Terrific text.
Iteration 2.
At the beginning of a new iteration, the contents of the previous iteration are erased. Regardless of whether there are copies of it anywhere, or whether it is the last copy. Everything, sooner or later, goes into oblivion.

You know what I want?
I want one to draw a squiggle and the other to draw a broken squiggle.

The new squiggle must have an angle, so you can't draw a clean circle.
A new polygon may be drawn from a corner.

A polygon may be drawn from a corner and tangentially to an arc.

An arc cannot have more than two bends. A broken arc may not have more than one fracture.

If the arc of one corner touches the arc of another corner, these arcs are considered "out of work". They cannot be tangent.

If a squiggle flies into (out of) a corner of a broken arc such that that corner "disappears," that corner really disappears. Now there is only an arc.

If a new, broken line or arc flies out of the corner, that corner is considered "trapped" (read, "not working"). Nothing else can be taken out of it, or pounded into it. The same thing happens if something is poked into the corner.
Similarly for arcs: if an arc is tangent, it can no longer be engaged.

The arc cannot touch the corner. If a player has done so, he has lost.

No lines can be crossed. If a player crosses a line, he loses.

Players act step by step, one after another.

The task of each player is to drive the other into a stalemate, not to leave him any moves.

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This is the game I came up with. I don't know why. I don't know what the point of it is. I don't even know if I can win it.

I recommend playing it on a small piece of paper.

P.S.: Sorry about the mistakes, it's the translator! XD