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Moss's Egg
The highlighted curve is called Moss's Egg, though I'm not certain who Moss is. (The link cites Dixon's Mathographics—I ordered a copy, perhaps it'll tell me.) I was surprised to find a way to construct it using only 8 circles! This is because circles naturally want to generate √3's, so usually it'd take extra effort to generate the √2 proportions the egg demands, but it turned out to only require going a single circle out of my way.
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Maybe you've addressed already, but wondering if you are "tracing" this back to the "shape" of the soul, according to the ancients, as a kind of ritual meditative technique...?
Axé
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If you're asking "why circles?" the answer is that I was really surprised to discover the Mohr-Mascheroni Theorem—anything you can do with compasses and straightedge, you can do with compasses alone—and wanted to develop an intuition for why. But in playing around with it, I've found that compass-only constructions have a very different feel to them; normal construction feel like you're using a set of tools in known ways; but compass-only constructions are much more mysterious, more like a puzzle. I suppose the mystery of it fascinates me.
Also, I like how the space is less explored. People far smarter than I have been doing "normal" geometry for millennia: it's very well understood, and there's 40 constructions for anything. If you search around for compass-only constructions, though, it's hard to find much of anything. So if you want to know how to draw something, the only way is to sort it out yourself!
also like any straight male I like round things more than straight things