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

Martin Davis mbdavis at refractions.net
Tue Jun 3 09:07:40 PDT 2008

```Andy,

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

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 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.

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

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.

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.

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 "spatially"
>> is too vague and overloaded.  How about "point-set 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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>>>>>
>>>>
>>>> --
>>>> Martin Davis
>>>> Senior Technical Architect
>>>> Refractions Research, Inc.
>>>> (250) 383-3022
>>>>
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>>
>> --
>> Martin Davis
>> Senior Technical Architect
>> Refractions Research, Inc.
>> (250) 383-3022
>>
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--
Martin Davis
Senior Technical Architect
Refractions Research, Inc.
(250) 383-3022

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