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How can I split all the black circles shapes using the greens ones ? I want to get this result, but I cannot find the good tool using boolean operations.

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Numerous possibilities!

Visually right appearance can be got by simply adding separate background colored shapes on the circles. Or as well you can group the green shapes and use it as inverse clipping mask:

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The green circles in the left are a group which is lifted to the foreground in the layering order. The black circles are another group.

Both groups are selected and Object > Clip > Set inverse clip is applied. It causes no harm that the green circles have half-transparent fill color.

These methods are visually OK, but the full black circles are still there, they are only made invisible on the screen. The methods are useless for engraving machines which expect simple paths and nothing invisible. The existence of invisible parts is no harm if the image is only watched on the screen or it will be printed or converted to a raster image; for ex. to PNG.

Another possibility is to combine the green circles to one path by applying Path > Combine. Then each black circle should be converted to strokeless filled areas By applying to each of them Path > Stroke to path. The resulted black rings should be united (Path > Union) and the combined green shape should be subtracted from the black union.

The result again looks right, but the black shapes are not curves, they are a complex filled shape. It also is useless if single stroke simple curves are a must.

You get simple single stroke curves (as user Billy Kerr has six hours later written, too) if you combine the black circles to one by applying Path > Combine and then combine the green circles. Use the combined green shape to split the black circles by applying Path > Cut Path. The unwanted pieces must be deleted manually, but that's easy if you have a spare copy of the green circles below.

In this phase people probably think "how to select (for deletion) the unwanted pieces as a single selection by using the spare green shapes as a selection mask". That problem has appeared several times in this site, but generally valid methods have not appeared for "selection by touching by another shape", no matter how useful it would be.

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  • Thank you fir the detailed answer !
    – Jean
    Commented Dec 24, 2023 at 11:34
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If you first combine the black circles, then combine the green circles, you could then do a boolean Cut operation which will cut the black paths at the intersections with the green circles.

An extra trick: if you copy the combined green circles before doing the boolean Cut operation, you can then Paste in Place, send to the bottom of the stack to use as a guide, then simply delete the black segments which overlie the green circles. Finally delete the green circles.

An example of the process

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Another more manual method is also possible. Simply add nodes, select the segment, and use the Delete Segment button.

An example

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  • Thank you fir the detailed answer ! My problem as you pointed out was that the two groups were not combined before the operation.
    – Jean
    Commented Dec 24, 2023 at 11:35
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    @Jean Yeah, boolean operations only work on two objects (two paths, or two combined paths). Groups are not combined paths.
    – Billy Kerr
    Commented Dec 24, 2023 at 11:58

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