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[1] 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.
[2] 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.
The Hjelmslev transformation is a function designated as
which operates upon all points
in hyperbolic (Lobachevskian) space.
, this mapping yields images
where the following properties are preserved: This function is useful in the studies of hyperbolic (Lobachevskian) space because it produces characteristic figures of parallel lines.
Given a set of two parallel lines
{\displaystyle {\overline {AB}}\parallel {\overline {CD}}}
with an imaginary vertex
in their direction of parallelism.
, in order to find the
First draw the line segment
Next, construct an auxiliary line
is only necessary to define the line
Now construct the perpendicular line
as a radius, construct a circle with center
such that the circumference of said circle intersects
By performing the transformation for every point on the two parallel lines, we yield the Hjelmslev circle: The circumference of the circle created does not have a corresponding location within the plane, and therefore, the product of a Hjelmslev transformation is more aptly called a Hjelmslev Disk.
Likewise, when this transformation is extended in all three dimensions, it is referred to as a Hjelmslev Ball.
If we represent hyperbolic space by means of the Klein model, and take the center of the Hjelmslev transformation to be the center point of the Klein model, then the Hjelmslev transformation maps points in the unit disk to points in a disk centered at the origin with a radius less than one.
Given a real number k, the Hjelmslev transformation, if we ignore rotations, is in effect what we obtain by mapping a vector u representing a point in the Klein model to ku, with 0 It is therefore in terms of the model a uniform scaling which sends lines to lines and so forth. To beings living in a hyperbolic space it might be a suitable way of making a map.
