How to create position-cells from scratch?

Question

The geometry proximity-node position output is quite mysterious to me, it is able to dynamically split space into position cells, as shown in the gif below. I would like to have a better understanding of how it is mathematically possible to achieve such a result.

In this gif example I'm using vertices with the geometry proximity node, however, these vertices are only but simple Location vectors after all! Could we replicate this behavior only using math nodes & 3 location vectors? thus the title, can we reproduce such position-cells from scratch?

Here's a setup, the goal is to reproduce the behavior of the gif

Answer 1

The node Geometry Proximity basically provides you with the position of the closest element of a certain domain at the output Position.

Since you only have three points here, you can also capture these nearest positions with Transfer Attribute.

However, you would first have to convert your objects, which in your case have no geometry, into points.

To do this, you must first retrieve these objects with Object Info and the option As Instance. Then you join these instances and convert them into points.

From these points you can then retrieve the position of the nearest one, and the result should be identical to Geometry Proximity:

And here's another variant that, with a little logic, comes up with the right vector:

_{(Blender 3.1+)}