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I have a layer X with road geometries and want to calculate the overlap area with region Y (the overlap of X and Y). However, the geometries within layer X are already overlapping.

Since the tool description doesn't explicitly state it: Will the tool "Overlap Analysis" count these areas (overlapping feature geometries in layer X) twice per region Y?

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I performed a quick test, the simple and short answer is: No, it won't consider any area twice. It will only calculate the "flat" overlap of the layer, i.e. overlap can never be larger than 100% or in absolute numbers never larger than the area of the polygon for which the overlap is analyzed.

example showing area and calculated overlap

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