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?

Thank you for your help


Write the conic in matrix notation:

$\mathbf{x}^T \mathbf{Q} \mathbf{x}=0$

$\mathbf{Q}$ 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.


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