I thought about this issue a bit more over the weekend.
I still think you're right about continuous transformations not altering
the topology of geometries, Andrea. The apparent counterexample that
you proposed has a flaw in it. This is that the preservation of
topology applies to the *exact* image of the geometry under the
transformation. In particular, you have to work with the image of the
line segments. Most geodetic arcs have curved images under planar
projections - and I think if you inspected the curved images you would
see that the original topology was preserved.
The basic problem is that we are used to being able to linearly
interpolate between vertices of geometries in planar space. This is no
longer the case in geodetic space - the interpolation has to follow an
arc of a great circle. As long as all the implications of this are
properly implemented (e.g. correct coordinate for arc intersection) the
structures modelling topology should still work. (I still think there's
places in JTS where linearity is assumed - these would have to be
enhanced/removed. A fundamental example is the concept of Envelope -
it's used everywhere, and would have to be enhanced to support
geodetic. Or maybe redefined - an nice way of modelling geodetic
coordinates is using direction cosines - essentially 3D points on the
sphere. The envelope then becomes a 3D box).
The other key point is the one raised by Michael. You need to have a
more rigorous definition of geometry topology in a spherical model.
There's standard techniques for doing this - arc is assumed to be the
smaller of the two possible semiarcs between two points, geometry is
oriented with inside to the right of a ring, etc. These are a bit fussy
but I think in practice aren't much of a problem.
Andrea Aime wrote:
Martin Davis ha scritto:
I agree with Paul - it's not just distance and angle, but also the
actual location of intersections which is affected by working in
geodetic.
I think Andrea's basically correct about the *topology* of operations
not be affected.
Hum... consider two lines that do barely touch. You have an intersection
point. If the transformation changes the intersection points, it would
mean that it's possible that after reprojection the two lines do
not touch anymore, thereby changing their topological relationship.
I have vague memories of continuous transformations never altering
the topological relationships between the transformed geometries,
but I may have dreamt about it :)
Cheers
Andrea
--
Martin Davis
Senior Technical Architect
Refractions Research, Inc.
(250) 383-3022
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