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 AB C D What is “total” population of the solid box?

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 blocks1990 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 blocks1990 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 Two entities One entity “Donut” entities Multiple discs that overlap in space

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

11. 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 A 2. Identify intersections 3. Create new nodes A B

12. 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

15. Types of transformations

16. 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