How to determine effective angular range in an area of a sphere I have populated a sphere using the Lambert's azimuthal projection method to obtain a uniform grid/area - see link for reference to the paper (https://www.aanda.org/articles/aa/full_html/2010/12/aa15278-10/aa15278-10.html).
The generated sphere is shown in the image attached 1, and I would like to calculate the angular range that each area occupies. I have identified 4 points on the sphere and their respective azimuthal and elevation angles. So my question is - how do I calculate the effective angular range that each area on the sphere contains?
Hope that is clear. Thank you.
*Edit: Added another image of the sphere from a different angle 2.
 A: We have a set of "pixels" or points on the sphere arranged in rings,
that is, the pixels can be partitioned into subsets so that the pixels in each subset lie along one line of latitude at equal intervals.
I will call one such subset a ring of pixels.
I will suppose that the sphere is divided into zones between lines of latitude so that each zone contains one ring of pixels.
Each zone is then divided into bounding boxes by a set of lines of longitude so that each bounding box contains exactly one pixel.
We require that every bounding box in every zone have the same area.
We assume the pixels are placed in such a way that this is possible.
Within these constraints, we arrange the bounding boxes so that each pixel is as close to the center of its box as possible.
Let there be $n$ rings, with $p_k$ pixels in the $k$th ring, $1\leq k \leq n,$
numbered in sequence of latitude from one pole to the other.
The total number of pixels is
$$ P = \sum_{i=1}^n p_i = p_1 + p_2 + \cdots + p_n. $$
For simplicity, we suppose the radius of the sphere is $1.$
Then the area of the sphere is $4\pi$ and the area of the bounding box of each pixel is $4\pi/P.$
The combined area of all zones up to and including the one occupied by the $k$th ring is
$$ \frac{4\pi}{P} \sum_{i=1}^k p_i = \frac{4\pi}{P}(p_1 + p_2 + \cdots + p_k). $$
These zones occupy a spherical cap;
the area of a spherical cap is proportional to its height,
with the entire sphere considered as a cap of height equal to the sphere's diameter,
so the height of this spherical cap is
$$
\frac{\frac{4\pi}{P} \sum_{i=1}^k p_i}{4\pi} \cdot 2
= \frac 2P \sum_{i=1}^k p_i.
$$
This implies that the boundary of the cap (the line of latitude between ring number $k$ and ring number $k+1$) is at colatitude
$$ \arccos \left(1 - \frac 2P \sum_{i=1}^k p_i\right).$$
(Colatitude is like latitude, but measured starting from zero the pole instead of at the equator.)
The other boundary of the zone occupied by the $k$th ring is the boundary of the spherical cap occupied by the first $k-1$ rings, which is at colatitude
$$ \arccos \left(1 - \frac 2P \sum_{i=1}^{k-1} p_i\right).$$
So that is the range of colatitudes of the bounding box of a pixel in the $k$th ring; subtract from $\frac\pi2$ radians to get the range of latitudes.
Since there are $p_k$ equal-sized bounding boxes in a ring which is a total of $2\pi$ radians around, the angular "width" of each box (measured in radians) is $2\pi/p_k.$
If the longitude of the pixes is $\lambda$, then the range of longitudes of the bounding box is
$$\left[\lambda - \frac{2\pi}{p_k}, \lambda + \frac{2\pi}{p_k} \right]. $$
