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For example use the olympics circles logo. Add 7 circles like in that logo. Now how do I make it into a one object like this: enter image description here

This is an ugly manual example tho. How do I get intersections with vertices that follows the shape 100%.

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There's an addon called TinyCAD, which I think is implemented in Blender by default, so you just have to activate it in the Preferences.

Once activated, select all your vertices in Edit mode, then right click > TinyCAD > XALL Intersect Selected Edges. It will create a common vertex everywhere the edges are crossing. Then you just need to remove the doubles with AltM > Merge By Distance, and remove all the inner vertices.

enter image description here

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