Similarly to the earlier post “Different projections of spherical kaleidoscopic images” I am now showing the same kaleidoscopic image using different projections you can use in my kaleidoscope browser app http://geometricolor.ch/sphericalKaleidoscopeApp.html. It primarily generates images as Poincaré discs. A typical result looks like this:
This is a decoration of a {5,4} hyperbolic tiling made of pentagons, four of them meeting at each corner; or you might see an equivalent tiling of squares, five meeting at each corner. Mirror lines are geodesic lines. Here they are circles that intersect the border of the disc at right angles. Be careful if you follow these lines across the centers of five-fold rotational symmetry.
The Poincaré disc is a particular projection of the hyperboloid model of hyperbolic space. Other projections give different related images. However, I have difficulties to work with the hyperboloid model and I prefer the hemisphere model for looking at different projections. The hemisphere model results from an inverse stereographic projection of the Poincaré disc to the upper hemisphere of a sphere with the same radius as the disc. The center of projection is the south pole of the sphere. The projected boundary of the Poincaré disc lies at the equator of the sphere. Obviously, we can then use all spherical projections available for a sphere. This allows for a lot of experiments.
An orthographic projection of the hemisphere gives the Beltrami-Klein disc. A gnomonic projection of the hyperboloid model gives the same result. The hyperbolic tiling then looks like this:
The orthographic projection magnifies the center of the image and squashes outer regions towards the border. This makes that we see a decorated sphere. Here the mirror lines are straight lines. Again, be careful not get confused by the centers of five-fold symmetry. I prefer this projection to the Poincaré disc because it is more dramatic.
A gnomonic projection of the hemisphere model or an orthographic projection of the hyperbolid model gives the Gans model. The tiling fills the entire plane:
It has strong distortions outside the center as expected for a gnomonic projection. The mirror lines are now hyperboles. This image makes me dizzy. It somehow looks like the inside of a vault.
Rotating the hemisphere such that its border is perpendicular to the xy-plane and then doing a stereographic projection we get the Poincaré plane. This is equal to doing a Cayley transform on the Poincaré disc and gives such an image:
As for the Poincaré disc, geodesics and mirror lines are circles that intersect the border at right angles. The become half-circles and have their centers at the border. This image is more impressive than the Poincaré disc because it is less uniform.
Geometry in color 