Consider a space curve which its projection onto $xy$-plane is a circle. Is it true necessarily for that curve $x = R\sin t$ and $y = R\cos t$? It seems reasonable but I don't whether there is a proof for it.One obvious example is the helix. I think it is related to the definition of projection.
2026-03-27 16:20:44.1774628444
A space curve with the circle projection
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If the curve projects to a circle centered at the origin (possibly partially, but with no point elsewhere), then yes, for any point of the curve there are $r$ and $t$ such that
$$x=r\cos t, y=r\sin t.$$