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

Spatial Concepts Euclidean space –Regular coordinate space that allows for the measurement of distance and bearing

Spatial Concepts Euclidean space –geometric primitives: points lines polygons

Spatial Concepts Network space –connectors between points –can be directed network

Spatial Concepts Metric space –distance between 2 points is the same no matter the direction of measurement

Spatial Concepts Measuring distance is different in Euclidean space than in network space EuclideanNetwork

Spatial Concepts Measuring distance is different in Euclidean space than in network space Distance(1,1 to 4,4) = sq. root of [(4-1) sq. + (4-1) sq.] = 4.24 Distance(1,1 to 4,4) = = EuclideanNetwork

Spatial Concepts Is this example of Network space a metric space? Distance(4,4 to 1,1) = ? Distance(1,1 to 4,4) = =

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

Spatial Concepts Topologic space

Spatial Concepts Spatial Measurement Levels

Spatial Concepts Spatial Measurement Levels

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

Spatial Concepts Spatial pattern concepts: –spatial arrangement regularclusteredrandom

Spatial Concepts Spatial pattern concepts: –spatial orientation

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

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

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)

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

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

Mercator projection Equal area projection

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

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

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

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

Universal Transverse Mercator (UTM) Projections and Coordinate Systems

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

United States State Plane Coordinate System Projections and Coordinate Systems