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Union and Intersection of Polygons
• Union =
• Intersection =
Areal interpolation
• How to compute new values for polygons formed of preexisting ones– Simple when polygons are nested
A B
C D
What is “total” populationof the solid box?
Areal interpolation
– More difficult when new units are not nest-able– typical example 1980 census units compared to
1990 -• because of population growth the 1990 units are
smaller than the 1980 units and overlap boundaries
1980 blocks 1990 blocks
Methods to interpolate– Use proportionate areas
• e.g. convert total population to population density
• compute area for new units and reallocate
– use centroids of (a large number) of locations as x, y and z - interpolate z and reallocate into new polygons
– use other factors to model interpolation• e.g. consider land use for area and distribute original
population differentially in unit before re-allocation
1980 blocks 1990 blocks
Measurement of shapes of polygons– In addition to attributes and areas of polygons measures
reflecting shapes may also be important• examples
– congressional districts– package delivery, newspaper zones
– A variety of possible measures• measures of compactness (area to perimeter measures) etc.
Polygon clipping
• Creation or retrieval o polygons “included” in a second polygon– examples
• Parcels in city limits
• soils in river floodplain
Buffering
– General process• creation of new entities defined by boundaries (inside or
outside or both) of existing entities offset by a particular distance and parallel to the original boundary
Buffering continued– Ways in which multiple ‘starting entities are considered is important– ways in which buffer zones are viewed
Two entities
One entity
“Donut” entities
Multiple discs thatoverlap in space
More on buffering
– Other issues • termination of buffer entities
Multiple entities can lead to very complex polygon structures
Polygon overlay
• probably the most challenging computational problem in spatial systems
• general process– identify line segments involved - preferably having topology
– establish minimum enclosing rectangle
– determine if line segments in one polygon are inside the other by a point-in-polygon
– find intersections of segments that represent boundaries
– create records for new line segments and associated topology
– re-label all polygons
Examples
Draw boxes around polygons boxes don’t overlap polygons don’t
Boxes overlap - 1. check for point in polygon node/vertex of B is in A2. Identify intersections3. Create new nodes
A
B
Problems with polygon overlay
• non-coincidence of lines
• slivers
• line coincidence
Pre-post topology and polygon overlay
• different systems deal with topology (and latter polygon overlay) differently
• topology as part of data development– most GIS systems Arc, GRASS etc.
• topology built as analysis needed– MGE
• No clipping/overlay– ArcView 3.0 now in 3.1
– GeoMedia 2.0 but in Geomedia 2.0
• strengths and weaknesses
Data transformations
• changes in dimensionality– ex. Polygon to line to point as scale changes
• changes in position– ex; map projections,
– scaling,
– rotation,
– rubber sheeting
– affine
– projective (as in some map projection)
– topological
Types of transformations
Geometry properties Permitted
Equi-area area preserved rotation
similarity shape preserved scaling
affine angular distortion parallelismpreserved
projective angular and lengthdistorted
rubber sheeting
topology topo propertiespreserved
a hole is still ahole
Conflation
• “merging” of two data sources often from two scales or level of detail– ex: detailed transportation map and street names from TIGER
Easy Street
North Street
100 200 300