Spatial Concepts • Mathematical Types of Space – Euclidean – Network – Metric – Topologic
Dec 26, 2015
Spatial Concepts
• Euclidean space– Regular
coordinate space that allows for the measurement of distance and bearing
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Spatial Concepts
• Euclidean space– geometric
primitives:
• points
• lines
• polygons
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Spatial Concepts
• Network space– connectors
between points
– can be directed network
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Spatial Concepts
• Metric space– distance
between 2 points is the same no matter the direction of measurement
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Spatial Concepts• Measuring distance is different in Euclidean space
than in network space
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Euclidean Network
Spatial Concepts• Measuring distance is different in Euclidean space
than in network space
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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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Euclidean Network
Spatial Concepts
• Is this example of Network space a metric space?
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Distance(4,4 to 1,1) = ?5
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Distance(1,1 to 4,4) = 2 + 2 + +1 + 2 + 1 = 8
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Spatial Concepts
• Topologic space
– generally, spatial relationships between geometric elements
adjacency
containment
neighborhood
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Spatial Concepts
• Spatial pattern concepts:– proximity - the quality of being near something
1 mile
Spatial Concepts
Spatial pattern concepts: diffusion of things (objects, people, styles) moving over time
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)
• Conformal projection– preserves direction– e.g. Mercator
• Equal area projection– preserves area– e.g. Alber’s equal area
Projections and Coordinate Systems
• 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
• 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