Hyperbolic geometry is an example of a geometry where the parallel postulate fails in the sense that given a line L and a point P not on that line there are infinitely many lines through P that are parallel to L. The parallel postulate stood out from Euclid's other postulates for geometry in that it was less elementary than the others.

For almost 2000 years people attempted to resolve this by showing that it followed from the other axioms of geometry. This finally came to an end in the beginning of the 19th century. A model of geometry where the parallel postulate does not hold was claimed by Lobachevsky, though later it was learned that Karl Friederich Gauss had come up with one earlier.

One of the basic forms of non-euclidean geometry. In hyperbolic geometry, any given line has infinitely many parallels through a given point.

Hyperbolic geometry is not as convenient to model as spherical geometry. David Hilbert proved that it is impossible to map the entire hyperbolic plane isometrically (that is, in a manner that preserves all distances) onto any surface in Euclidean 3-space. However, there are several non-isometric models used.

The most commonly used model of the hyperbolic plane is probably the Poincare disk. In this model, the "plane" is the unit disk, "lines" are circular arcs (or line segments) that meet the unit circle at right angles at both ends, and the distance function is given by:

dh(P, Q) = ln((d(P, A) d(Q, B))/(d(Q, A) d(P, B)))

where d is the Euclidean distance function (sqrt((x1-x2)^2+(y1-y2)^2), and A and B are the endpoints of the "line" joining P and Q.

That's a pretty complicated distance function, but what it essentially means is this: the closer things get to the periphery of the disk, the larger distances become. Poincare explained it with a metaphor involving insects that become smaller the farther they get from the origin, so that they move slower and slower and can never achieve a distance of 1 or more from it. In effect, the unit circle is infinity. While it is possible for parallel lines to meet there, they needn't. If they do, they are said to have a common direction.

For further help in visualizing the Poincare disk, see M.C. Escher's Circle Limit series, a set of woodcuts based on tesselations that are only possible on the hyperbolic plane.

One of the advantages of this model is that it is conformal - that is, although it does not preserve distance, it does preserve angles. Thus, we can measure them with an ordinary protractor, or just eyeball them for a rough idea. Because of this, it is pretty easy to see that the angles of a triangle in hyperbolic geometry do not add up to 180 degrees. In fact, the larger the triangle, the less the sum of its angles will be. It's even possible to compute the area of a triangle from its angles. Thus, it is impossible for two triangles to be similar without also being congruent.

Another interesting thing about hyperbolic geometry concerns uniform curves - which is to say, figures which are in all places identical, so that any region of the curve is congruent to any other region of the same size. In Euclidean geometry, the only such figures possible are the circle and the line. Hyperbolic geometry has a couple of other types. First, there's the hypercycle, which is a locus of points that are a given constant perpendicular distance from a given line. In Euclidean geometry, this definition would simply yield another line parallel to the first, but not in hyperbolic geometry, where parallel lines diverge. Two hypercycles are congruent if and only if they have the same distance from their base lines.

The other uniform curve, the horocycle, is a bit more complicated. It can be defined as a locus of corresponding points in all the lines with a given common direction, where a point P on line A corresponds to a point Q on line B if the line joining P to Q makes the same angle with A as it does with B. The tangent line to any point on a horocycle will be perpendicular to the line with the common direction through that point.

Now here's the kicker. All the different kinds of uniform curves - straight lines, circles, hypercycles, horocycles - are represented in the Poincare disk model by Euclidean circles (or straight lines, which are degenerate circles). If it is contained entirely within the unit disc, it's a hyperbolic circle. If it's tangent to the unit circle, it's a horocycle. If it's perpendicular to the unit circle, it's a straight line. Anything else is a hypercycle, and the angle with which it meets the unit circle is determined by its distance from its base line. Veryfying this is left as an exercise for the reader.

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