At 5:07 PM +0000 1/28/08, Barker, Sean (UK) wrote:
>Pat  See below
>....
><SB>
>CAD is computationally intensive, and one doesn't want to carry round
>error intervals with every point, line etc. The problem is that many
>algorithms that make geometric calculations do so on the basis of
>topological properties (usually ones which map to the integers  is
>topology an ontology of geometry?)  for example, that you cannot go
>from the inside of a circle to the outside without crossing the
>boundary. (01)
Right. (02)
>The problem I was looking at was not how to represent points accurately,
>but in the presence of (unquantified) inaccuracy, to ensure that
>topological validity was preserved. (03)
Ah, that is hard. Arbitrarily small 'errors' can completely change
the topology, is the problem. (04)
>In practice, the geometrical errors
>were too small to worry about, but the topological errors were
>intractable, at least with the tools we had at the time.
>
>In the resulting system, intervals did not have endpoints, though you
>could separate two intervals by a point. Alternatively, you could say
>that all intervals were open sets, AS WERE THEIR COMPLIMENTS. (05)
There is a classical way to do this, which is to restrict oneself to
open regular subsets of points, and use as an addition operation the
interior of the unions of the closures. This 'fills up' any
singlepoint 'cracks' or breaks in the solids. Or use the dual:
closed sets and the addition is the closure of the union of the
interiors. This eliminates 'foils' one point thick. Sounds like one
of these might have been what you needed (?) (06)
>Or rather,
>a point could be classified against an interval as being IN, OUT or ON,
>with ON corresponding to the indeterminate value of the Logic of Partial
>Functions. (07)
Which logic of PFs? Do you mean 3valued logic? (Blech) (08)
>
>I should add, in the system I was working on, in 2D we did distinguish
>between points, lines and circles, but that was a matter of
>representation.
></SB>
>
> The discussion reminds me of the story of a man looking
>for his keys by
> a street lamp. When asked where he had dropped them, he
>pointed to a
> place a little way off  "but the light is much better
>over here".
>
>
> Old story, but I fail to see the relevance. The, er, point of
>points is that they are useful.
><SB>
>Its not that points are not useful, the problem is, if we both use the
>term "point", how much work do we need to do before the conversation
>makes sense. (09)
Well, see my recent post. If you agree with all the axioms, we can
communicate usefully without necessarily talking 'about' the same
notion of point. (010)
></SB>
>And, by the way, you can't not have them, in a sense. One can construct
>the points from the intervals. So in a sense they have to be there,
>whether you talk about them or not.
>
><SB>
>Not necessarily in the system I was using. (011)
If the system allowed intervals to meet and satisfied this axiom: (012)
IF a meets b and a meets c and d meets b
THEN d meets c (013)
then one can construct the points from the intervals mathematically. (014)
Pat (015)
></SB>
>
>
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