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Map Projection and Coordinates

The shape of the earthModels

Sphere with radius of ~6378 km Ellipsoid (or Spheroid) with equatorial radius (semi-major axis) of ~6378 km and polar radius (semi-minor axis) of ~6357 km

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Difference of ~21 km usually expressed as “flattening” (f ) ratio of the ellipsoid: f = difference/major axis

= ~1/300 for earth

The Shape of the Earth

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a = Semimajor axis b = Semiminor axis X, Y, Z = Reference

frame

Spheroid is the result of rotating an ellipse around an axis


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Ellipsoid– The Earth revolves

easterly on its axis, generating centrifugal force, causing a bulge in the middle and flattening at the poles

Geoid – Representation of

the Earth as an equigravitationalsurface

Due to variations in gravity, the geoid does not follow the ellipsoid exactly (100 m deviation)

Geoid height –difference between ellipsoid and geoid(geoid undulation)

© Paul Bolstad, GIS Fundamentals


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Many attempts to measure the size and shape of the earth’s spheroid.An estimate of ellipsoid allows calculation of the elevation of every point on earth including the sea level.A base reference for the adjustment of geodetic measurements is called datum.

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Why different EllipsoidsBetter methods for approximating the surface of the EarthDeveloped from different areas of the EarthJust an approximation…


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We have to know datum and ellipsoid when using a GPS (Global Positioning System) receiver.In Thailand, a topographic map is based on Indian datum, and Everest Spheroid.


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Datum

The ellipsoid is a mathematical model that describes the shape of the earth. A datum defines the position of the ellipsoid relative to the center of the earthProvides a frame of reference for measuring locations on the surface of the earthAligns its spheroid to closely fit the earth’s surface in a particular area

© ESRI, ARC/INFO Help

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Horizontal Datum - A base reference for a coordinate system. It includes the latitude and longitude and orientation of an initial point of origin and an ellipsoid that models the surface of the earth in the region of interest.

Datum

Y

Prime Meridian

X

Z

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Traditional Horizontal Datums

Limitations to the Traditional Approach

NAD 27(Clarke Ellipsoid )

ED 50(International Ellipsoid)

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Datums

Global replaces regional datums with a common, accurate standardOne system for maps of the entire planet

Regional vs. Global Approach

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DatumsNorth American Datums– NAD27 (North American Datum of 1927)– NAD83 (North American Datum of 1983)

World Datums – WGS84 (World Geodetic System of 1984)

• Used for GPS

Other Datums– European Datum of 1979– Ordnance Survey Datum of Great Britain, 1936– Indian Datum 1975

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The Graticule

Graticule: the pattern of meridians and parallels on the earth based on– Equally spaced

parallels between the equator and the poles

– Parallels are always parallel to one another, so they are always the same distance apart

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Parallels and Meridians

Meridians are also known as lines of longitude– Example is the prime

meridianParallels are also known as lines of latitude– Example is the equator


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The Graticule– Meridians are spaced farthest apart on

the equator and converge to a single point at the poles

– Parallels and meridians cross one another at right (90o) angles

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Concept of a Projection

A systematic rendering of a graticuleProjection Surfaces– Plane– Developable Surfaces

• Cone• Cylinder

Different projections create different distortions


© ESRI, ArcView Help

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AGS 722 (2-2003) © ESRI, Understanding Map Projections

ProjectionsConformal Projections – Conformal projections preserve

local shape. They show the perpendicular graticule lines intersecting at 90 degree angles

– However, an area may become more distorted to maintain the angles

Equal Area Projections – preserve the area of features.

Other properties such as shape, angle, and scale are distorted.

Equidistant projections– preserve distances between

certain points. Scale is not maintained correctly, however, typically one or more lines has its scale maintained.

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Planar Projections

Project map data onto a flat surface touching the globeUsually tangent to the globe, but may be secantTangent point is focus of the projection

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Conic Projections Conic Projection - tangent to the globe along a line of latitude– distortion increases away

from the standard parallelConic projections are used for mid-latitude zones that have an east-to-west orientationSomewhat more complex conic projections contact the global surface at two locations– called secant conic

projections


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Cylindrical Projections

Mercator is most common cylindrical projectionEquator is typically the line of tangencyMeridians are of equal space, lines of latitude increases toward polesDisplays true direction along straight linesTransverse projections use meridian lines as tangent point, therefore, North/South lines are preserved


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Azimuthal Projections

Gnomonic projectionviews the surface data from the center of the Earth, Stereographic projection views it from pole to pole. Orthographic projection views the Earth from an infinite point, as if viewed from deep space.

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Major Properties of Projections

Conformality – retention of correct angles. Preserves shape, but not size. Also allows for accurate directionsEquivalence – retains unit area. Good for measuring area phenomena (amount of arable land). There are scale changes however, and changes in shape

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Projection Effects

Conformal projections– Mercator, Lambert conformal

Equivalence– Sinusoidal

Equidistance– Azimuthal

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Compromise Projections

Miller Cylindrical – similar to Mercator, but distorts size somewhat less– However, its not conformal, and lack the

directional integrityRobinson – minimizes visually disturbing distortions

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Different Projections

Mercator projection – conformality

Sinusoidal – equivalence

Millers Cylindrical – compromise

Robinson - compromise

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Basic Coordinate-System Concepts:

The fundamental spherical coordinate system used to locate positions on the earth’s surface.A way of representing the earth’s spherical shape as a flat surface.A smooth approximation of the earth’s irregular surface.

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What is a Coordinate System

A coordinate system defines the location of a point on a planar or spherical surfaceA geographic coordinate system is a three–dimensional reference system that locates points on the Earth’s surface. – The unit of measure is usually

decimal degrees. A point has two coordinate values: latitude and longitude.

– Latitude and longitude measure angles.

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Cartesian VS. Geographic Coordinate Systems


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Facts About Latitude

Lines of latitude (parallels)are evenly spaced (small circles) from 0o at equator (a great circle) to 90o at poles. 60 nautical miles (~ 110 km)/1o, ~1.8 km/minute and ~ 30 m/second of latitude. N. latitudes are positive (+φ)S. latitudes are negative (-φ).

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Longitude (λ)

Longitude is the angle (λ) between the plane of the prime meridian and the meridianalplane containing the point of interest (P).

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Facts About Longitude (λ)

Lines of longitude (meridians) converge at the poles.The distance of a degree of longitude varies with latitude. Zero longitude is the Prime (Greenwich) Meridian (PM); longitude is measured from 0-180o east and west of the PM. East longitudes are positive (+λ), west longitudes are negative (−λ).

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Example:

Given the following DMS coordinate, 45o 25’ 15.3” convert it to DD.

Going the other way…

DD – trunc(DD) * 3600 = total seconds / 60 = total minutes.Total minutes - Trunc(total minutes) * 60 = seconds

4209.453600

3.15602545

360060

=

++=

++=

DD

SMDDD

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Coordinate Systems cont.

A projected coordinate system is a two-dimensional planar surface. However, the Earth’s surface is three-dimensional. Transforming three-dimensional space onto a two-dimensional surface is called projection.

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Projection formulas are mathematical expressions that convert data from a geographical location (latitude and longitude) on a sphere or spheroid to a corresponding location (x and y) on a flat, two–dimensional surfaceThe projected coordinate system uses two axes: the x-axis, representing east-west, and the y-axis, representing north-south.They intersect at the origin, (0,0)

Coordinate Systems (cont.)

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Universal Transverse Mercator (UTM)

Based on transverse MercatorprojectionCovers Earth surface between 80o

South and 84o North60 north-south zones 6 degrees of longitude wide

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Universal Tranverse MercatorThe UTM graticule coverage

180o 180o0o

1 30 60

84o N

80o S

Equator

Each belt is 6O in longitude wide

0 meters10,000,000m

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Projections

UTM is an international metric coordinate system that covers the entire earth. It has the advantage of being mathematically consistent and well defined for the entire earth. Local coordinate systems are often used to fit mapping needs for a particular region.

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Each zone overlaps ½ degree into the adjoining zonesFalse origin of 500,000 meters west of the central meridian of each UTM zone


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Projection :Transverse MercatorEllipsoid : EverestCentral Meridian: 105EReference Latitude: 0False Easting: 500,000False Northing: 0Scale Factor: .9996

Ex: UTM Zone 48N


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map projection and coordinates - chiang mai university projection an… · 1 ags 722 (2-2003) map...

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