The revolution in geometry begun by Gauss, Bolyai, and Lobachevsky in the early to mid 1800's felt incomplete to many mathematicians. They had discovered a different and strange new geometry, but it was purely an axiomatic, abstract geometry. Geometers felt that Euclidean geometry was "real" in that it seemed to be connected with their perception of the universe. Lines, points, and circles could be drawn on paper and these "actual" objects seemed to behave as their abstract cousins said they should. Hyperbolic geometry had no such environment in which to "live". What was needed was a "model" in which lines, points, circles, etc could be interpreted in such a way that the hyperbolic fifth axiom would hold.
Henri Poincare, in about 1880, came up with exactly such a
model.
The hyperbolic "universe" was interpreted to be the interior of a unit
circle. The points of the geometry would be points in the "usual" sense
inside the unit circle. Lines, however, were quite
different.
A line in this model would be an arc of a circle that met the unit
circle
at right angles. Consider the diagram below. We have drawn in the
unique "line" through points A and B. Point C is a point not on
this
line. It is clear that there will be many parallels (i.e.lines that do
not intersect) to line AB through C. Two special parallels called
"ultraparallels"
are shown. (Remember that the unit circle itself is outside the
hyperbolic
space. Thus the three lines can intersect on the unit circle, but still
be parallel in the actual hyperbolic space)
It is clear then that this model satisfies axioms 1 and 4 (existence of lines and right angles are congruent). As to axioms 2 and 3, we need a definition of length that will make these axioms hold. The definition of length in the Poincare model is fairly technical, and will not be explained here. For a good reference, see the text Modern Geometries, by Michael Henle, Prentice-Hall, 1997. The new definition of length has the property that lengths approach infinity as we approach the boundary of the disk. This makes both axioms 2 and 3 hold.
In conclusion, since the first four Euclidean axioms hold in the Poincare model, and the hyperbolic fifth axiom holds as well, then exploring hyperbolic geometry in the Poincare model is just as rigorous and valid as exploring in the Euclidean plane!