spherical | j symbols , and spherical harmonics

I am playing with React Fiber and React Drei but i do not want to use Typescript like in the examples i have found in their git

I have converted the following example Stars.tsx with typescriptlang tool

This is the output

I have (a bunch of 3D) stacks of tomographic data, in those I have deduced a certain (3D) coordinate around which I need to cut out a spherical region.

My code produces me the following image which gives an overview of what I do. I calculate the orange and green points, based on the dashed white and dashed green region. Around the midpoint of these, I'd like to cut out a spherical region, the representation of it is now marked with a circle in the image (also drawn by my code).

Constructing a sphere with skimage.morphology.ball and multiplying this with the original image is easy to do, but how can I set the center of the sphere at the desired 3D location in the image? The ~30 3D stacks of images are all of different size with different regions, but I have all the necessary coordinates ready for further use.

Scenario

I'm using unity c# to re-invent a google-earth like experience as a project. New tiles are asynchronously loaded in from the web while a user pans the camera around the globe. So far I'm able to load in all the TMS tiles based on their x & y coordinates and zoom level. Currently I'm using tile x,y to try and figure out where the tile should appear on my earth "sphere" and it's becoming quite tedious, I assume because of the differences between Euler angles and quaternions.

• I'm using the angle of Camera.main to figure out which tiles should be viewed at any moment (seems to be working fine)
• I have to load / unload tiles for memory management as level 10 can receive over 1 million 512x512 tiles
• I'm trying to turn a downloaded tile's x,y coordinates (2d) into a 3d position & rotation

Question

Using just the TMS coordinates of my tile (0,0 - 63,63) how can I calculate the tile's xyz "earth" position as well as its xyz rotation?

Extra

• in the attached screenshot I'm at zoom level 4 (64 tiles)
• y axis 0 is the bottom of the globe while y axis 15 is the top
• I'm mostly using Mathf.Sin and Mathf.Cos to figure out position & rotation so far

I've figured out how to get the tile position correct. Now I'm stuck on the correct rotation of the tiles.

The code that helped me the most was found with a question about generating a sphere in python.

I modified to the code to look like so:

Why won't a MeshPhongMaterial's envMap property work on polygonal faces when viewed through an orthographic camera?

It works on spheres but not an IcosahedronGeometry, for example. If I set the detail parameter of the IcosahedronGeometry to 2+ (more faces), the envMap begins to show. But if I switch to perspective cam, the envMap is fully visible even with detail of 0.

This is what it looks like with perspective cam, note the cubemap reflection of some clouds:

This is what it looks like with orthogonal cam and detail is 0, note the lack of cubemap reflection (please ignore the warping of the image):

Orthogonal cam, detail is 1; cubemap reflection is back:

The only difference between these two versions of the script is the camera.

Here's the code I'm using to create this object:

I have a set of data values for a scalar 3D function, arranged as inputs x,y,z in an array of shape (n,3) and the function values f(x,y,z) in an array of shape (n,).

EDIT: For instance, consider the following simple function

I have got a mostly functioning self made system for handling google maps in Vue3. I haven't used libraries because none have quite the functionality that I want (eventually), and it seemed relatively straightforward to implement myself (which it has been up until this issue).

The Map component is as follows:

I am trying to get transparency working within a wavy plane terrain

Here is my demo:

I have a big spherical Gameobject which moves forward in 3D with constant velocity. I have other spherical objects that other big object needs to attract to itself. I am using Newton's law of universal gravitation formula to attract other objects, but as expected, other objects are doing a slingshot movement much like the space shuttles doing when needed with other planets' orbits to accelerate.

I actually want a magnetic effect that without taking the masses into account, all other objects will be catched by the big object. How can I do that? Do I need a different formula? Or do I need to change the movement behavior of the objects altogether?

I'm trying to write two functions for converting Cartesian coordinates to spherical coordinates and vice-versa. Here are the equations that I've used for the conversions (also could be found on this Wikipedia page):

And

Here is my spherical_to_cartesian function: