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The HYG star database can be found here

From the hygdata_v3.csv file, how are the cartesian (x,y,z) coordinates calculated? (Is there a mathematical formula for this?)

And in the dso.csv file, there aren't any x,y,z coordinated. How can I calculate that?

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    $\begingroup$ It's a simple conversion from spherical coordinates (J2000 right ascension and declination plus distance) to Cartesian coordinates. There does not appear to be a way to calculate Cartesian coordinates for the objects in dso.csv $\endgroup$
    – user21
    Commented Apr 24, 2020 at 16:12
  • $\begingroup$ @barrycarter, yes but doesn't the cartesian coordinates in the database take into account the parallax angles? $\endgroup$
    – SidS
    Commented Apr 24, 2020 at 16:25
  • $\begingroup$ Parallax angle is just a measure for distance, hence distance is given often in parsec $\endgroup$
    – planetmaker
    Commented Apr 24, 2020 at 16:55
  • $\begingroup$ What is your question? How the distance is found or how X,y,z is found? $\endgroup$
    – ProfRob
    Commented Apr 24, 2020 at 18:22
  • $\begingroup$ @RobJeffries - I would like to know how x,y,z, is found for the hygdata csv file and how the distance is found from the dso csv file $\endgroup$
    – SidS
    Commented Apr 25, 2020 at 8:26

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The transformation to $x,y,z$ coordinates, starting from the RA, Dec and a distance (as given in the HYG catalogue) is a simple trigonometric exercise, since the $x,y,z$ (in this case) is referred to a coordinate system aligned with the equatorial coordinate system.

$$x = d \cos({\rm RA}) \cos({\rm DEC}),$$ $$y = d \sin({\rm RA}) \cos({\rm DEC}),$$ $$z = d \sin({\rm DEC}),$$ where $d$ is the distance. How that distance has been obtained is not clear from the README file accompanying the database. It may have been by (incorrectly) inverting Hipparcos parallaxes.

The dso.csv appears to be a catalogue of deep sky objects. There is no information in the catalogue itself which allows you to calculate or estimate a distance.

You could try cross-referencing the catalogue against the NASA/IPAC Extragalactic Database (NED). https://ned.ipac.caltech.edu/ If the recession velocity there is bigger than about 1000 km/s, then you will get a rough distance by using Hubble's law $$ d = v/H_0$$ with $H_0 \sim 70$ km/s per Mpc (or whatever your preferred value is). If the recesson velocity is smaller than this then the object is too close for Hubble's law to be valid.

e.g. The first line inthe catalogue appears to be a galaxy known as IC 5370, that has a recession velocity of 10372.00 km/s (no error bar given and I doubt any more than 4 significant figures is warranted and in any case $H_0$ is not known accurately to more than 2 significant figures), which gives an approximate distance of 150 Mpc.

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