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A color diagram of the Lobachevsky upper half plane, drawn on a light blue-gray grid. The x and y axes are drawn in black. A vertical red line labeled L extends upwards from the x-axis. A point P, indicated with a small blue circle, is shown to the right of L. Four different semi-circle geodesics pass through P but do not intersect L. They are drawn in red, orange, green, and purple. Below the diagram the distance rule and geodesic equations are shown.

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A color diagram of the Lobachevsky upper half plane, drawn on a light blue-gray grid. The x and y axes are drawn in black. A vertical red line labeled L extends upwards from the x-axis. A point P, indicated with a small blue circle, is shown to the right of L. Four different semi-circle geodesics pass through P but do not intersect L. They are drawn in red, orange, green, and purple. Below the diagram the distance rule and geodesic equations are shown.
https://files.mastodon.social/media_attachments/files/109/439/239/488/490/901/original/54f07ef087d88c7e.jpeg

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Robert McNees (mcnees@mastodon.social)'s status on Thursday, 01-Dec-2022 16:01:13 UTC Robert McNees
in reply to

And this means that given a straight line L like the vertical geodesic shown here, and a point P not on L, there are an infinite number of semicircle geodesics that pass through P but never intersect L.
It's an unusual geometry!
Source code for this diagram, made in LaTeX with pgf/tikz, can be found here: https://gist.github.com/mcnees/c8e9840c39c5d9202a509c46c2ad4c91

In conversation Thursday, 01-Dec-2022 16:01:13 UTC from mastodon.social permalink