Union and Intersection of Polygons

1
Union and Intersection of Polygons

• Union
• Intersection

2
Areal interpolation

• How to compute new values for polygons formed of
  preexisting ones
• Simple when polygons are nested

A
B
What is total population of the solid box?
C
D
3
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
4
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
5
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.

6
Polygon clipping

• Creation or retrieval o polygons included in a
  second polygon
• examples
• Parcels in city limits
• soils in river floodplain

7
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

8
Buffering continued

• Ways in which multiple starting entities are
  considered is important
• ways in which buffer zones are viewed

Donut entities
Two entities
Multiple discs that overlap in space
One entity
9
More on buffering

• Other issues
• termination of buffer entities

Multiple entities can lead to very complex
polygon structures
10
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

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Examples
Draw boxes around polygons boxes dont overlap
polygons dont
Boxes overlap - 1. check for point in polygon
node/vertex of B is in A 2. Identify
intersections 3. Create new nodes
A
B
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Problems with polygon overlay

• non-coincidence of lines
• slivers
• line coincidence

13
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

14
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

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Types of transformations
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Conflation

• merging of two data sources often from two
  scales or level of detail
• ex detailed transportation map and street names
  from TIGER

North Street
300
200
100
Easy Street