Consider the following interpretations, and in each case, decide whether I1, I2, I3, PA, EPP and HPP are true or false. (Six “yes” or “no” answers in case (a) and (b), three “yes” or “no” answers in case (c); with short justification.)
(a) Take a usual circle C on the usual Euclidean plane. The “points” of the model are those usual points of the plane which are strictly inside the circle C (points of the circle and points outside the circle are not considered to be points of the model). The “lines” of the model are (open) chords of the circle C (without the endpoints). “Incidence” is the usual incidence.
(b) Consider the usual Cartesian system of coordinates, with perpendicular x and y axis. Let us call the set of points with positive second coordinate y > 0 (points “above the x-axis”) the upper halfplane. “Points” of the model are usual points of the upper halfplane. “Lines” of the model are semicircles in this upper half plane with centers being on the x-axis. Incidence is the usual incidence (points being on the semicircle).
(c) Define some additional lines in the above interpretation (b) such that it becomes a model of incidence geometry (I1, I2, I3 holds). Decide about PA, EPP, HPP in this new model.
I1: For any two points, there is a unique line incident to both.
I2: For every line, there are at least two points incident to it.
PI: (“stronger I2”) For every line, there are at least three points incident to it.
I3: There exist three non-collinear points.
PA: For any two lines, there is a point incident to both.
EPP: (Euclidean Parallel Postulate) For any line l and any point P not incident to it, there is exactly one line incident to the point P and parallel to the line l.
HPP: (Hyperbolic Parallel Property) For any line l and any point P not incident to it, there are at least two lines incident to the point P and parallel to the line l.