TITLE: What is the equation of a general circle in 3-D space?
QUESTION [31 upvotes]: I know that $(x-x_0)^2+(y-y_0)^2-r^2=0$ is a general planar circle and $(x-x_0)^2+(y-y_0)^2+(z-z_0)^2-r^2=0$ is a general sphere.
I want to know the general expression of a circle in space. 
Can anyone please give me some advice here?

REPLY [3 votes]: The equation of a circle in $3$-space is given by
$$
|x|^2 + 2\xi \cdot x + \xi^2 = 0
$$
where $\xi = (\xi_1, \xi_2, \xi_3) \in {\mathbb C}^3$, $x\in {\mathbb R}^3$, $\xi \cdot x = \xi_1x_1+\xi_2x_2+\xi_3x_3$ and $\xi^2 = \xi_1{}^2 + \xi_2{}^2+\xi_3{}^2$.
To see this, write $\xi = - a+ i n$ where $a\in {\mathbb R}^3$ is the centre of the circle and $n\in {\mathbb R}^3$ is the normal to the plane of the circle with $|n| = r$ where $r$ is the radius of the circle. 
This representation goes back to Laguerre : 
E. N. Laguerre, Sur l’emploi des imaginaires dans la g´eom´etrie de l’espace, Nouvelles Annales de Math., Series 2, vol. xi, (1872).
See also
P. Baird, Conformal foliations by circles and complex isoparametric functions on Euclidean 3-space. Math. Proc. Cambridge Philos. Soc., 123, (1998) 273–300.