Models for nine different Cayley-Klein geometries
These are arranged in order of familiarity and comprehensibility.  For each, here are the questions to answer:
1.	Given a “line” l and a “point” P not on l, how many “points” on l cannot be joined to P by a “line?”
2.	Given a “line” l and a “point” P not on l, how many “lines” through P do not intersect l?
3.	Given three “points” on a “line,” is one unambiguously between the other two?
4.	Given three “lines” that intersect at a “point,” is one of them unambiguously between the other two?
A: “Points” are ordered pairs of real numbers (x,y).
    “Lines” are sets of points that satisfy a linear equation ax+by=c
     where a and b are not both zero.
B: “Points” are events (t,x) in the motion of a train along a straight track.
   “Lines” are sets of points that correspond to motion with constant velocity.
   Newtonian relativity: there is no speed limit imposed by the speed of light
C: “Points” are events (t,x) in the motion of a train along a straight track.
   “Lines” are sets of points that correspond to motion with constant velocity.
   Einsteinian relativity: there is a speed limit of c, the speed of light
D: “Points” all have coordinates (x,y) with x2 + y2 <= 1
     “Lines” are sets of points that lie on a diameter of the unit circle or on a circular arc that intersects the unit circle at the ends of a diameter.
E: “Points” all have coordinates (x,y) with x2 + y2 < 1
    “Lines” are ordinary straight line segments inside the unit circle.
F: “Points” all have coordinates (x,y) with x2 + y2 > 1
     “Lines” are ordinary straight lines outside the unit circle that do not intersect the unit circle. Two lines that appear parallel are regarded as meeting at a “point at infinity.” 
G: “Points” all have coordinates (x,y) with x2 + y2 > 1
    “Lines” are ordinary straight lines outside the unit circle that intersect the unit circle. The two halves of such a line are regarded as sharing a “point at infinity” that lies between the two halves.
H: “Points” are number pairs (v,s) where s can have any value but –c<v<c.
  Such a point describes a motion along a line with speed v, initial position s.
     “Lines” are non-vertical straight lines, specified by s = -x + tv.
I: “Points” are lines through the origin in three-dimensional space that intersect the cylinder x2 + y2 = 1 (this excludes the z axis).
   “Lines” are planes through the origin that intersect the cylinder x2 + y2 = 1 in ellipses (this excludes planes that contain the z axis)