1. Spatial Concepts Mathematical Types of Space –Euclidean –Network –Metric –Topologic

2. Spatial Concepts Euclidean space –Regular coordinate space that allows for the measurement of distance and bearing 543210543210 0 1 2 3 4 5 2

3. Spatial Concepts Euclidean space –geometric primitives: points lines polygons 543210543210 0 1 2 3 4 5

4. Spatial Concepts Network space –connectors between points –can be directed network 543210543210 0 1 2 3 4 5

5. Spatial Concepts Metric space –distance between 2 points is the same no matter the direction of measurement 543210543210 0 1 2 3 4 5 33

6. Spatial Concepts Measuring distance is different in Euclidean space than in network space 543210543210 543210543210 0 1 2 3 4 5 EuclideanNetwork

7. Spatial Concepts Measuring distance is different in Euclidean space than in network space 543210543210 543210543210 0 1 2 3 4 5 Distance(1,1 to 4,4) = sq. root of [(4-1) sq. + (4-1) sq.] = 4.24 Distance(1,1 to 4,4) = 2 + 2 + +1 + 2 + 1 = 8 3 3 22 1 2 1 EuclideanNetwork

8. Spatial Concepts Is this example of Network space a metric space? 543210543210 0 1 2 3 4 5 Distance(4,4 to 1,1) = ? 543210543210 0 1 2 3 4 5 Distance(1,1 to 4,4) = 2 + 2 + +1 + 2 + 1 = 8 22 1 2 1

9. Spatial Concepts Topologic space –generally, spatial relationships between geometric elements adjacency containment neighborhood A A B B a a’

10. Spatial Concepts Topologic space

11. Spatial Concepts Spatial Measurement Levels

12. Spatial Concepts Spatial Measurement Levels

13. Spatial Concepts Spatial pattern concepts: –proximity - the quality of being near something 1 mile

14. Spatial Concepts Spatial pattern concepts: –spatial arrangement regularclusteredrandom

15. Spatial Concepts Spatial pattern concepts: –spatial orientation

16. Spatial Concepts Spatial pattern concepts: diffusion of things (objects, people, styles) moving over time

17. Urban density Percent Minority Spatial Concepts Spatial pattern concepts: spatial association

18. Projections and Coordinate Systems Map Projections –a means to depict the spherical earth on a two dimensional medium –cannot simultaneously preserve all of these properties of the earth in two dimensions: shape (of a region) distance (between two points) direction (bearing from one point to another) area (of a region)

19. Three families of map projections planar cylindrical conical Projections and Coordinate Systems

20. Conformal projection –preserves direction –e.g. Mercator Equal area projection –preserves area –e.g. Alber’s equal area Projections and Coordinate Systems

21. Mercator projection Equal area projection

22. Coordinate (grid) systems –provide a spatial referencing system to locate points on the earth surface –spherical coordinate system coordinates that describe locations on a sphere –planar coordinate system coordinates that describe locations within a two dimensional Cartesian space Projections and Coordinate Systems

23. Spherical coordinate system: latitude and longitude –parallel: lines parallel to equator –meridian: lines going from pole to pole Projections and Coordinate Systems

24. –rectangular grid –X and Y axes –X: easting –Y: northing –point is defined by X,Y coordinate Projections and Coordinate Systems Planar Coordinate System

25. Planar coordinate system: Universal Transverse Mercator (UTM) –units in meters –transverse cylindrical projection –60 vertical zones cover globe –northing in meters from equator –easting in meters from false origin 500,000 meters west of zone central meridian Projections and Coordinate Systems

26. Universal Transverse Mercator (UTM) Projections and Coordinate Systems

27. Planar coordinate system: United States State Plane Coordinate System –units in feet –projection in transverse Mercator or Lambert’s conformal conic –tied to U.S. datum –two or more zones for each state with a false origin southwest of the zone Projections and Coordinate Systems

28. United States State Plane Coordinate System Projections and Coordinate Systems