Written by Mouna Deubler, Yunus Sahin and Juliane Stehle

During the summer term 2022, our group worked on a painting app in Hyperbolic Space of dimension two.
Our goal was to create an editor for drawing, but not in the Euclidean Geometry we all know. We designed a Poincaré Disk Model raster graphics editor using p5.js and implemented different shapes and features like movements and rotations.

First, we programmed circles. A special characteristic about circles in
hyperbolic geometry is that a circle of fixed radius appears smaller near the edge than in the center.
The Poincaré Disk Model illustrates hyperbolic space of dimension two on a unit disk. Geodesics (the equivalents of straight lines in Euclidean Geometry) are diameters of the unit circle or arcs that are perpendicular to the unit circle. A key difference between Euclidean geometry and hyperbolic geometry is this: assuming there is (in hyperbolic space) a geodesic \(g\) and a point \(P\) that is not on \(g\). Then there are infinitely many geodesics that intersect \(P\) but do not intersect \(g\). In Euclidean geometry however, given a straight line \(g\) and a point \(P\) that is not on \(g\), there is exactly one straight line \(h\) that intersects \(P\) but not \(g\) (\(h\) is called parallel to \(g\) in this case).

All in all, we implemented tools to draw circles, lines, triangles, squares and stars.
Similar to popular paining apps, we can choose between different colors and sizes
and have an eraser to correct the painting.
Future steps could be to create a hyperbolic reflection and to add other objects in
the editor.

Have fun Painting!

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