# [postgis-users] How to tell if 2 geometries are spatially equal

Andy Anderson aanderson at amherst.edu
Tue Jun 3 10:48:02 PDT 2008

```On Jun 3, 2008, at 12:07 PM, Martin Davis wrote:

> Andy,
>
> Can you send me the PDF that you found?  That link appears to be dead.

Sent directly.

> I realize that the SFS doesn't actually define equal.  I was the
> designer of JTS and GEOS, so the definition is due to me.  I
> followed what I thought was a logical extension of the other
> definitions, and something that was expressible in terms of the
> DE-9IM.  I also provided "equalsExact", which corresponds to the
> other proposed definition that you give.

I don't have a problem with how they function, it makes sense to me as
well.

> I will note that in fact according to the SFS, in the geometry
> LINESTRING(0 0, 5 5, 10 10)' the point (5,5) is *not* a boundary
> point.  In fact, even in the geometry MULTILINESTRING((0 0, 5 5), (5
> 5, 10 10)) the point (5 5) is not on the boundary, due to the
> (slightly bizarre) 'mod-2' rule used by the SFS.

Let me just provide an "operational" explanation of why I don't
consider them topologically equivalent. LINESTRING(0 0, 5 5, 10 10)
could be turned into LINESTRING(0 0, 5 5, 10 15) with a single
operation ("rotate second line segment" or "move third point") while
LINESTRING(0 0, 10 10) would require *two* operations, beginning with
"insert point (5 5)".

I'll also note that, to be consistent, this must also mean that
MULTIPOLYGON((0 0, 5 5, 5 0, 0 0), (0 0, 0 5, 5 5, 0 0)) doesn't have
a boundary line at LINESTRING(0 0, 5 5).

> There's plenty of references that support the definition I chose (in
> fact, I suspect I chose it based on other references I scanned at
> the time):
>
> * The IBM Spatial Datablade manual says: Using the *ST_Equals()*
> function is functionally equivalent to using
> *ST_IsEmpty*(*ST_SymDifference*(/a/,/b/))  (http://publib.boulder.ibm.com/infocenter/idshelp/v10/index.jsp?topic=/com.ibm.spatial.doc/spat122.htm
> )
>
> ESRI gives the DE-9IM pattern of T*F**FFF* for ST_Equals, which is
> what JTS uses
> http://webhelp.esri.com/arcgisdesktop/9.2/index.cfm?TopicName=Spatial_relationships

Ah, this is actually a more complete description than what I found.
Interestingly, the matrix they use:

IBE
I T*F
B **F
E FF*

is the same matrix used in the mlblog.osdir.com document by Rueben
Schulz I quoted, right after the claim is made about requiring the
"same number of vertices". However, looking more carefully at the
matrix, it says that the intersection of the boundaries and interiors
can be anything, meaning that an interior boundary point doesn't
exclude equality. So either he doesn't define that as a vertex or he
meant to say "endpoints" (or he made an error :-).

> Egenhofer has a paper on "Point-Set Topological Spatial Relations" (http://www.spatial.maine.edu/~max/pointset.pdf
> )
>
> There's also all the references given in the SFS paper.

I'm not sure which one that is.

> In the end it all comes down to naming.  There are various kinds of
> equality, which are useful for different things.  Different systems
> name them differently, which is ok as long as the semantics are
> documented.  Of course, it's nice to have some standard names - and
> it looks to me like ST_equals is pretty well defined.

And I like it like that! :-)

-- Andy

> Andy Anderson wrote:
>> On Jun 2, 2008, at 4:29 PM, Martin Davis wrote:
>>> I use "topologically equal" because the OGC SFS specification uses
>>> the term "topology" extensively in their discussion of the meaning
>>> of the DE-9IM model, on which the semantics of ST_equals is based.
>>
>> No doubt they do, because the spatial relationships that can be
>> determined by the DE-9IM are, generally speaking, topological in
>> nature:
>>
>>    Disjoint, Touches, Crosses, Within and Overlaps
>>
>> These are the ones defined here:
>>
>>    http://portal.opengeospatial.org/files/?artifact_id=829
>>
>> Oddly, in this document they also mention "Equals" but they never
>> explicitly define it (though they claim to later in the document).
>> However, I did find a very complete description of DE-9IM here:
>>
>>    http://mlblog.osdir.com/gis.postgis/2004-02/pdfHaVE9FZPMj.pdf
>>
>> and they say that for "equals", the "Geometries must be identical:
>> – Same dimension
>> – Same geometry type
>> – Same number of vertices
>> – All x,y coordinates must be identical"
>>
>> So this definition would seem to agree with me that LINESTRING(0 0,
>> 10 10) and LINESTRING(0 0, 5 5, 10 10) are not topologically equal.
>> The best way to think of this is that the second is a *polyline*,
>> and the vertex (5, 5) is a boundary between two lines, and that
>> changes the DE-9IM.
>>
>> As you point out, however, ST_equals('LINESTRING(0 0, 10
>> 10)','LINESTRING(0 0, 5 5, 10 10)') returns true; perhaps this was
>> by design to distinguish it from 'LINESTRING(0 0, 10 10)'::geometry
>> ~= 'LINESTRING(0 0, 5 5, 10 10)'::geometry .
>>
>> If you can point to further discussion on this issue, I would be
>> interested.
>>
>>> I don't like the term "spatially-equal", because I think
>>> equal"?  The idea is that  A = B iff every point of A is in B and
>>> every point of B is in A.
>>
>> Personally, I don't have a problem with "spatially equal", it says
>> to me "they fill space in the same way", which in fact they do
>> because a point has no extent.
>>
>> -- Andy
>>
>>> Andy Anderson wrote:
>>>> I wouldn't call this example "topologically" equal; one has two
>>>> vertices and the other has three, and that's the only
>>>> characteristic that's relevant in topology (not even their
>>>> positions :-)
>>>>
>>>> "Coincident" is probably a better term, though "spatially equal"
>>>> is probably just as good, and contrasts well with the term
>>>> "geometrically equal" that the manual uses to describe the ~=
>>>> operator.
>>>>
>>>> -- Andy
>>>>
>>>> On May 30, 2008, at 7:54 PM, Martin Davis wrote:
>>>>
>>>>> Will ST_equals do what you want?  It reports whether two
>>>>> geometries are topologically equal.
>>>>>
>>>>> (So for example, ST_equals('LINESTRING(0 0, 10
>>>>> 10)','LINESTRING(0 0, 5 5, 10 10)') is true)
>>>>>
>>>>> Obe, Regina wrote:
>>>>>> I recall this having come up before.  I always thought that ~=
>>>>>> would
>>>>>> tell me if 2 geometries are spatially equal but it doesn't seem
>>>>>> to.
>>>>>>
>>>>>> The only way I can figure to determine spatial equality is if
>>>>>> ST_Within(A,B)  And  ST_Within(B,A)  (or ST_Difference(A,B) AND
>>>>>> ST_Difference(B,A) both return an empty geometry collection)
>>>>>>
>>>>>> --So case in point
>>>>>>
>>>>>> SELECT geom1 ~= geom2 as what, ST_Within(geom1, geom2) AND
>>>>>> ST_Within(geom2, geom1) As spatial_equal,
>>>>>>   ST_AsText(ST_Difference(geom1, geom2)) as diffgeom12,
>>>>>> ST_AsText(ST_Difference(geom2, geom1)) as diffgeom21 FROM
>>>>>> (SELECT 'LINESTRING(1 1, 1 2, 1 3)'::geometry As geom1,
>>>>>> 'LINESTRING(1 1, 1 3)'::geometry As geom2) As foo
>>>>>>
>>>>>> Results:
>>>>>>
>>>>>> what | spatial_equal |        diffgeom12        |
>>>>>> diffgeom21
>>>>>> ------+---------------+--------------------------
>>>>>> +----------------------
>>>>>> ----
>>>>>> f    | t             | GEOMETRYCOLLECTION EMPTY |
>>>>>> GEOMETRYCOLLECTION
>>>>>> EMPTY
>>>>>>
>>>>>>
>>>>>> Is there a function / operator that does that (also what does
>>>>>> geom1 =
>>>>>> geom2 compare - is it just bounding boxes or is that the
>>>>>> spatially equal
>>>>>> operator I am looking for?)
>>>>>>
>>>>>> Thanks,
>>>>>> Regina
>>>>>>
>>>>>> -----------------------------------------
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>>>>>>
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>>>>>>
>>>>>>
>>>>>
>>>>> --
>>>>> Martin Davis
>>>>> Senior Technical Architect
>>>>> Refractions Research, Inc.
>>>>> (250) 383-3022
>>>>>
>>>>> _______________________________________________
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>>>>
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>>>
>>> --
>>> Martin Davis
>>> Senior Technical Architect
>>> Refractions Research, Inc.
>>> (250) 383-3022
>>>
>>> _______________________________________________
>>> postgis-users mailing list
>>> postgis-users at postgis.refractions.net
>>> http://postgis.refractions.net/mailman/listinfo/postgis-users
>>
>> _______________________________________________
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>
> --
> Martin Davis
> Senior Technical Architect
> Refractions Research, Inc.
> (250) 383-3022
>
> _______________________________________________
> postgis-users mailing list
> postgis-users at postgis.refractions.net
> http://postgis.refractions.net/mailman/listinfo/postgis-users

```