<p>I'm trying to build a headrest for my Sayl office chair. For that, I'm designing a 3d-printed part that's going to fit on one of the existing rods of the chair.</p>

<p>Check out this picture:</p>

<p><a href="https://i.stack.imgur.com/PMzGy.jpg" rel="noreferrer"><img src="https://i.stack.imgur.com/PMzGy.jpg" alt=""></a></p>

<p>How would you go about in getting the exact measurements of that white rod? I tried a caliper, and I'm able to get the width and depth, and I can just assume that the rod's profile is a perfect ellipse, which is probably a close estimate. But say that I want to get a more precise measurement. Is there any technique to do that? </p>

You could pull it apart and have it 3d scanned if you want to know the exact dimensions. There are companies that can do that for you at a certain price. Our company has used such services in scanning various parts before we obtained our own laser scanning device.
The question is whether you want exactly the same (dimension wise) part (maybe you do for ecstatic reasons) while a part that is a little more beefier would work also.
Edit:
Although tagged with 3D scanning, the OP did not mention whether he would be able to disassemble the part. 3D scanning is an option when taken apart. Another solution has been posted since.