Blue line is a straight line. The picture is the trace of a moving circle. Red and Green lines are diameters. Their intersection is hyperbolic center. 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.