Circle
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a circle $$\mathbf c$$ is a trivector with ten components having the general form
- $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$ .
If the $$gx$$, $$gy$$, $$gz$$, and $$gw$$ components are all zero, then the circle contains the point at infinity, and it is thus a straight line.
Given a center $$\mathbf p = (p_x, p_y, p_z)$$, a plane normal $$\mathbf n = (n_x, n_y, n_z)$$, and a radius $$r$$, a circle can be formulated as
- $$\mathbf c = n_x \mathbf e_{423} + n_y \mathbf e_{431} + n_z \mathbf e_{412} + (p_yn_z - p_zn_y) \mathbf e_{415} + (p_zn_x - p_xn_z) \mathbf e_{425} + (p_xn_y - p_yn_x) \mathbf e_{435} + (\mathbf p \cdot \mathbf n)(p_x \mathbf e_{235} + p_y \mathbf e_{315} + p_z \mathbf e_{125} - \mathbf e_{321}) - \dfrac{p^2 - r^2}{2}(n_x \mathbf e_{235} + n_y \mathbf e_{315} + n_z \mathbf e_{125})$$ .
The various properties of a circle are summarized in the following table.
Constraints
A valid circle $$\mathbf c$$ must satisfy the following constraints, where $$\mathbf g = (c_{gx}, c_{gy}, c_{gz})$$, $$\mathbf v = (c_{vx}, c_{vy}, c_{vz})$$, and $$\mathbf m = (c_{mx}, c_{my}, c_{mz})$$ are treated as ordinary 3D vectors.
- $$\mathbf g \times \mathbf m - c_{gw}\mathbf v = \mathbf 0$$
- $$\mathbf g \cdot \mathbf v = 0$$
- $$\mathbf v \cdot \mathbf m = 0$$
The last two constraints are not independent since they can be derived from the first constraint by taking dot products with the vectors $$\mathbf g$$ and $$\mathbf m$$.
Carrier and Cocarrier
The carrier of a circle $$\mathbf c$$ is the plane
- $$\operatorname{car}(\mathbf c) = \mathbf c \wedge \mathbf e_5 = c_{gx} \mathbf e_{4235} + c_{gy} \mathbf e_{4315} + c_{gz} \mathbf e_{4125} + c_{gw} \mathbf e_{3215}$$ .
The cocarrier of a circle $$\mathbf c$$ is the line
- $$\operatorname{ccr}(\mathbf c) = \mathbf c^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5 = -c_{gx} \mathbf e_{415} - c_{gy} \mathbf e_{425} - c_{gz} \mathbf e_{435} - c_{vx} \mathbf e_{235} - c_{vy} \mathbf e_{315} - c_{vz} \mathbf e_{125}$$ .
The carrier and cocarrier meet at the flat center of the circle, which is given by the flat point
- $$\operatorname{car}(\mathbf c) \vee \operatorname{ccr}(\mathbf c) = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{15} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{25} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{35} + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{45}$$ .
Center and Container
The round center of a circle $$\mathbf c$$ is the round point having the same center and radius as $$\mathbf c$$, and it is given by
- $$\operatorname{cen}(\mathbf c) = \operatorname{ccr}(\mathbf c) \vee \mathbf c = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_1 + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_2 + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_3 + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_4 + (c_{vx}^2 + c_{vy}^2 + c_{vz}^2 + c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})\mathbf e_5$$ .
The container of a circle $$\mathbf c$$ is the sphere
- $$\operatorname{con}(\mathbf c) = \mathbf c \wedge \operatorname{car}(\mathbf c)^\unicode["segoe ui symbol"]{x2606} = -(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{1234} + (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{4235} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{4315} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{4125} + (c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz} - c_{gw}^2)\mathbf e_{3215}$$ .
A unitized circle is equal to the meet of its carrier and container, a relationship that can be expressed as
- $$\mathbf c = \operatorname{car}(\mathbf c) \vee \operatorname{con}(\mathbf c)$$ .
Norms
The radius of a circle $$\mathbf c$$ is given by
- $$\operatorname{rad}(\mathbf c) = \dfrac{\left\Vert\mathbf c\right\Vert_R}{\left\Vert\mathbf c\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{c_{vx}^2 + c_{vy}^2 + c_{vz}^2 - c_{gw}^2 + 2(c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})}{c_{gx}^2 + c_{gy}^2 + c_{gz}^2}}$$ .
Contained Points
The attitude of a circle $$\mathbf c$$ is given by
- $$\operatorname{att}(\mathbf c) = \mathbf c \vee \overline{\mathbf e_4} = c_{gx} \mathbf e_{23} + c_{gy} \mathbf e_{31} + c_{gz} \mathbf e_{12} + c_{vx} \mathbf e_{15} + c_{vy} \mathbf e_{25} + c_{vz} \mathbf e_{35}$$ .
The set of round points contained by a circle $$\mathbf c$$ can be expressed parametrically in terms of the center and attitude as
- $$\mathbf p(\boldsymbol \alpha) = \operatorname{cen}(\mathbf c) + \operatorname{att}(\mathbf c) \vee (\alpha_x \mathbf e_1 + \alpha_y \mathbf e_2 + \alpha_z \mathbf e_3)^\unicode["segoe ui symbol"]{x2605}$$ .
That is, $$\mathbf c \wedge \mathbf p(\boldsymbol \alpha) = 0$$ for all vectors $$\boldsymbol \alpha = (\alpha_x, \alpha_y, \alpha_z)$$. In particular, points on the surface of a circle are given when $$\boldsymbol \alpha$$ is parallel to the attitude and has magnitude $$\left\Vert\mathbf c\right\Vert_R$$ (the weighted radius). When $$\mathbf c$$ is a real circle, this is precisely where the radius of $$\mathbf p(\boldsymbol \alpha)$$ is zero. For smaller magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is real, and for larger magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is imaginary. When $$\mathbf c$$ is an imaginary circle, the round point $$\mathbf p(\boldsymbol \alpha)$$ is always imaginary, and it has an absolute radius at least as large as the circle itself.