We can’t see all of the Earth at once, and we had really good maps of the whole world even before satellites.
We can look into space in all directions with pretty much nothing getting in our way.* Map all of the stars (direction and distance) in every direction and you get a Milky Way map.
The semi-automated Gaia Catalogue has mapped around 2 billion stars, out of the estimated 100 billion in total. That’s still enough to have a pretty detailed map.
*The galactic center is too dense with stars to see *through*, so we have a blind spot there.
The galaxy is rich with the fundamental gas Hydrogen.
It was discovered that a special condition called hyperfine splitting produces a radio wave emission of particular nature, which can penetrate dust, allowing observations at long galactic distances.
This allows radio astronomers to observe not just the direction and density of emissions, but to also gauge the velocity difference.
From computations on observations, we get a very good understanding of our side of the galaxy, and can model that against millions of other observable galaxies.
That is just one of the more sophisticated measures we have. A fun fact is that a spiral armed galaxy like our own is a common idea of what a galaxy is like, but they are less common in total surveys than one might expect.
https://www.radio2space.com/study-of-the-galactic-hydrogen-distribution-with-spider-300a-radio-telescope/
http://article.sapub.org/10.5923.j.astronomy.20130203.03.html
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