Using the well known mirror reflection equation in Gaussian Geometric Optics relating $(u,v)$ object(blue)/ image (red) distances:
$$\frac{1}{u(\theta)}+\frac{1}{v(\theta)}=\frac{1}{f} \tag1 $$
Gaussian form of reflection between object/image:
$$ [u(\theta)-f]\cdot [v(\theta)-f)] = f^2 \tag2 $$
Reflection of object point to a diametrically opposite image point by mapping: $$ r\rightarrow \dfrac{1}{1/f-1/r} \tag3 $$
Green circle is $r=f$ and brown $r=2f$ correspond to focal and curvature central positions.
Plotted below is a circle $ ( u = 2 \cos \theta ) $ diameter $D=2$ units through the origin "reflected" about an origin centered unit focal circle $f=1$ with the above polar transformations :
A bizarre reflection curve with asymptotes at angles $ \cos\theta_{asymptote}=\frac{f}{D}$.
A straight line and and an ellipse respectively "reflect" to an ellipse and hyperbola.A Hyperbola reflects to another hyperbola.
Is this known in Optics or Geometry? Please give references if available. No such reference made in elementary physics text-books like those authored by Halliday /Resnick.
In this Research
We have image formation in the same way as in paraxial geometric optics by means of mapping at (3) as happens with mirrors/lenses. If object is between $(f,2 f)$ the image is outside $2f$ and vice-versa. If the object is on $2f$ circle the image is also on the same circle. Object at $\infty$ maps to focal circle $f=1$ and vice- versa.
The difference that may be noted here is that the object /image need not be at a constant distance/radius on a single concentric circle from the origin but can extend freely across $ ( r=f,2f )$ radii circles. A polar generalization is seen here.
When object extends across $2f$ brown circle the object, image and 2f circle are always concurrent.
Picture below shows the image (red) of an eccentric Circle object (blue). Concurrency of the three lines are clearly seen. It is not yet clear how eccentric circles reflect to secondary tiny islands/ovals seen at left and in the first picture.
In view of the above where we are obeying Gauss geometric optics, for time being dropping out quotes for reflection sounds appropriate. Due to the polar placement of object/image curves a distinction between reflective (mirrors) and refractive (lenses) ray tracing can be also overlooked.
