Hjelmslev transformation

In mathematics, the Hjelmslev transformation is an effective method for mapping an entire hyperbolic plane into a circle with a finite radius. The transformation was invented by Danish mathematician Johannes Hjelmslev. It utilizes Nikolai Ivanovich Lobachevsky's 23rd theorem from his work Geometrical Investigations on the Theory of Parallels.

Lobachevsky observes, using a combination of his 16th and 23rd theorems, that it is a fundamental characteristic of hyperbolic geometry that there must exist a distinct angle of parallelism for any given line length. Let us say for the length AE, its angle of parallelism is angle BAF. This being the case, line AH and EJ will be hyperparallel, and therefore will never meet. Consequently, any line drawn perpendicular to base AE between A and E must necessarily cross line AH at some finite distance. Johannes Hjelmslev discovered from this a method of compressing an entire hyperbolic plane into a finite circle.