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Boolean Operatorsand
Topological OVERLAY FUNCTIONS IN GIS
Query – asking a question of the attribute data
Standard Query Language (SQL) is used to query the data
There are 4 basic statements used to get information from 2 (or more) datasets
AND – if you are desiring the subset of each dataset that is ‘true’ of both datasets
OR – if you are desiring the subset of each dataset that is ‘true’ of either one or both datasets
NOT – if you are desiring the subset of one dataset that is only true of one dataset
OR, BUT NOT BOTH (XOR) – if you are seeking the subset of data that is ‘true’ of one and another dataset, but not both datasets
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Conceptual Design: Overlay Operations
In this design, each data set is represented by a circle:
A B
Topological Overlay Operations
Querying the dataset databases can be done several different ways, but they always use the same type of query language.
A BANDOR
NOTXOR
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Boolean Operators: AND
A B
A AND B = True if Both
Boolean Operators: AND
A AND B = True if Both
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Topological Overlay Operations
Querying the dataset databases can be done several different ways, but they always use the same type of query language.
A BANDOR
NOTXOR
Boolean Operators: OR
AA B
A OR B = True if one or other
A
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Boolean Operators: OR
A
A OR B = True if one or other
Topological Overlay Operations
Querying the dataset databases can be done several different ways, but they always use the same type of query language.
A BANDOR
NOTXOR
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Boolean Operators: NOT
AA
A NOT B = True if Neither
B
Boolean Operators: NOT
AA
A NOT B = True if Neither
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Topological Overlay Operations
Querying the dataset databases can be done several different ways, but they always use the same type of query language.
A BANDOR
NOTXOR
Boolean Operators: A XOR B
A B
A OR B, but not both (XOR)
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Boolean Operators: A XOR B
A OR B, but not both (XOR)
Union is an AND operation that produces a 3rd output dataset
BCA
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Polygon on Polygon Vector Overlay Operations
CLIP
SELECT (NOT)
SPLIT
XOR
UNION
INTERSECT
INPUT LAYER 1 INPUT LAYER 2 OUTPUT LAYER
Splits 1 into manylayers based on 2
Overlays polygons and keeps all of both [1 OR 2]
Overlays but keeps onlyportions of layer 1that fallwithin layer 2 [1 AND 2]
Cuts out a piece of layer 1using layer 2 as cookie cutter[1 AND 2]
Erases (deletes) part of layer 1using layer 2 [1 NOT 2]
Layer 1 or 2, but not both[1 XOR 2]
Dissolve Operation
Change in geometry based on common attribute values
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Clip Operation
The “Clip feature” is used as a cookie cutter
Buffer Operation
Proximity is measured from target features
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Union and Intersect
Union
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Union
Line in Polygon VectorOverlay Operations
Point in Polygon Vector Overlay Operations
INPUT LAYER 1
INPUT LAYER 2
OUTPUT LAYER
INPUT LAYER 1
INPUT LAYER 2
OUTPUT LAYER
Even or Zero intersects to each side = NO
Odd intersects to each side = YES
MinimumBounding Box –
MinimumBounding Box –
Even or Zero intersects to each side = NO
Odd intersects to each side = YES
Also called the ‘even-odd rule algorithm’
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Representing The Shape of the Earth
•Geoids•Ellipsoids •Datums•Coordinate Systems•Projections
The Shape of the Earth3 ways to model it
• Topographic surface• the land/air interface
• complex (rivers, valleys, etc) and difficult to model
• Geoid• a theoretical, continuous surface for the earth which is perpendicular at every point
to the direction of gravity (surface to which plumb line is perpendicular)
• approximates mean sea‐level in open ocean without tides, waves or swell
• satellite observation (after 1957) showed it to be quite irregular because of local variations in gravity.
• Spheres and spheroids (3‐dimensional circle and ellipse)• mathematical models which can be used to approximate the geoid and provide the
basis for accurate location (horizontal) and elevation (vertical) measurement
• sphere (3‐dimensional circle) with radius of 6,370,997m considered ‘close enough’ for small scale maps (1:5,000,000 and below ‐ e.g. 1:7,500,000)
• spheroid (3‐dimensional ellipse) should be used for larger scale maps of 1:1,000,000 or more (e.g. 1:24,000)
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Spheroid, Ellipsoid, and Geoid• Spheroid is a solid generated by rotating an ellipse about either the major
or minor axis
• Ellipsoid is a solid for which all plane sections through one axis are ellipses and through the other are ellipses or circles
• If any two of the three axes of that ellipsoid are equal, the figure becomes a spheroid (ellipsoid of revolution)
• If all three are equal, it becomes a sphere
• Geoid is the equipotential gravity surface of the earth at mean sea level. At any point it is perpendicular to the direction of gravity
Reference: Smith, James R., Introduction to Geodesy: The history and concepts of modern geodesy. John Wiley & Sons, 1997
What is an Oblate Ellipsoid (Spheroid)?
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Which Spheroid to use?
Hundreds have been defined depending upon:
• Available measurement technology
• Area of the globe – e.g North America, Africa
• Map extent – country, continent or global
• Political issues – e.g Warsaw pact versus NATO
• ArcGIS supports 26 different
spheroids! – conversions via math formulae
Most commonly encountered are:
• Clarke 1866 for North America
• basis for USGS 7.5 Quads
• a=6,378,206.4m b=6,356,583.8m
• GRS80 (Geodetic Ref. System, 1980)
• current North America mapping
• a=6,378,137m b=6,356,752.31414m
• WGS84 (World Geodetic Survey, 1984)
• current global choice
• a=6,378,137 b=6,356,752.31
Latitude and Longitude: location on the spheroid
Longitude meridiansPrime meridian is zero: Greenwich, U.K.International Date Line is 180° E&W
1 degree=69.17 mi at Equator53.06 mi at 40N/S
0 mi at 90N/S
Latitude parallels equator is zero
1 degree=68.70 mi at equator69.41 mi at poles
(1 mile=1.60934km=5280 feet)
Lat / long coordinates for a location change dependingon spheroid chosen!
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graticule: network of lines on globe or map representing latitude and longitude.Origin is at Equator/Prime Meridian intersection (0,0)
grid: set of uniformly spaced straight lines intersecting at right angles.(XY Cartesian coordinate system)
Latitude normally listed first (lat,long), the reverse of the convention for X,Y Cartesian coordinates
Latitude and Longitude Graticule
Latitude and Longitude
• The most comprehensive and powerful method of georeferencing
• Metric, standard, stable, unique
• Uses a well‐defined and fixed reference frame• Based on the Earth’s rotation and center of mass, and the Greenwich Meridian
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Definition of longitude. The Earth is seen here from above the North Pole, looking along the Axis, with the Equator forming the outer circle. The location of Greenwich defines the Prime Meridian. The longitude of the point at the center of the red cross is determined by drawing a plane through it and the axis, and measuring the angle between this plane and the Prime Meridian.
Definition of Latitude
• Requires a model of the Earth’s shape
• The Earth is somewhat elliptical• The N‐S diameter is roughly 1/300 less than the E‐W diameter
• More accurately modeled as an ellipsoid than a sphere
• An ellipsoid is formed by rotating an ellipse about its shorter axis (the Earth’s axis in this case)
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The Public Land Survey System (PLSS)
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The (Brief) History of Ellipsoids
• Because the Earth is not shaped precisely as an ellipsoid, initially each country felt free to adopt its own as the most accurate approximation to its own part of the Earth
• Today an international standard has been adopted known as WGS 84• Its US implementation is the North American Datum of 1983 (NAD 83)
• Many US maps and data sets still use the North American Datum of 1927 (NAD 27)
• Differences can be as much as 200 m
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Latitude and the Ellipsoid
• Latitude (of the red point) is the angle between a perpendicular to the surface and the plane of the Equator
• WGS 84• Radius of the Earth at the Equator 6378.137 km
• Flattening 1 part in 298.257
Geoid
• A geoid is a representation of the Earth which would coincide exactly with the mean ocean surface of the Earth, if the oceans were to be extended through the continents
• A smooth but highly irregular surface that corresponds but to a surface which can only be known through extensive gravitational measurements and calculations, not to the actual surface of the Earth's crust
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Geoid
Geoid vs. an Ellipsoid
1. Ocean2. Ellipsoid3. Local plumb4. Continent5. Geoid
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Datums:all location measurement is relative to a specific datum
For the Geodesist
• a set of parameters defining a coordinate system, including:
• the spheroid (earth model)
• a point of origin (ties spheroid to earth)
For the Local Surveyor
• a set of points whose precise location and /or elevation has been determined, which serve as reference points from which other point’s locations can be determined (horizontal datum)
• a surface to which elevations are referenced, usually ‘mean sea level’ (vertical datum)
• points usually marked with brass plates called survey markers or monuments whose identification codes and precise locations (usually in lat/long) are published
North American Datums
• NAD27• Clark 1866 spheroid
• Meades Ranch origin
• visual triangulation
• 25,000 stations • (250,000 by 1970)
• NAVD29 (North American Vertical Datum, 1929) provided elevation
• basis for most USGS 7.5 minute quads
NAD83
• satellite (since 1957) and laser distance data showed inaccuracy of NAD27
• 1971 National Academy of Sciences report recommended new datum
• used GRS80 spheroid
(functionally equivalent to WGS84)
• origin: Mass‐center of Earth
• 275,000 stations
• “Helmert blocking” least squares technique fitted 2.5 million other fed, state and local agency points.
• NAVD88 provides vertical datum
• points can differ up to 160m from NAD27, but seldom more than 30m, and data from digit. map more inaccurate than datum diff.
• no universal mathematical formulae for conversion from NAD27: See USGS Survey Bulletin # 1875 for conversion tables (in ARC/INFO). Transformations are preformed to local coordinates
• http://www.ngs.noaa.gov/cgi‐bin/nadcon.prl
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NAD27 and NAD83 Ellipsoids (Canadian Spacial Reference System, 2006)
Ground‐zero for Geo‐nerds everywhere
Meades Ranch, KS (12 miles north of Lucas, KS) is the designated geodedetic base point for the North American Datum 1927 (NAD 27)
Owner of ranch is now Mr. Kyle Brant•access with permission only
http://www.scottosphere.com/history/meades‐ranch.html
The NGS data sheet is here:http://www.ngs.noaa.gov/cgi‐bin/ds2.prl?retrieval_type=by_pid&PID=KG0640
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State Plane & the NAD 27Calculations for map projections are performed using the parameters of the ellipsoid
Roadside Marker for the Geodetic Center of North America
Meades Ranch, KS (12 miles north of Lucas, KS) is the designated geodedetic base point for the North American Datum 1927 (NAD 27)
http://www.worldslargestthings.com/kansas/geodetic.htm
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Continuously Operating Reference Stations (CORS)
• http://www.ngs.noaa.gov/CORS/cors‐data.html
• Each CORS site provides Global Navigation Satellite System (GNSS ‐ GPS and GLONASS) carrier phase and code range measurements in support of 3‐dimensional positioning activities throughout the United States and its territories
• Surveyors, GIS/LIS professionals, engineers, scientists, and others can apply CORS data to position points at which GNNS data have been collected
• The CORS system enables positioning accuracies that approach a few centimeters relative to the National Spatial Reference System, both horizontally and vertically
High Accuracy Reference Network (HARN)
• The generic acronym HARN is now used for both HARN and the High Precision Geodetic Network (HPGN)
• A HARN is a statewide or regional upgrade in accuracy of NAD 83 coordinates using Global Positioning System (GPS) observations
• HARNs were observed to support the use of GPS by Federal, state, and local surveyors, geodesists, and many other applications
• Horizontal relative accuracies range from 5mm± to 8mm± at 1:1,000,000
• Of these 16,000 stations, NGS has committed to maintaining about 1,400 survey stations, named the Federal Base Network, and the various states will maintain the remainder
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Map Projections: the concept
A method by which the curved 3‐D surface of the earth is represented on a flat 2‐D map surface
a two dimensional representation, using a plane coordinate system, of the earth’s three dimensional sphere/spheroid
location on the 3‐D earth is measured by latitude and longitude
location on the 2‐D map is measured by x,y Cartesian coordinates
unlike choice of spheroid, choice of map projection does not change a location’s lat/long coords, only its XY coords.
Map projections
• Earth spherical ‐maps flat!
• Thus all maps have distortions
• A good map has distortions that are predictable and systematic
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Map projection
Why is it called a ‘projection’?
Because we ‘project’ the earth’s spherical surface on a flat surface‐
As if we were shining a light from center of earth:
Maps can be:
1. Conformal: Shape correct
2. Equivalent (Equal area): Area correct
3. Azimuthal: Direction correct
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Physical Surface
Use a physical surface to project the sphere
1. Plane2. Cone3. Cylinder
Note differences between projections by comparing distortions of the lines of latitude and longitude.
1. Plane projection
Hold plane against the surface of the globe (typically the pole)
Lines of longitude straight, radiating
Lines of latitudes are circles
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1. Plane projection
4. Distortion increases away from the center ("principal point")
5. Good for polar regions
6. But can't show more than half the world.
2. Conic projection
• Hold cone over pole
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2. Conic projection
2. Lines of latitude curve
3. Line of longitude are straight, and convergetowards top
4. Distortion increases away from standard parallel
5. Good for Mid‐latitudes
3. Cylindrical projection
1. Wrap cylinder around the earth
2. Lines of latitude and longitude are straight, intersect at 90°
3. Distortion greatest at poles
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3. Cylindrical projection
4. Good for low‐latitude areas
5. Poor representation of poles
Good for navigation
Mercator’s Projection
Other ‐mathematical
• Condensed and interrupted projections.