### Figures of Hyperbolic Geometry in the Poincaré Plane

These graphs were created by custom Igor procedures that draw and construct all types of figures of hyperbolic geometry in the Poincaré plane. Hyperbolic lines are either upright, or semi-circles, on a dotted horizon.

These Igor procedures were written by Thomas Braun from byte physics and Annette Huck in the course of investigations about hyperbolic geometry following the book of George E. Martin "The Foundations of Geometry and the Non-Euclidean Plane".

Pictures drawn with it form a beautiful part of the diploma thesis "Zur Existenz und Eindeutigkeit der ebenen hyperbolischen Geometrie" by Annette Huck in 2017, Humboldt-Universität zu Berlin, 209 pages.

The procedures are currently not publicly available.

Graphs submitted by:

• Thomas Braun
• () byte physics
• Schwarzastraße 9
• 12055 Berlin
• Germany
• www.byte-physics.de

Angles are measured by means of the usual tangents - imagine to zoom in and you just see them. They can be marked differently. This picture shows some corresponding angles.

Distance increases as points approach the horizon. As the horizon itself is infinitely far away it’s not part of the Poincaré plane anymore. The red one is the midpoint of the pink circle. The picture shows that possibly you turn three times right and yet never meet your line again.

In a cyclic quadrilateral the sum of opposed angles is equal. The red one is the midpoint of the pink circle. It’s not obvious at all why the curve of fixed distance to a point M turns out to look like a “normal” circle!

Length is measured along hyperbolic lines using special logarithmic formulas that ensure the horizon being infinitely far away. Like the circle, there are other special curves, that are not hyperbolic lines! To measure their length is possible by approximation.

Segments of equal length can be marked accordingly. The picture shows multiple reflections of a triangle which is leading to congruent triangles. The yellow ones are curves, called hypercircles - these are no hyperbolic lines!

Given a segment AB, watch every circle that has AB as a chord, and realize that two horocircles are the limiting curves as these circles get bigger and bigger. In normal (meaning Euclidean) geometry this process would result in one line containing A and B - not here!

Rays can be drawn, as well as equidistant curves. In normal geometry this construction of equidistance would lead to parallel lines - again, not in hyperbolic geometry!

There is a pentagon having five right angles!!!

The perpendicular bisectors of corresponding chords AB and A'B' of concentric circles happen to be identical. You know these kind of things to be true from the axiomatic method and proofs. But seeing it work in a “real” hyperbolic plane does give you some new intuition and confidence. In the end, geometry is about drawing.

Every picture drawn in the Poincaré plane is simultaneously transformed into the Cayley-Klein plane by a custom Igor extension. Lines are “straight” in this representation of hyperbolic geometry, but angles can no longer be measured by just zooming in and looking at them. This turns out to be more of an obstacle to intuition than accepting “bent” lines. And arising out of this: what does “straight” mean anyway?