Show Poincare Disk is incidence geometry

murmillo

1. The problem statement, all variables and given/known data
I have to show that the Poincare disk satisfies the incidence axiom that any line contains at least two points.


2. Relevant equations
There are two kinds of lines on the Poincare disk. I've found 2 points for the first kind, which are straight lines going through the origin. The second kind are lines
L a,b = {(x,y) | (x-a)² + (y-b)² = a² + b² - 1, with x² + y² < 1}

3. The attempt at a solution
Given a and b, I need to find (x, y) and (x',y') that lie on the line.
I've shown that x^2+y^2+1 = 2ax+2by, but I don't know if I'm headed in the right direction.

HallsofIvy

I'm confused as to what you are doing here. In the "Poincare' disk model for hyperbolic geometry", a "line" is either a Euclidean straight line throug the center of the disk or an arc of a Euclidean circle contained in the disk. In either case, it follows immediately from Euclidean geometry that they contain an infinite number of points. Are you required to actually calculate coordinates of two points in the "line"? If so, what coordinate system are you using?

murmillo

Yes, I have to show that there exist at least two points on the line. I need to find coordinates in terms of a and b. I'm using the normal coordinate system for points in R2.