Touche Bill :-)
You're absulutely right!
Roeland
----- Original Message -----
From: Bill Huber <[EMAIL PROTECTED]>
To: Mapinfolist <[EMAIL PROTECTED]>
Sent: Tuesday, April 17, 2001 5:30 PM
Subject: MI-L Re: Calculating distance from a point to bodies of water
> At 04:22 PM 4/17/01 +0200, Roeland van der Spek wrote:
> >1. create points from the nodes in the polygons that represent the bodies
of
> >water
> >2. use Triangulator to create Thiessen polygons around these newly
created
> >nodes/points
> >3. calculate the distance to the node that is in the same Thiessen
> >polygon as the point you want to calculate the distance from....
>
> Unfortunately, this nice idea doesn't really work either.
>
> Recall that the question was to assign to each of many points its distance
> to the nearest "body of water"--a polygon, presumably. One approach is to
> divide the plane into regions; each region consists of all points closest
> to a given body of water. The problem is then solved with a
> point-in-polygon solver followed by a quick point-to-polygon distance
> calculation.
>
> This idea extends the notion of Thiessen (Voronoi, Dirichlet)
> polygon. However, it is a distinctly different construct: in particular,
> the region boundaries usually contain parts of parabolas and so are not
> even polygons in the usual sense.
>
> For example, consider a long straight stretch of river, part of whose
> boundary is represented by a line segment between two widely separated
> points. Let's call this segment "D". Suppose that nearby is a pond or
> lake (such as a cutoff oxbow). To simplify the analysis, imagine this
pond
> is so small that we can practically represent it as a single point
> "F". (This simplification still produces the correct conclusions.)
>
> Locally, the generalized Thiessen region corresponding to D and F is
> bounded by the locus of points equidistant from D and F. This, by
> definition, is a parabola with directrix D and focus F. Since parabolas
> are everywhere non-linear, and (the usual) Thiessen polygons always have
> piecewise linear boundaries, NO Thiessen polygon software will generate a
> perfectly correct solution.
>
> In particular, the "Triangulator" approach described above will produce
> highly erroneous results in all such areas where a long line segment
> represents a portion of a polygon's boundary.
>
> One work-around is to represent polygon boundaries not just by their
nodes,
> but by their nodes together with a collection of tightly-spaced points
> marching around their perimeters. This in effect creates piecewise linear
> approximations to the parabolic arcs in step 2 above. It will also
greatly
> increase time and RAM requirements for the solution.
>
> It's no longer as "easy as 1-2-3," but it can be done. (I have taken this
> approach in ArcView, which doesn't do a bad job at all.) However, the
> approximation inherent in the method, together with its complexity,
suggest
> that a raster solution will be just as good as this vector approach.
>
> Preparata and Shamos (Computational Geometry: An Introduction,
> Springer-Verlag, 1985) indicate that an algorithm has been developed to
> compute the generalized Thiessen regions, but they do not provide
> references. I would be grateful to hear from anyone who can supply a
> reference or code, in any language.
>
> --Bill Huber
> Quantitative Decisions
>
>
>
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