I am trying to make an algorithm which will calculate N zones on sphere surface which are related with given sequence (f.e. $4$ zones with $1:3:2:4$ sizes) and returns all $\theta$ and $\phi$ values for each zone.
This solution should fix my problem where I am trying to divide a space into parts that are related to each others with given sequence to create a 3d representation of data tree like star cluster.
I've started solving it on lower dimension - on circle and did it, but couldn't carry the solution to the 3d space since it is easy to calculate edges of cyclic polygon but not faces of spherical Polyhedron.
Sorry for bad explanation, hope you got my answer correctly. Thank you!
EDIT: I need the shape of the sections be maximum identically to each others. All triangles are not possible as I understood, it is OK if there will be other shapes also. Like 3 triangles and 1 square for the N = 5.
If you don't particularly care what the sections look like, then you can just take proportional wedges of the sphere (like sections of an orange). In your example, the angles would be $$0 \to 2\pi/10\\ 2\pi/10 \to 8\pi/10\\ 8\pi/10 \to 12\pi/10\\ 12\pi/10 \to 20\pi/10.$$