Red diameter is intentionally drawn in horizontal at the center. Green diameter is perpendicular to Red diameter. The nearer a circle gets to the model edge, the more the center slips to the circumference. This figure is easy to see how diameters look. Snapshots of a moving circle are shown in small picture windows.
Distance shown in each picture is a hyperbolic distance from the origin.
And magnification in small picture windows is for your visual convenience. It is a ratio of visual size, a circle in the small picture window to corresponding circle in the big picture window. However, I do not know anything at infinity. Interval of circles on Blue line are not constant. They are taken just for our observation.