In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Hjelmslev transformation is an effective method for mapping

Map (mathematics)

In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...

 an entire hyperbolic plane

Hyperbolic space

In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...

 into a circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

 with a finite radius

Radius

In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...

. The transformation was invented by Danish mathematician Johannes Hjelmslev

Johannes Hjelmslev

Johannes Trolle Hjelmslev was a mathematician from Copenhagen, Denmark. Hjelmslev worked in geometry and history of geometry...

. It utilizes Nikolai Ivanovich Lobachevsky

Nikolai Ivanovich Lobachevsky

Nikolai Ivanovich Lobachevsky was a Russian mathematician and geometer, renowned primarily for his pioneering works on hyperbolic geometry, otherwise known as Lobachevskian geometry...

'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

Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

 that there must exist a distinct angle of parallelism

Angle of parallelism

In hyperbolic geometry, the angle of parallelism φ, also known as Π, is the angle at one vertex of a right hyperbolic triangle that has two asymptotic parallel sides. The angle depends on the segment length a between the right angle and the vertex of the angle of parallelism φ...

 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

Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

, 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

Johannes Hjelmslev

Johannes Trolle Hjelmslev was a mathematician from Copenhagen, Denmark. Hjelmslev worked in geometry and history of geometry...

 discovered from this a method of compressing an entire hyperbolic plane in to 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.

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

Klein model

In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of n-dimensional hyperbolic geometry in which points are represented by the points in the interior of the n-dimensional unit ball and lines are represented by the...

, 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 0The source of this article is wikipedia, the free encyclopedia.  The text of this article is licensed under the GFDL.