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| GeoCommunity Mailing List |
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| Subject: | Re: [gislist] 90 degrees |
| Date: |
07/08/2004 12:45:01 PM |
| From: |
Quantitative Decisions |
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At 07:05 AM 7/8/2004 +0000, Pradeep Kumar wrote:
>I am trying to find a tool which makes all the corners of polygons or
>polylines 90 degrees in ArcView.
Easy enough: rasterize these features, then convert the raster image back
to vector format by tracing the outlines of the cells. ArcView with its
Spatial Analyst extension will do this quickly.
That's usually not a nice solution, though: it forces all lines to be
isothetic (parallel to the coordinate axes, meaning they must be exactly
right-left or up-down), and it can add many unwanted vertices.
To clean up digitized vector features, one wants a solution that
* Maintains the shape as closely as possible
* Moves the vertices as little as possible
* Does not create or remove vertices
* Allows for non-isothetic right angles.
This can be accomplished by solving a constrained optimization problem. As
examples of constraints, one could require that the low-order moments (such
as the centroid of the vertices and their second moments) not be changed,
or that the area be preserved. These constraints are needed because the
angles alone do not determine the position or size of the feature. The
function to optimize must take its minimum when all angles are multiples of
90 degrees. A good choice is the sum of squares of sines of twice the
angles. To this you can add some multiple of the sum of squared distances
the vertices are moved so that the overall appearance of the shape is
preserved as much as possible. The multiple is a "tuning parameter" that
lets you balance the twin objectives of creating right-angled vertices
while only slightly moving the vertices. Although this problem is strongly
nonlinear, it is pretty well behaved (convex, differentiable, etc.), and so
is not difficult to solve, despite the potentially large number of
variables (two for each vertex).
Others have asked this question before (twice in 2001 on the ArcView-L list
server). At the time I created an Excel spreadsheet to test and illustrate
the optimization approach. The spreadsheet uses Excel's Solver add-in to
find a solution. In tests with small realistic figures, and with some
artificial figures (like circles, so that we are attempting to "square the
circle"), it does very well. For your benefit, it will be available
briefly on my Web site at
http://www.quantdec.com/misc/arcview/squaring_polygons.xls .
For a full-blown industrial-strength implementation I recommend using a
nonlinear solver, such as that available from IMSL, and packaging it as a DLL.
--Bill Huber
Quantitative Decisions
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