# What is the proper name for the “Venn diagram” of a set of geometries?

Sometimes, you have a set of overlapping geometries, and you want to generate an output like this:

The resulting polygons have no overlaps; each region of space is covered by only one polygon. All boundaries are preserved; every combination of intersecting input polygons results in a separate polygon in the result. This is similar to a Venn diagram in that result is all the separate relations of the polygons.

What is the proper name for this operation or for the output? "Intersection" does not seem appropriate, as that might be only the region where all the inputs intersect. "Overlay" seems inappropriate, since that involves combining multiple sets of input (layers) and does not self intersect a single input layer.

I'm really at a loss at what to call this thing, but I feel confident it must have a formal name.

• I'd welcome any tag suggestions. I really couldn't think of or find anything else to add. – jpmc26 Jan 17 '19 at 17:44
• That's called a Union. See here: desktop.arcgis.com/en/arcmap/10.5/tools/analysis-toolbox/… – Tom Jan 17 '19 at 18:20
• @Tom ESRI's Union toolbox is inappropriately named. Even in the UI, you can see the tool is inside an "Overlay" grouping. It's actually an overlay operation (which takes multiple layers) that just includes areas covered by any input set in the result. Contrast with the Overlay Intersect tool, which only includes areas that are included in all inputs. Even that aside, a geometric union is not mathematically defined the way ESRI is using it. A mathematical union will generate a single result set (which would corrspeond to a single multipolygon geometrically), losing the boundaries between. – jpmc26 Jan 17 '19 at 18:51
• A geometric union is, in fact, mathematically defined the way ESRI uses it; the discrepancy appears to be your conflation of features vs feature classes. ESRI's Union tool unions features across datasets. E.g., if you had multiple feature classes, each with a single feature, and you unioned those feature classes, then the output feature class (as a whole) contains the geometric union of all inputs, while each feature contains the geometric union of a given pair of inputs. Maybe that's what you're after: there's no official name, it's simply a union subsequently split by the input boundaries. – Tom Jan 17 '19 at 19:33
• @Tom No. Mathematically, a union is what ESRI calls a dissolve. It removes the boundaries between all geometries. That is not what their Union tool does at all. The Overlay Union tool generates intersections between geometries and then also includes the symmetric difference from the two input layers as well. (Conceptually. I'm sure the implementation is not like that.) As you seem to understand, since you point out that it's the union split by the input boundaries. – jpmc26 Jan 17 '19 at 20:20

Mathematically, in terms of set theory, your output is a combination of intersections (`∩`), unions (`∪`) and differences (`-`). Note than the intersection and union operators have the conmutative property, but the difference operator has not.

``````A = A - (B ∪ C)
B = B - (A ∪ C)
C = C - (A ∪ B)
AB!C = (A ∩ B) - C
AC!B = (A ∩ C) - B
BC!A = (B ∩ C) - A
ABC = A ∩ B ∩ C
``````

That is a particular (and useful) combination of those operators on the original sets. But I don't think that it has a formal name.

It is easier to tackle this question in two folds:

• i) which tool (if any) does such type of GIS operation in one step;
• ii) what is the standard terminology for it.

i) Tools:

The Union tool from ArcGIS Desktop does exactly what is described in the Venn diagram when the input features belong to different/separate files/layers. For example, polygon A is in one file, polygon B is in another file. It will create from those two overlapping polygons, tree polygons in the output:

• A without AB intersection,
• B without AB intersection and,
• AB intersection.

The attributes from them will be joined, meaning AB will have attributes from both polygons A and B. See from How Union works article:

The output feature class will contain polygons representing the geometric union of all the inputs as well as all the fields from all the input feature classes. ...

On the other hand, the `Union` tool will work in a different way if overlapping features belong to the same input file/layer. In this case, it will create four polygons in the output:

• polygon A without the AB intersection,
• polygon B without the AB intersection,
• AB intersection with attributes from polygon A,
• AB intersection with attributes from polygon B.

So, in other to achieve the result expected from the Venn diagram, one would need to stack attributes from intersection, and delete one of the duplicates. Still from How Union works:

Tip: Union can run with a single input feature class or layer ... The area of overlap will always generate two identical overlapping features, one for each of the features that participates in that overlap.

ii) Terminology (of a GIS operation/task):

On the other hand, it seems there is not a standard name adopted to produce that output of a Venn diagram.

Union is not consistent at least across some GIS software. For example, `ST_Union` from PostGIS will work like tools `Dissolve` or `Merge` from ArcGIS where there are overlapping polygons.

The most close standard terminology for a GIS task I can think of to describe a Venn diagram output are the complementary tasks of Intersect and Symmetrical Difference on overlapping polygons.