What is the 3D representation of a self-contained disc?

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If there is a disc where a vector that crosses its edge re-enters the same disc on the opposite side, what is the 3D representation of this disc?

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If there is a circle where a vector that crosses its edge re-enters the circle on the opposite side, what is the 3D representation of this circle?

I assume that by "circle" you mean a circular disc. In mathematics, the word "circle" refers to the "outer rim" only; if you want to talk about the "inside" of a circle, you need to call it a disc.

If you take a circular disc and glue together opposite points on its edge, the resulting surface is called the real projective plane. There doesn't seem to be a "standard" or "usual" way of representing the real projective plane in 3 dimensions, and there is no representation of it which doesn't intersect itself.

That said, the Wikipedia article "Real projective plane" has pictures and descriptions of various "immersions" of the real projective plane in 3 dimensions.