Cube Map Query Question

I have a series of vectors, each of which is assigned to one of 6 cube map walls and an x/y value on those walls, depending on each vector’s direction.

The full set of possible vector directions corresponds to a sphere, while the walls of the cube map are each 2D.

If I want to search for vectors pointing in directions along a circular region along the surface of this sphere, this circle on the surface of the sphere would have to be projected or otherwise transformed onto the surface of the cube map, in order to account for the distortion. I assume it would be a derived form of a point in circle test ultimately.

Likewise, if I query a rectangular region on the surface of the sphere, this too much be projected or otherwise transformed to account for distortion.

What kind of techniques or relationships might be helpful in transforming this test? I can’t seem to find any lead on it.