The Poincare model is based on a mapping of points in the Euclidean plane to the interior of a circle. The following sketch shows one way to do this mapping using GSP[1].
Definitions of Euclidean geometry, as expounded by Playfair [3], are visually displayed using GSP on Paul Godfrey's homepage under Something COO-EL. One of these definitions about lines is "If two lines are such that they cannot coincide in any two points, without coinciding altogether, each of them is called a straight line." A common extension of this definition is about parallel lines. If two straight lines do not coincide, i.e., cross each other, they are parallel.
This is sometimes applied to the Poincare disk to demonstrate the differences in Geometries. Using GSP, Figure 3 is a picture of two lines in the Poincare disk.
Figure 4 adds another line that must be a "straight" line as it applies to M. But is M a straight line?
This relationship has been shown by Juraschek [1] and others many times. It is pretty straight forward. We see here that M is not a straight line as it applies to a redrawn L because they intersect in more than one point.
