Formalisation

The Hjelmslev transformation is a function designated as ${\displaystyle H(P)}$ which operates upon all points ${\displaystyle P_{n}}$ in hyperbolic (Lobachevskian) space. Given an angle ${\displaystyle \alpha }$ such that ${\displaystyle 0<\alpha <{\frac {\pi }{2}}}$, and an origin ${\displaystyle O}$, this mapping yields images ${\displaystyle P'_{n}}$ where the following properties are preserved:

1. The image of a circle with a center at ${\displaystyle O}$ is a circle with a center at ${\displaystyle O}$.
2. The image of a rectilinear angle is a rectilinear angle.
3. Any angle with vertex ${\displaystyle O}$ is mapped onto itself, i.e. any angle with vertex ${\displaystyle O}$ is preserved.
4. The image of a right angle with one side passing through ${\displaystyle O}$ is a right angle with one side passing through ${\displaystyle O}$.
5. The image of any straight line will be a finite straight line segment.
6. Finally, 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.

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 ${\displaystyle {\overline {AB}}}$, ${\displaystyle {\overline {CD}}}$ such that ${\displaystyle {\overline {AB}}\parallel {\overline {CD}}}$, the resulting images ${\displaystyle {\overline {A'B'}}}$, ${\displaystyle {\overline {C'D'}}}$ will form a triangle ${\displaystyle \triangle A'B'IC'D'}$ with an imaginary vertex ${\displaystyle I}$ in their direction of parallelism.

Transformation of a single point ${\displaystyle P}$ into image ${\displaystyle P'}$

Given ${\displaystyle \alpha }$, ${\displaystyle O}$, ${\displaystyle P}$, in order to find the ${\displaystyle P'}$ (image) of ${\displaystyle P}$. First draw the line segment ${\displaystyle {\overline {OP}}}$, connecting the point ${\displaystyle P}$ to the origin ${\displaystyle O}$. Next, construct an auxiliary line ${\displaystyle {\overline {OQ}}}$ such that ${\displaystyle \angle POQ=\alpha }$. Point ${\displaystyle Q}$ is only necessary to define the line ${\displaystyle {\overline {OQ}}}$.

Line ${\displaystyle {\overline {OQ}}}$ at angle ${\displaystyle \alpha }$ from ${\displaystyle {\overline {OP}}}$.

Now construct the perpendicular line ${\displaystyle {\overline {PP''}}}$ passing through the point ${\displaystyle P}$, perpendicular to ${\displaystyle {\overline {OQ}}}$. This will form the right angle ${\displaystyle \angle OP''P}$ at point ${\displaystyle P''}$:

Perpendicular ${\displaystyle {\overline {PP''}}}$ onto ${\displaystyle {\overline {OQ}}}$.

Using line segment ${\displaystyle {\overline {OP''}}}$ as a radius, construct a circle with center ${\displaystyle O}$ such that the circumference of said circle intersects ${\displaystyle {\overline {OP}}}$ at a point ${\displaystyle \color {red}P'}$. Thus, we obtain the point ${\displaystyle \color {red}P'}$ on line segment ${\displaystyle OP}$, which is the Hjelmslev transformation ${\displaystyle H(P)}$ of the given point ${\displaystyle P}$. ${\displaystyle H(P)=\color {red}P'}$:

Point ${\displaystyle P'}$ from radius ${\displaystyle {\overline {OP''}}}$.

The Hjelmslev Disk

Let ${\displaystyle {\overline {P_{1}P_{2}}}}$ be parallel to ${\displaystyle {\overline {P_{3}P_{4}}}}$, where ${\displaystyle \angle P_{3}P_{1}P_{2}}$ is the angle of parallelism. 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.

A completed Hjelmslev disk representing two intersecting lines

A completed Hjelmslev disk representing two hyperparallel lines

A completed Hjelmslev disk representing two ultraparallel lines