One last comment on this. I notice that the OGC SFS document describes
these relationships at the beginning of the chapter as "topological
spatial". In other words, they aren't purely topological in nature,
they take account of spatial (positioning) characteristics, too.
In fundamental topology, any two finite curves are identical. When
your curves are based on point set definitions, I see where you could
go either way in defining a topology, where coincident boundary points
are distinguished from each other or cancel out their "boundariness".
The latter is the mod-2 rule you mention, and the SFS uses it for
linestrings and multilinestrings. They also use it for multipolygon
boundary curves in 2D, but there by enforcing their avoidance during
construction. But they *don't* use it for tangent polygon boundary
points, i.e. two polygons that intersect at a single point are still
considered to be two polygons with a shared boundary. They also don't
use it for polygon boundary curves in 3D (when expressed as polyhedral
surfaces).
Again, it's all to make things work the way we expect (for the most
part :-O)
-- Andy
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.
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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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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