When point K moves on a straight line, point P draws a circular arc. You can freely exchange Poincare's disc for Klein's disc, and viceversa. Of course, hyperbolic value does not change by this exchange. Change appears on visual drawing only. Green curve is to show how ordinary values P , K correspond to given hyperbolic value.
Gray line is the shadow of the green curve.
It is impossible that you go to the Hyperbolic NonEuclidean World and mesure with an ordinary ruler and come back here in Euclidean World. Formula left shows relation between ordinary value for drawing of Poincare's disc and hyperbolic value. When you draw on a disc, you must get ordinary value for drawing. Let us call this value hyperbolic value for drawing . Formula left is applicable to distance from the origin only.