Geodesic Dome Types for skylights
Basically, Geodesic domes are produced by subdividing the faces of a regular polyhedron that fits onto a sphere. Geodesic spheres and domes are then categorized according to the base polyhedron, the number of subdivisions and the type of subdivisions.
There are several regular polyhedrons that can be used as a basis for subdividing into geodesic spheres but 3 are more common.
The most common one is the icosahedron, the second one is the octahedron and then the less commonly used is the tetrahedron.
Classes and frequencies
Class refers to the way each base triangle is subdivided. Class I are domes where the subdivisions are from edge to edge or, put another way parallel (turned 0°) to the edges. Class II are domes where the subdivisions are from edge to vertex or, put another way, perpendicular (turned 90°) to the edges. Class III are domes where the subdivision are other angles than 0° or 90° to the edges.
Frequency refers to the number of subdivisions that occur at each edge. As seen in the above illustration, no matter the class, a frequency 4 dome divides the edges into 4 segments. Because of the way subdivision is performed on class II domes, there can only be even frequencies class II. Also note that a frequency 1 dome is the base polyhedron.
Choice of dome type for skylighting
Different combinations of polyhedron and classes can produce quite different quality of domes. Those qualities are all compromises and the best compromise is a matter of application. For general skylights for instance, the best compromises are offered by domes based on icosahedron and of class II. Icosahedron of class I are also usefull. Those are the ones I decided to generate in my packages. Here is why :
The subdivision of a Icosahedron offers the most regular structure and so the lights are the most uniformly distributed.
It can be observed, however, that the Icosahedron tends to produce light alignments along the edges of the triangles.This is not necessarily an interesting feature since it will most probabl;y produce shadows alignment in the scene.
The trick, when using an icosahedron subdivision in a skylight arrangement is to orient them in such a way that the alignment of lights is not parallel or perpendicular to the majority of the edges in the scene.
As can be observed from the illustrations above, domes based on subdividing the Octahedron produce much less regular structures and so, when used for a skylight rig, the lights are less well distributed although the class II subdivision produces a slightly better distribution.
Apart from that, the same light alignment artifacts are produced by both class I and class II domes. Actually, because of the base octahedron structure, the light alignment is the worse in the class I subdivision because it follows the 3 total circular belts around the 3 X, Y and Z axises. The class II is just a little bit less worse because it produces light belts around 2 axises instead of 3.
Concerning distribution of lights, tetrahedron based domes with class I subdivisions are really the worst choice. It produces concentration of lights in some spots and sparseness of lights in other spots.
Class II subdivision, though, produces an interestingly uniform distribution with not too much light alignment.
There are three possible views when looking at a dome. The vertex view, the edge view and the face view. For skylights, it is useful to consider which dome view is used when we look at the dome from the top view in A:M. The vertex view generate several horizontal light alignments around the Y axis. The edge view produces only 2 vertical alignments of lights. The face view produces both vertical alignments and horizontal alignments. An optimal view would be a face view but with a slight rotation toward an edge and a vertex but that is for a later version.
Because the Icosahedron based domes of class I and II are the most useful for building skylights, those are the ones I supply here. I generated the dome data from the edge view in order to produce as less alignments of lights as possible. A reorientation of 18° around the Y axis when the skylight is placed in a choreography can help reduce shadow edges. However much better results can be achieved with multi ray casts soft shadow lights as demonstrated in the Technical analysis.