# Circle homography

I'm attending a 3d-graphics course and I want to figure out which homograpic transformations conserve a circle's equation.

The circle's equation is given as:

Circle = $x^2 + y^2 + Ax + By + C = 0$

So in an effort to find out some transformations, I though about creating a matrix

$H=\big( \matrix{h_1 & h_2 & h_3 \\h_4 & h_5 & h_6\\ h_7 & h_8 & h_9}\big)$

and then do right side multiplication with Circle, i.e $H*Circle=Circle*H=Circle$

But I am having a problem figuring out how to multiply the equation with H

Any ideas? Maybe I'm looking at it the wrong way, and there is a simpler solution out there?

$$\mathbf{x}^T \mathbf{Q} \mathbf{x}=0$$
$$\mathbf{Q}$$ is a 3x3 matrix encoding your $$A$$, $$B$$ and $$C$$ (see wikipedia for details).
Under the point homography $$\mathbf{x}^{\prime}=\mathbf{H}\mathbf{x}$$ a conic $$\mathbf{Q}$$ transforms to $$\mathbf{Q}^{\prime}=\mathbf{H}^{-T}\mathbf{Q}\mathbf{H}^{−1}$$. See for example Algorithms for computing a planar homography from conics in correspondence by Juho Kannala, Mikko Salo, and Janne Heikkilä.