mathbf p \cdot \mathbf m = 0$$
- $$\mathbf v \cdot \mathbf m = 0$$
The last two constraints are not independent since they can be derived from the first constraint by taking dot products with the vectors $$\mathbf p$$ and $$\mathbf v$$.
Carrier and Cocarrier
The carrier of a dipole $$\mathbf d$$ is the line
- $$\operatorname{car}(\mathbf d) = \mathbf d \wedge \mathbf e_5 = d_{vx} \mathbf e_{415} + d_{vy} \mathbf e_{425} + d_{vz} \mathbf e_{435} + d_{mx} \mathbf e_{235} + d_{my} \mathbf e_{315} + d_{mz} \mathbf e_{125}$$ .
The cocarrier of a dipole $$\mathbf d$$ is the plane
- $$\operatorname{ccr}(\mathbf d) = \mathbf d^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5 = d_{vx} \mathbf e_{4235} + d_{vy} \mathbf e_{4315} + d_{vz} \mathbf e_{4125} - d_{pw} \mathbf e_{3215}$$ .
The carrier and cocarrier meet at the flat center of the dipole, which is given by the flat point
- $$\operatorname{car}(\mathbf d) \vee \operatorname{ccr}(\mathbf d) = (d_{vz} d_{my} - d_{vx} d_{pw} - d_{vy} d_{mz})\mathbf e_{15} + (d_{vx} d_{mz} - d_{vy} d_{pw} - d_{vz} d_{mx})\mathbf e_{25} + (d_{vy} d_{mx} - d_{vz} d_{pw} - d_{vx} d_{my})\mathbf e_{35} - (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_{45}$$ .
Center and Container
The round center of a dipole $$\mathbf d$$ is the round point having the same center and radius as $$\mathbf d$$, and it is given by
- $$\operatorname{cen}(\mathbf d) = \operatorname{ccr}(\mathbf d) \vee \mathbf d = (d_{vy} d_{mz} - d_{vz} d_{my} + d_{vx} d_{pw})\mathbf e_1 + (d_{vz} d_{mx} - d_{vx} d_{mz} + d_{vy} d_{pw})\mathbf e_2 + (d_{vx} d_{my} - d_{vy} d_{mx} + d_{vz} d_{pw})\mathbf e_3 + (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_4 + (d_{pw}^2 - d_{vx} d_{px} - d_{vy} d_{py} - d_{vz} d_{pz})\mathbf e_5$$ .
The container of a dipole $$\mathbf d$$ is the sphere having the same center and radius as $$\mathbf d$$, and it is given by
- $$\operatorname{con}(\mathbf d) = \mathbf d \wedge \operatorname{car}(\mathbf d)^\unicode["segoe ui symbol"]{x2606} = (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_{1234} + (d_{vz} d_{my} - d_{vy} d_{mz} - d_{vx} d_{pw})\mathbf e_{4235} + (d_{vx} d_{mz} - d_{vz} d_{mx} - d_{vy} d_{pw})\mathbf e_{4315} + (d_{vy} d_{mx} - d_{vx} d_{my} - d_{vz} d_{pw})\mathbf e_{4125} + (d_{mx}^2 + d_{my}^2 + d_{mz}^2 + d_{vx} d_{px} + d_{vy} d_{py} + d_{vz} d_{pz})\mathbf e_{3215}$$ .
A unitized dipole is equal to the meet of its carrier and container, a relationship that can be expressed as
- $$\mathbf d = \operatorname{car}(\mathbf d) \vee \operatorname{con}(\mathbf d)$$ .
Norms
The radius of a dipole $$\mathbf d$$ is given by
- $$\operatorname{rad}(\mathbf d) = \dfrac{\left\Vert\mathbf d\right\Vert_R}{\left\Vert\mathbf d\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{d_{pw}^2 - d_{mx}^2 - d_{my}^2 - d_{mz}^2 - 2(d_{px} d_{vx} + d_{py} d_{vy} + d_{pz} d_{vz})}{d_{vx}^2 + d_{vy}^2 + d_{vz}^2}}$$ .
Contained Points
The attitude of a dipole $$\mathbf d$$ is given by
- $$\operatorname{att}(\mathbf d) = \mathbf d \vee \underline{\mathbf e_4} = d_{vx} \mathbf e_1 + d_{vy} \mathbf e_2 + d_{vz} \mathbf e_3 + d_{pw} \mathbf e_5$$ .
The set of round points contained by a dipole $$\mathbf d$$ can be expressed parametrically in terms of the center and attitude as
- $$\mathbf p(\alpha) = \operatorname{cen}(\mathbf d) + \alpha \operatorname{att}(\mathbf d)$$ .
That is, $$\mathbf d \wedge \mathbf p(\alpha) = 0$$ for all real numbers $$\alpha$$. In particular, the two points on the surface of a dipole are given by the parameter $$\alpha_R = \pm \left\Vert\mathbf d\right\Vert_R$$ (the weighted radius). When $$\mathbf d$$ is a real dipole, this is precisely where the radius of $$\mathbf p(\alpha)$$ is zero. For smaller absolute values of $$\alpha$$, the round point $$\mathbf p(\alpha)$$ is real, and for larger absolute values of $$\alpha$$, the round point $$\mathbf p(\alpha)$$ is imaginary. When $$\mathbf d$$ is an imaginary dipole, the round point $$\mathbf p(\alpha)$$ is always imaginary, and it has an absolute radius at least as large as the dipole itself.