On integer sequences for rendering limit sets of Kleinian groups

Abstract

We present a technique for rendering limit sets for kleinian groups, based upon the base transformation of integers and which aims at saving memory resources and being faster than the traditional dictionary based approach.

Figures (20)

• Figure 1: Tessellation by circle inversions. The top diagram il-lu-stra-tes how cir-cle in-ver-si-on works. At the bottom, the disc images under the action of a subgroup whose generators have no self-intersecting inversion circles and known as of Schottky type.
• Figure 2: Simple examples of limit sets. (a) Generators are four and mutually tangent inversion circles. The limit set is a circle; the subgroup is defined Fuchsian. (b) If not being exactly a geometrical, but a Jordan curve, such groups are known as quasi-fuchsian.
• Figure 3: Circle packings. Many limit sets for subgroups of Möbius maps spread their point around circles, which tend to pack bounded surfaces, like disks, or infinite strips.
• Figure 4: Tessellation via circle inversion. (a) A different disposition of four mutually tangent circles than fig. \ref{['fig_simple_examples']}/a was adopted here to work inversion maps (\ref{['eq_03']}). The tangency condition is preserved along the construction and it prevents images circles from overlapping each other. The gradient of lighter shades enhances convergence as well as position and shape of the limit set.
• Figure 5: Choice of proper strategy. (a) Generators are parabolic. The subgroup action has been rendered through (b) circles (the limit set is barely recognizable) and (c) points/pixels. Colors in (c) are associated to the starting ge-ne-ra-tor of each orbit.