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

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 North Street 300 200 100 Easy Street