I was browsing "Thinking Geometricly: A Survey in Geometries" by Thomas Q. Sibley, 2015 and on page 388 it mentions a finite hyperbolic geometry of order 3 (3 points per line) consisting of 13 (ordinary) points.
I was wondering are there finite hyperbolic geometries that also contain ideal points? (for exploring parallel and hyperparallel lines, parallel lines share an ideal point, hyperparallel lines are not sharing any point)
So I guess this would mean:
- There are two types of points
- ordinary points
- ideal points
Each lines contains
- 3 ordinary points and
- 2 ideal points
Each pair of two points (ideal or ordinary) are on one and only one line.
- Given an ordinary point P not on line l there are at least two lines not containing an ordinary point of l. (is this even needed?)
Does such a finite geometry exist?
any references welcome
following the remarks (thanks)
i guess the last condition " given an ordinary point P not on line l there are at least two lines not containing an ordinary point of l." is not needed.