Pseudosphere (or something similar) in MetaPost

I want to construct a graph similar to a pseudosphere, but I keep running into many errors. Since this is my first time using MetaPost, I wanted to start from a pseudosphere and then keep changing it to my needed graphic. Unfortunately, no AI has been much help (I've been trying many different variations for the past two days). I've understood that MetaPost isn't the best with 3D graphics, but that's what I unfortunately have to use. Here is the code that won't work:

Is there even any point in trying to construct my needed graphic from a pseudosphere? The graphic that I need is here:

It's the last part of my LaTex assignment and as a struggling uni student, I'm thankful for any help.

You are right that Metapost is not the easiest tool for working in three dimensions. But here is an effort, based on the fact that the pseudosphere is the solid of revolution of the tractrix.



To get a PDF that looks like this, compile this source file with lualatex.

(BTW I strongly suggest that you switch to using luamplib + lualatex instead of plain mpost)