Map Projections & Coordinate Systems

• How does a cartographer deal with the translation of a round-ish Earth to a flat map/screen?

• How and where are 2-dimensional coordinates assigned?

Datums - Spheroids - Ellipsoids - Geoids…

• The Earth’s shape is not truly spherical: There is a slight bulging at the equator and flattening at the poles due to the centrifugal force generated by the Earth’s rotation

• The closest mathematical approximation of Earth’s shape is an oblate spheroid or an ellipsoid

• Better is a Geoid (not a mathematical model but a model of mean sea level based on survey measurements taken across the planet)

• Not a problem for small scale maps of the Earth - a sphere is sufficient

• In order to be accurate, larger scale maps must use an ellipsoid (or geoid) as a base (earth model)

• Datums are built upon an ellipsoid (or a geoid) in conjunction with local/regional survey control points (Ex: North American Datum 1927 (NAD27); Kertau 1948 )

Turns out Columbus was wrong…

An example of an ellipsoid

Map Projections• Methods for flattening

(unrolling) a roundish earth onto a flat surface

• Based on a Datum (which is based on an ellipsoid)

• Ellipsoid (earth model) changes, the datum changes – (Clarke 1866 = NAD 1927) v. (GRS 1980 = NAD 1983)

• The shapes of the earth land features are ‘projected’ onto a flat surface – as if a light were aimed at the planet casting a shadow

Datum ShiftDatum Shift

700m

4789

541

4790

5424788

543

Datum cornerNAD27

275m

1000m

Datum ShiftDatum Shift

Datum cornerNAD83

600m

4789

541

4790

5424788

543

1000m

350m

Basic Types of Map Projection

Plane Cone

Cylinder

Common Developable Surfaces

Basic Types of Map ProjectionCommon Developable Surfaces

Plane Cone

Cylinder

Map Projections - CylindricalTend to be ConformalGlobe is projected onto

a cylinder tangent at equator (typically)

Low distortion at equator

Higher distortion approaching poles

A good choice for use in equatorial and tropical regions, e.g., Ecuador, Kenya, Malaysia

A cylindrical projection

Mercator Projection

Invented by Gerhardus Mercator - Flemish cartographer - in 1569

A special purpose projection intended as a navigational tool

A straight line between two points gives a navigator a constant compass bearing to the destination - not necessarily the fastest route

Map Projections - Conic

Tend to be Equal AreaSurface of globe projected onto cone tangent at standard

parallelDistorts N & S of standard parallel(s)Normally shows just one semi-hemisphere in middle

latitudes

Map Projections – Planar or Polar

Conformal

Surface of globe is projected onto a plane tangent at only one point (frequently N or S pole)

Usually only one hemisphere shown (often centered on N or S pole)

Works well to highlight an areaSometimes used by airports

Shows true bearing and distance to other points from center/point of tangency

Planar or Polar Projection -- Conformal

Map Projections – DistortionConformal vs. Equal-area

(The Great Debate)Preserve true shapesPreserve anglesExaggerate areasGraticules perpendicular

Show true size (area)Distorts shapes, angles and/or scale (squish/stretch shapes)Graticules not perpendicular

OR

Distortion: direction and distanceN

orth

Nor

thea

stN

orth

wes

t

Conformal vs. Equal-area

Conformal

Equal Area

Map Projections - families & examples

Tend to be equivalent (equal-area)Not bad for world maps

Elliptical/Pseudocylindrical (football) Projection

Mollweide projection

Map Projections - families & examples

• Mild distortion of shapes• Interrupts areas - oceans,

Greenland, Antarctica - sometimes reversed

• Equivalent/equal area• Good for climate,

soils, landcover - latitude and area comparisons

Goode’s Homosoline Interrupted Elliptical Projection

Map Projections - families & examples

Waterman Polyhedron “Butterfly” Projection

Good approximation of continents’:

• size• shape• position

Map Projections

Johann Heinrich Lambert (1728-77)

Lambert invented two of the most important and popular projections in use today

Two to Remember

Transverse Mercator

Lambert Conformal Conic

1. Conformal ConicA conic with two standard parallels (used for some State Plane systems)

2. Transverse Mercator A rotated cylindrical with the tangent circle N-S instead of along the Equator (used for UTM & some State Plane systems)

Map ProjectionsCoordinate Systems

For a spatial database to be useful, all parts must be registered to a common coordinate system.

Coordinate Systems (other than latitude-longitude) use a particular Projection, as well as a particular Datum (which is based upon a particular Ellipsoid

or Geoid)…

Coordinate SystemsThree to Remember

3. Latitude-Longitude(a spherical coordinate

system)

2. Universal Transverse

Mercator (UTM)

1. State Plane

• Created in the 1930’s, zones follow state/county boundaries

• Each zone uses a projection: Lambert’s Conformal Conic (E-W zones) Transverse Mercator (N-S zones)

• Each zone has a centrally located origin, a central meridian and a false origin established to the W and S

– Don’t have to deal with negative numbers

• Uses planar coordinates (instead of Lat./Long. spherical coordinates)

– Square grid with constant scale - distortion over small areas is minimal

• USA only

Zones of the SPCS for the

contiguous US

Coordinate Systems: State Plane Coordinate System

False origin for WA. N. zone

Coordinate Systems: State Plane Coordinate System

• Convenience of a plane rectangular grid on a global level

• Popular in scientific research• A section from a transverse

Mercator projection is used to develop separate grids for each of 60 zones

• Low distortion along the tangent central meridian, increasing E & W

• Works great for large scale data sets and satellite image rectification though some areas cross zones (WA, TN, etc.)

Beginning at 180o, Transverse Mercator projections are obtained every 6 degrees of longitude along a central meridian

Coordinate Systems: Universal Transverse Mercator

• 60 N-S zones each spanning 6o of longitude (0.5o overlap each side) from 84o N - 80o S

• In polar regions the Universal Polar Stereographic grid system (UPS) is used

• Each zone has an origin, central meridian, and false origin, just as with SPCC

• Coordinates read similar to SPCC but in meters:

UTM zones (10-19 North) covering the lower 48 states

Coordinate Systems: Universal Transverse Mercator

Coordinate Systems: Lat./Long. (Geographic Coordinates)

• Works for a sphere or spheroid

• Lines of latitude begin at the equator and increase N and S toward the poles from 0o to 90o

• Degrees of Latitude are constant

• Lines of Longitude begin at some great circle (prime meridian) passing through some arbitrary point

• 1o of Longitude = 1o of Latitude only at the Equator. Degrees of Longitude get smaller (converge) towards poles

• Technically, Unprojected (a spherical coordinate system)

NOT Projected“Geographic Coordinate System”

So why does this matter?

GIS programs must know what projection was used for data creation and where the Coordinate System’s point of origin is…

Map data or satellite images in different projections, coordinate systems, or referenced to different datums may not overlay properly…

So why does this matter?

Things work best in ArcMap if the Data and the Data Frame use the same coordinate system, projection and datum…

ArcMap can project data (using one coordinate system) to another, different coordinate system (e.g., that of the Data Frame) if the coordinate systems of the data and the data frame are properly defined

If the Data and Data Frame use different Datums, a Datum Transformation must be chosen

Summary A round-ish earth must be ‘projected’ onto a developable surface in

order to make a flat map. Common developable surfaces are: Plane Cylinder Cone

Two common Projections used in the USA are: Lambert’s Conformal Conic Transverse Mercator (Cylindrical, a variant of the classic Mercator

projection)

Coordinate Systems assign a unit of measurement and a point of origin. These require a projection as well as a datum (earth model).

Three common Coordinate Systems used in the USA are: Latitude-Longitude (which is technically unprojected (or ‘geographic’),

but still requires a datum, and uses speherical coordinates as opposed to planar coordinates)

UTM (Universal Transverse Mercator) State Plane