The Hyperbolic Plane and its Immersions into R3

The hyperbolic plane is an example of a geometry where the first four of Euclid’s Axioms hold but the fifth, the parallel postulate, fails and is replaced by a hyperbolic alternative. We discuss some basic properties of hyperbolic space, including how to coordinatize and measure lengths. We then consider the possibility of isometrically immersing the hyperbolic plane into 3 in such a way that lengths of curves are preserved. This is possible on small pieces, for example mapping to the pseudosphere. However, Hilbert’s Theorem says it is impossible for the whole hyperbolic space. Euclid’s Postulates. Euclid began with at least an intuitive description of points, lines, rays, a line segment has two endpoints, straight lines, circles, angles, lengths, triangles and the other sets and objects which are taken to be understood as usual. He tacitly assumes that points and lines exist, not all points are on the same line, two distinct lines have no more than one point in common, a straight line that contains the vertex B and an interior point of a triangle ABC also contains a point of the segment AC, things which are equal may be made to coincide (for example by a Euclidean motion, congruent figures are equal and conversely,) all sets of objects are finite, a line segment joining the center of a circle to a point outside the circle must contain a point of the circle (continuity,) a point on a line separates the line into two rays, and the existence of an order relation on the line. [M] These assumptions validate straightedge and compass constructions.