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

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4

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2

1

0

0 1 2 3 4 5

2

Spatial Concepts

• Euclidean space– geometric

primitives:

• points

• lines

• polygons

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0 1 2 3 4 5

Spatial Concepts

• Network space– connectors

between points

– can be directed network

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0 1 2 3 4 5

Spatial Concepts

• Metric space– distance

between 2 points is the same no matter the direction of measurement

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0 1 2 3 4 5

33

Spatial Concepts• Measuring distance is different in Euclidean space

than in network space

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00 1 2 3 4 50 1 2 3 4 5

Euclidean Network

Spatial Concepts• Measuring distance is different in Euclidean space

than in network space

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00 1 2 3 4 50 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

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2 2

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Euclidean Network

Spatial Concepts

• Is this example of Network space a metric space?

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00 1 2 3 4 5

Distance(4,4 to 1,1) = ?5

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00 1 2 3 4 5

Distance(1,1 to 4,4) = 2 + 2 + +1 + 2 + 1 = 8

2 2

1

2

1

Spatial Concepts

• Topologic space

– generally, spatial relationships between geometric elements

adjacency

containment

neighborhood

A

A

B

B

aa’

Spatial Concepts

• Topologic space

Spatial ConceptsSpatial Measurement Levels

Spatial ConceptsSpatial Measurement Levels

Spatial Concepts

• Spatial pattern concepts:– proximity - the quality of being near something

1 mile

Spatial Concepts

• Spatial pattern concepts:– spatial arrangement

regular clustered random

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 SystemsPlanar 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