In section 2.4 an expression is given to find the hyperbolic distance between two points A and B on a Poincare disc representation of hyperbolic space.



Where P and Q are points on the boundary circle of the disc where the hyperbolic line through points A and B terminate, and QA, QB, PA, and PB are the Euclidean distances between points Q and A, Q and B etc.

Is the Euclidean distance mentioned here the Euclidean straight line distance between the various points, or the Euclidean arc length along the circle representing the hyperbolic line?

Since under diagram 2.12 Penrose refers to hyperbolic straight lines as being Euclidean circles or lines, which meet the bounding circle orthogonally, then the hyperbolic distance formula must be using QA etc as the distances along the hyperbolic straight lines (Euclidean circles).