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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. 5 4 3 2 1 0. 2. 0 1 2 3 4 5. Spatial Concepts. Euclidean space

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## Spatial Concepts

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**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 5 4 3 2 1 0 2 0 1 2 3 4 5**Spatial Concepts**• Euclidean space • geometric primitives: • points • lines • polygons 5 4 3 2 1 0 0 1 2 3 4 5**Spatial Concepts**• Network space • connectors between points • can be directed network 5 4 3 2 1 0 0 1 2 3 4 5**Spatial Concepts**• Metric space • distance between 2 points is the same no matter the direction of measurement 5 4 3 2 1 0 3 3 0 1 2 3 4 5**Spatial Concepts**• Measuring distance is different in Euclidean space than in network space Euclidean Network 5 4 3 2 1 0 5 4 3 2 1 0 0 1 2 3 4 5 0 1 2 3 4 5**Spatial Concepts**• Measuring distance is different in Euclidean space than in network space Euclidean Network 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 5 4 3 2 1 0 5 4 3 2 1 0 1 2 3 1 2 2 3 0 1 2 3 4 5 0 1 2 3 4 5**Spatial Concepts**• Is this example of Network space a metric space? Distance(1,1 to 4,4) = 2 + 2 + +1 + 2 + 1 = 8 Distance(4,4 to 1,1) = ? 5 4 3 2 1 0 5 4 3 2 1 0 1 2 1 2 2 0 1 2 3 4 5 0 1 2 3 4 5**Spatial Concepts**• Topologic space • generally, spatial relationships between geometric elements A B adjacency A B containment a’ a neighborhood**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 regular clustered random**Spatial Concepts**• Spatial pattern concepts: • spatial orientation**Spatial Concepts**Spatial pattern concepts: diffusion of things (objects, people, styles) moving over time**Spatial Concepts**• Spatial pattern concepts: spatial association Percent Minority Urban density**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)**Projections and Coordinate Systems**• 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**Mercator projection**Equal area projection**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**Planar Coordinate System • rectangular grid • X and Y axes • X: easting • Y: northing • point is defined by X,Y coordinate**Projections and Coordinate Systems**• 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

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