It is known that a homographic transformation of a circle is a conic. The first figure is an example that fix the notation.
Here the point $O$ is the center of the homography $h$. The axis of the homography is the line $CC_1$ and the point $A$ is transformed to the point $A'$. So, to find $D'=h(D)$ (the corresponding point of $D$) we take the line from $A$ and $D$ and the point $D_1$ intersection of this line with the axis $C C_1$, than we take the line $D_1 A'$ and $D' =h(D)$ is the intersection of this line with the line that contains $D$ and the center of the homography $O$.
In this case, with $A=(0,1)$, $A'=(0,-1)$, $O=(0,0)$ and the axis the line $x=2$, the homographic transformation of the circle $x^2+y^2=1$ is the parabola $y^2+2x-1=0$.
Now I want to generalize such construction, taking the points $A,O,A'$ not aligned ( so the transformation is no more a homography). The simpler situation is to displace the center $O$ to a new point $O=(\alpha, \beta)$. In this case we can see that the circle $x^2+y^2=1$ should be transformed in a quartic. An example is given in the second figure.
But we can have different situations changing the position of the axis or using a different conic as starting curve. You can experiment some situation using the file https://www.geogebra.org/geometry/vqgmdh5q where we can change the position of the points $A, A', O, C, C'$ and move the point $D$ on the circle (or another conic changing the equation). An analytic classification of all these possibilities is really tedious and, as far as I know, the classification of quartic curves is not a simple task. So I am interesting to know if there is some known use of this kind of transformation to classify quartic curves as transformations of a conic and if there is some published work about this topic.