In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Hjelmslev transformation is an effective method for
mapping an entire
hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
into a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
with a finite
radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
. The transformation was invented by Danish mathematician
Johannes Hjelmslev
Johannes Trolle Hjelmslev (; 7 April 1873 – 16 February 1950) was a mathematician from Hørning, Denmark. Hjelmslev worked in geometry and history of geometry. He was the discoverer and eponym of the Hjelmslev transformation, a method for mapp ...
. It utilizes
Nikolai Ivanovich Lobachevsky
Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russian mathematician and geometer, kn ...
's 23rd theorem from his work
Geometrical Investigations on the Theory of Parallels
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
.
Lobachevsky observes, using a combination of his 16th and 23rd theorems, that it is a fundamental characteristic of
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
that there must exist a distinct
angle of parallelism
In hyperbolic geometry, the angle of parallelism \Pi(a) , is the angle at the non-right angle vertex of a right hyperbolic triangle having two asymptotic parallel sides. The angle depends on the segment length ''a'' between the right angle an ...
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
Johannes Trolle Hjelmslev (; 7 April 1873 – 16 February 1950) was a mathematician from Hørning, Denmark. Hjelmslev worked in geometry and history of geometry. He was the discoverer and eponym of the Hjelmslev transformation, a method for mapp ...
discovered from this a method of compressing an entire hyperbolic plane into a finite circle. The method is as follows: for any angle of parallelism, draw from its line AE a perpendicular to the other ray; using that cutoff length, e.g., AH, as the radius of a circle, "map" the point H onto the line AE. This point H thus mapped must fall between A and E. By applying this process for every line within the plane, the infinite hyperbolic space thus becomes contained and planar. Hjelmslev's transformation does not yield a proper circle however. 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.
There are a few properties that are retained through the transformation which enable valuable information to be ascertained therefrom, namely:
#The image of a circle sharing the center of the transformation will be a circle about this same center.
#As a result, the images of all the right angles with one side passing through the center will be right angles.
#Any angle with the center of the transformation as its vertex will be preserved.
#The image of any straight line will be a finite straight line segment.
#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.
#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 may refer to:
People
*Klein (surname)
*Klein (musician)
Places
*Klein (crater), a lunar feature
*Klein, Montana, United States
*Klein, Texas, United States
*Klein (Ohm), a river of Hesse, Germany, tributary of the Ohm
*Klein River, a river ...
, 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
uniform scaling
In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a ''scale factor'' that is the same in all directions. The result of uniform scaling is similar ...
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.