Hjelmslev transformation
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In
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.
Construction
Formalisation
The Hjelmslev transformation is a function designated as which operates upon all points in hyperbolic (Lobachevskian) space. Given an angle such that , and an origin , this mapping yields images where the following properties are preserved:
- The image of a circle with a center at is a circle with a center at .
- The image of a rectilinear angle is a rectilinear angle.
- Any angle with vertex is mapped onto itself, i.e. any angle with vertex is preserved.
- The image of a right angle with one side passing through is a right angle with one side passing through .
- The image of any straight line will be a finite straight line segment.
- 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 , such that , the resulting images , will form a triangle with an imaginary vertex in their direction of parallelism.
Transformation of a single point into image
Given , , , in order to find the (image) of . First draw the line segment , connecting the point to the origin . Next, construct an auxiliary line such that . Point is only necessary to define the line .
Now construct the perpendicular line passing through the point , perpendicular to . This will form the right angle at point :
Using line segment as a radius, construct a circle with center such that the circumference of said circle intersects at a point . Thus, we obtain the point on line segment , which is the Hjelmslev transformation of the given point . :
The Hjelmslev Disk
Let be parallel to , where 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.
The Hjelmslev transformation and the Klein model
If we represent hyperbolic space by means of the
See also
References
- ^ "For every given angle a there is a line p such that Π(p) = a"
- ^ Lobachevski, Nicholas (1914). Geometrical Researches on The Theory of Parallels (PDF). Chicago, Illinois: The Open Court Publishing Company. pp. 13–14 (Theorem 16), 19–21 (Theorem 23).