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- Hjelmslev_transformation abstract "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.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.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.".
- Hjelmslev_transformation thumbnail Hjconstruction.svg?width=300.
- Hjelmslev_transformation wikiPageID "3535050".
- Hjelmslev_transformation wikiPageLength "3644".
- Hjelmslev_transformation wikiPageOutDegree "20".
- Hjelmslev_transformation wikiPageRevisionID "551540393".
- Hjelmslev_transformation wikiPageWikiLink Angle_of_parallelism.
- Hjelmslev_transformation wikiPageWikiLink Beltrami–Klein_model.
- Hjelmslev_transformation wikiPageWikiLink Category:Hyperbolic_geometry.
- Hjelmslev_transformation wikiPageWikiLink Circle.
- Hjelmslev_transformation wikiPageWikiLink Geometrical_Investigations_on_the_Theory_of_Parallels.
- Hjelmslev_transformation wikiPageWikiLink Hjelmslevs_theorem.
- Hjelmslev_transformation wikiPageWikiLink Hyperbolic_geometry.
- Hjelmslev_transformation wikiPageWikiLink Hyperbolic_space.
- Hjelmslev_transformation wikiPageWikiLink Johannes_Hjelmslev.
- Hjelmslev_transformation wikiPageWikiLink Klein_model.
- Hjelmslev_transformation wikiPageWikiLink Map_(mathematics).
- Hjelmslev_transformation wikiPageWikiLink Mathematics.
- Hjelmslev_transformation wikiPageWikiLink Nikolai_Ivanovich_Lobachevsky.
- Hjelmslev_transformation wikiPageWikiLink Nikolai_Lobachevsky.
- Hjelmslev_transformation wikiPageWikiLink Radius.
- Hjelmslev_transformation wikiPageWikiLink Scaling_(geometry).
- Hjelmslev_transformation wikiPageWikiLink Uniform_scaling.
- Hjelmslev_transformation wikiPageWikiLink File:Hjconstruction.svg.
- Hjelmslev_transformation wikiPageWikiLink File:Hjintersecting.svg.
- Hjelmslev_transformation wikiPageWikiLink File:Hjparallel.svg.
- Hjelmslev_transformation wikiPageWikiLink File:Hjultra.svg.
- Hjelmslev_transformation wikiPageWikiLinkText "Hjelmslev transformation".
- Hjelmslev_transformation hasPhotoCollection Hjelmslev_transformation.
- Hjelmslev_transformation wikiPageUsesTemplate Template:Unreferenced.
- Hjelmslev_transformation subject Category:Hyperbolic_geometry.
- Hjelmslev_transformation hypernym Method.
- Hjelmslev_transformation type Article.
- Hjelmslev_transformation type Software.
- Hjelmslev_transformation type Article.
- Hjelmslev_transformation type Surface.
- Hjelmslev_transformation comment "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.".
- Hjelmslev_transformation label "Hjelmslev transformation".
- Hjelmslev_transformation sameAs m.09k0jn.
- Hjelmslev_transformation sameAs Q5873234.
- Hjelmslev_transformation sameAs Q5873234.
- Hjelmslev_transformation wasDerivedFrom Hjelmslev_transformation?oldid=551540393.
- Hjelmslev_transformation depiction Hjconstruction.svg.
- Hjelmslev_transformation isPrimaryTopicOf Hjelmslev_transformation.
