Maps, Coordinate Systems and GIS -...
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GeodesyGeodesy
Map ProjectionsMap Projections
Geodesy Geodesy -- the shape of the earth and definition the shape of the earth and definition of earth of earth datumsdatumsMap Projection Map Projection -- the transformation of a curved the transformation of a curved earth to a flat mapearth to a flat mapCoordinate systems Coordinate systems -- (x,y) coordinate systems (x,y) coordinate systems for map datafor map data
GeodesyGeodesy
Determination the Earth’s size (geometry), shape (gravity) and figure (surface).
Determination of the Earth’s motions (in space: polar motion, variations in rotation rate), its deformations (e.g., plate tectonic motion, plate boundary deformation, volcanoes, land subsidence), and gravity variations.
Definition and maintenance of terrestrial reference frames (datums) for precise 3D positioning, thus providing the backbone for mapping, surveying, and GIS.
GeodesyGeodesy
GeodesyGeodesy
Triangulation from Dunkirk to Barcelona
Jean Baptiste Delambre measuredthe stations between Dunkirk and Rodez, France. The southern segment,from Rodez to Barcelona, was measured by Pierre Méchain. They began theproject in 1792.
GeodesyGeodesy
Spherical EarthSpherical Earth
Authalic SphereAuthalic SphereBasic figure for mappingBasic figure for mappingRadius = 6371 kmRadius = 6371 kmMeridians from pole to Meridians from pole to polepoleEquator and small circles Equator and small circles perpendicular to the perpendicular to the meridiansmeridiansGeographic grid of Geographic grid of meridians and small meridians and small circlescircles
GeodesyGeodesy
Latitude and Longitude on a SphereLatitude and Longitude on a Sphere
Meridian of longitude
Parallel of latitude
ϕ
λ
X
Y
N
E
ϕ
Z
W
ϕ=0-90°S
P
OR
λ=0-180°E
ϕ=0-90°N
•
Greenwichmeridian
λ=0°
•
Equator =0°
•
•λ=0-180°W
λ - Geographic longitudeϕ - Geographic latitude
R - Mean earth radius
O - Geocenter
GeodesyGeodesy
Spherical Earth ApproximationSpherical Earth Approximation
Y: Right-handed
Z = Pole of Rotation
X = Greenwich Meridian
GeodesyGeodesy
What kind of ellipsoid?What kind of ellipsoid?
Cassini's report to the Academy, that the length of a degree seemed to get shorter towards the pole, generated an intense controversy between French and English scientists and resulting in arc measurement expeditions to Lapland (1736/37, average latitude 66°20’) and Peru/Ecuador (1739-1743, (average latitude 1° 31' S).
Oblate? Prolate?
GeodesyGeodesy
Ellipsoid or SpheroidEllipsoid or SpheroidRotate an ellipse around an axisRotate an ellipse around an axis
O
X
Z
Ya ab
Rotational axis
The earth is flattened slightlyThe earth is flattened slightlyat the poles and bulges at the poles and bulges somewhat at the equatorsomewhat at the equator
GeodesyGeodesy
Definition of Latitude,Definition of Latitude, φφ
(1) Take a point S on the surface of the ellipsoid and define there the tangent plane, mn(2) Define the line pq through S and normal to thetangent plane(3) Angle pqr which this line makes with the equatorialplane is the latitude φ, of point S
O φ
Sm
nq
p
r
GeodesyGeodesy
By the early 19th century, scientists like Laplace (1802), Gauss (1828), Bessel (1837) recognized that the assumption of an ellipsoidal earth model was untenable under sufficiently high observational accuracy. One could no longer ignore the deviation of the physical plumb line, to which measurements refer, from the ellipsoidal normal.
CM
CE
Geoid – Figure of the Earth
GeodesyGeodesy
Standard EllipsoidsStandard Ellipsoids
Ellipsoid Major axis, a (m)
Minor axis, b (m)
Flattening ratio, f
Airy (1830)
6377,563 6356,257 1/299,32
Clarke (1866)
6,378,206 6,356,584 1/294.98
WGS 84 (1984)
6,378,137 6,356,752 1/298.57
Ref: Snyder, Map Projections, A working manual, USGSProfessional Paper 1395, p.12
GeodesyGeodesy
What is a Projection?What is a Projection?If you could project light from a source through the If you could project light from a source through the earth's surface onto a twoearth's surface onto a two--dimensional surface, you dimensional surface, you could then trace the shapes of the surface features could then trace the shapes of the surface features onto the twoonto the two--dimensional surface. dimensional surface. This twoThis two--dimensional surface would be the basis for dimensional surface would be the basis for your map.your map.
GeodesyGeodesy
Projections always distort the mapProjections always distort the map
Rectangularform
Sides of different lenght
GeodesyGeodesy
MapMap projectionsprojections
The The characteristicscharacteristics normallynormally consideredconsidered in in choosingchoosing a a mapmap projectionprojection are as are as followsfollows::
1. Area 1. Area -- equalequal--areaarea 2. 2. ShapeShape –– conformalconformal
3. 3. ScaleScale –– oneone or or moremore lineslines on the on the mapmap alongalong whichwhich thethescalescale remainsremains truetrue
4. 4. DirectionDirection –– conformalconformal, , azimuthalazimuthal
5. Special 5. Special –– gnomonicgnomonic 6. 6. MethodMethod of of constructionconstruction
GeodesyGeodesy
Earth to Globe to MapEarth to Globe to Map
Representative Fraction
Globe distanceEarth distance
=
Map Scale: Map Projection:
Scale Factor
Map distanceGlobe distance
(e.g. 0.9996)
=
(e.g. 1:50,000)
GeodesyGeodesy
Types of ProjectionsTypes of Projections
ConicConic (Albers Equal Area, Lambert (Albers Equal Area, Lambert Conformal Conic) Conformal Conic) -- good for Eastgood for East--West West land areasland areasCylindricalCylindrical (Transverse Mercator) (Transverse Mercator) -- good good for Northfor North--South land areasSouth land areasAzimuthalAzimuthal (Lambert (Lambert AzimuthalAzimuthal Equal Area) Equal Area) -- good for global viewsgood for global views
GeodesyGeodesy
Cylindrical ProjectionsCylindrical Projections(Mercator)(Mercator)
Transverse
Oblique
Normal
GeodesyGeodesy
Types of Coordinate SystemsTypes of Coordinate Systems
(1) Global Cartesian(1) Global Cartesian coordinates (X,Y,Z) coordinates (X,Y,Z) for the whole earthfor the whole earth(2) Geographic(2) Geographic coordinates (coordinates (φ, λφ, λ, z) , z) (3) Projected(3) Projected coordinates (x, y, z) on a coordinates (x, y, z) on a local area of the earthlocal area of the earth’’s surfaces surfaceThe zThe z--coordinate in (1) and (3) is coordinate in (1) and (3) is defined defined geometricallygeometrically; in (2) the z; in (2) the z--coordinate is defined coordinate is defined gravitationallygravitationally
GeodesyGeodesy
Global CartesianGlobal Cartesian Coordinates Coordinates (X,Y,Z)(X,Y,Z)
O
X
Z
Y
GreenwichMeridian
Equator
•
GeodesyGeodesy
Geographic CoordinatesGeographic Coordinates ((φ, λφ, λ, z), z)
Latitude (Latitude (φφ) and Longitude () and Longitude (λλ) defined ) defined using an using an ellipsoidellipsoid, an ellipse rotated about , an ellipse rotated about an axisan axisElevation (z) defined using Elevation (z) defined using geoidgeoid, a , a surface of constant gravitational potentialsurface of constant gravitational potentialEarth Earth datumsdatums define standard values of define standard values of the ellipsoid and the ellipsoid and geoidgeoid
GeodesyGeodesy
CoordinateCoordinate SystemSystem
(φo,λo)(xo,yo)
X
Y
Origin
A planar coordinate system is defined by a pairof orthogonal (x,y) axes drawn through an origin
GeodesyGeodesy
Cartesian Coordinate SystemCartesian Coordinate System
Planar coordinate systems are based on Planar coordinate systems are based on Cartesian coordinates.Cartesian coordinates.
GeodesyGeodesy
Any projected data that you add to Any projected data that you add to ArcMapArcMap, or that , or that you project within you project within ArcMapArcMap, is associated with a , is associated with a projected coordinate system (PCS) in addition to its projected coordinate system (PCS) in addition to its underlying Geographic Coordinate System (GCS). underlying Geographic Coordinate System (GCS).
GeodesyGeodesy
Universal Transverse MercatorUniversal Transverse Mercator(UTM)(UTM)
Divide the world into sixty vertical strips, each spanning Divide the world into sixty vertical strips, each spanning six degrees of longitude. Apply a custom Transverse six degrees of longitude. Apply a custom Transverse Mercator projection to each strip and use false Mercator projection to each strip and use false eastingseastingsand and northingsnorthings to make all projected coordinates positive. to make all projected coordinates positive.
Data that crosses zones is subject to distortion.Data that crosses zones is subject to distortion.
A comprehensive system for identifying locations and making measurements over most of the earth's surface.
GeodesyGeodesy
ArcGISArcGIS Reference FramesReference FramesDefined for a Defined for a feature feature datasetdataset in in ArcCatalogArcCatalogCoordinate SystemCoordinate System
ProjectedProjectedGeographicGeographic
X/Y DomainX/Y DomainZ DomainZ Domain
GeodesyGeodesy
Coordinate SystemsCoordinate Systems
GeographicGeographiccoordinates (decimal coordinates (decimal degrees)degrees)ProjectedProjected coordinates coordinates (length units, ft or (length units, ft or meters)meters)
GeodesyGeodesy
Datum Transformations7-parameter transformation
Geodetic tools available at: http://www.ngs.noaa.gov/TOOLS/
NAD27 to NAD 83
GeodesyGeodesy
Summary ConceptsSummary Concepts
Two basic Two basic locationallocational systems: systems: geometricgeometricor Cartesian (x, y, z) and or Cartesian (x, y, z) and geographicgeographic or or gravitational (gravitational (φ, λφ, λ, z), z)Mean sea level surface or Mean sea level surface or geoidgeoid is is approximated by an ellipsoid to define an approximated by an ellipsoid to define an earth earth datumdatum which gives (which gives (φ, λ) φ, λ) and and distance above distance above geoidgeoid gives (z)gives (z)