# The homogenous representation of a circle is given by x 2 </msup> + y

Craig Mendoza 2022-06-24 Answered
The homogenous representation of a circle is given by ${x}^{2}+{y}^{2}+2gxz+2fyz+c{z}^{2}=0$ (or, equivalently, if we set $z=1$, ${x}^{2}+{y}^{2}+2gx+2fy+c=0$). Now, given 3 points (in a homogenous form), we can solve a system of linear equations and retrieve the unknowns $f$, $g$ and $c$.
This is all very nice (because of linear algebra), but what do these unknowns actually represent with respect to the circle? Which of these numbers represent the x and y coordinates of a circle and which one represents the radius?
Apparently, $-f$ and $-g$ would be the $x$ and $y$ coordinate of the center of the circle? Why is that the case? I would like to see a proof/derivation of it. Also, what is the radius then?
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Braylon Perez
$\begin{array}{rl}0& ={x}^{2}+{y}^{2}+2gx+2fy+c\\ & ={x}^{2}+2gx+{g}^{2}+{y}^{2}+2fy+{f}^{2}+\left(c-{g}^{2}-{f}^{2}\right)\\ & =\left(x+g{\right)}^{2}+\left(y+f{\right)}^{2}-\left({f}^{2}+{g}^{2}-c\right)\end{array}$
This equation says that the squared distance of the point $\left(x,y\right)$ from the point $\left(-g,-f\right)$ is ${f}^{2}+{g}^{2}-c$, which describes a circle centered at $\left(-g,-f\right)$ with radius $\sqrt{{f}^{2}+{g}^{2}-c}$.