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.

The method for mapping an infinite line onto a finite one in hyperbolic geometry

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. By applying this process to every line within the plane, one could compress this plane so that infinite spaces could be seen as planar. Hjelmslev's transformation would not yield a proper circle however. The circumference of the circle 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.

A completed Hjelmslev disk representing two intersecting lines
A completed Hjelmslev disk representing two hyperparallel lines
A completed Hjelmslev disk representing two ultraparallel lines

There are a few properties that are retained through the transformation which enable valuable information to be ascertained therefrom, namely:

  1. The image of a circle sharing the center of the transformation will be a circle about this same center.
  2. As a result, the images of all the right angles with one side passing through the center will be right angles.
  3. Any angle with the center of the transformation as its vertex will be preserved.
  4. The image of any straight line will be a finite straight line segment.
  5. Likewise, the point order is maintained throughout a transformation, i.e. if B is between A and C, the image of B will be between the image of A and the image of C.
  6. The image of a rectilinear angle is a rectilinear angle.

The Hjelmslev transformation and the Klein model

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<k<1. 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.

gollark: ```print "Hacked with Python 2 or Lua"```
gollark: (produced by the common Unix tool `haxxdump`)
gollark: 011d3b0 ecda fe42 f33d d112 2b8c 7e1d 24d2 11e5011d3c0 2475 ae6a bb0f 0c59 592b 3e75 6074 5f61011d3d0 ff42 a907 c773 c81f 3095 97ba 7fe2 5270011d3e0 c021 d886 1dfc 01eb f22a 0174 38cb ab3e011d3f0 2476 6efa 2bb0 6dde cd92 0222 5467 7221011d400 bb13 2647 77f7 8c51 6206 e40d 3c85 117c011d410 86bb 928f 2234 bb31 298e dd89 7209 6a00011d420 49b1 182b 52fc 6659 f720 c14c 7064 213c011d430 be13 5b7f 36db 9228 232a be39 1c9e 4065011d440 3e92 3fa8 a538 8a60 c599 7c88 9f72 9748011d450 8a5d fc83 b21b e48d 666a 8670 3d61 0225
gollark: I have made many a useless side project.
gollark: I mean, there's a difference between programming and, say, sysadmin stuff, but yes.

See also
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