Map projection is the way we Map projection is the way we fit earth’s three-dimensional fit earth’s three-dimensional surface onto flat paper or a surface onto flat paper or a

screenscreen

On a map, when lines of latitude and On a map, when lines of latitude and longitude cross what is the resulting longitude cross what is the resulting angle?angle?

Pencils Down – The following Pencils Down – The following information is for background information is for background

knowledge only.knowledge only.

Map ProjectionsMap Projections

Think of an transparent globe Think of an transparent globe w/ an imagined light source insidew/ an imagined light source inside

What type of shadow What type of shadow would be cast?would be cast?

Shadow cast would depend on Shadow cast would depend on light location…light location…

Gnomonic – light source Gnomonic – light source at centerat center

Stereographic – light at Stereographic – light at point opposite of point opposite of tangent of globe tangent of globe meeting mapmeeting map

Orthographic – light Orthographic – light source at infinitysource at infinity

Onto What Onto What do you do you projectproject

An An azimuthazimuth is the angle is the angle formed at the beginning formed at the beginning point of a straight line, in point of a straight line, in relation to the meridianrelation to the meridian

Position of the surface

The Math…The Math… Derivation of the Projection: Derivation of the Projection:

cosφ=dR⇒d=Rcosφ cosφ=dR⇒d=Rcosφ cosλ=p2d⇒p2=dcosλ=Rcosφcosλ cosλ=p2d⇒p2=dcosλ=Rcosφcosλ sinλ=p1d⇒p1=dsinλ=Rcosφsinλ sinλ=p1d⇒p1=dsinλ=Rcosφsinλ

Derivation of the Inverse: Derivation of the Inverse: d=∥p∥ d=∥p∥ cosλ=p2∥p∥⇒λ=cos-1(p2∥p∥) cosλ=p2∥p∥⇒λ=cos-1(p2∥p∥) cosφ=dR⇒φ=cos-1(∥p∥R)cosφ=dR⇒φ=cos-1(∥p∥R)

Just Kidding

The Most Common: The Most Common: Conformal (i.e., angles are preserved) Conformal (i.e., angles are preserved) Equal Area (i.e., areas are in constant proportion) Equal Area (i.e., areas are in constant proportion) Equidistant (i.e., distances are in constant Equidistant (i.e., distances are in constant

proportion)proportion) An Important Mathematical Result: An Important Mathematical Result:

A single projection can not be both conformal A single projection can not be both conformal and equal areaand equal area

Polar Azimuthal Polar Azimuthal OrthographicOrthographic

Sinusodial ProjectionSinusodial Projection

Equatorial Cylindrical Equal Equatorial Cylindrical Equal AreaArea

Equatorial Cylindrical Equatorial Cylindrical ConformalConformal

Conical Equal AreaConical Equal Area

Three sources of map Three sources of map distortiondistortion

Map scaleMap scale – most maps are smaller – most maps are smaller than the reality they represent. Map than the reality they represent. Map scales tell us how much smaller.scales tell us how much smaller.

Map projectionMap projection – this occurs because – this occurs because you must transform the curved surface you must transform the curved surface of the earth on a flat plane.of the earth on a flat plane.

Map typeMap type – you can display the same – you can display the same information on different types of maps.information on different types of maps.

Now…You need to know the Now…You need to know the following projections!following projections!

Mercator ProjectionMercator Projection

Distorts size Distorts size not shapenot shape

Mercator ProjectionMercator Projection Stretches the poles from one length to the size Stretches the poles from one length to the size

of the equator. The north-south scale is of the equator. The north-south scale is constant, but east-west scale increases to twice constant, but east-west scale increases to twice the north-south scale at 60 degrees N and the north-south scale at 60 degrees N and infinitely at the poles.infinitely at the poles.

Shapes are correct for all areas, and map has Shapes are correct for all areas, and map has correct directional relationships.correct directional relationships.

Look at the size of Greenland and Antarctica.Look at the size of Greenland and Antarctica. Map exaggerates the distance between Chicago Map exaggerates the distance between Chicago

and Stockholm, both in northern latitudes.and Stockholm, both in northern latitudes. Created in 1569Created in 1569

Equal Area ProjectionEqual Area Projection

Distorts shapes, not areaDistorts shapes, not area

Equal Area ProjectionEqual Area Projection Represents areas correctly, but Represents areas correctly, but

distorts shapes.distorts shapes. If South America is 8 times larger If South America is 8 times larger

than Greenland on the globe, it will than Greenland on the globe, it will be 8 times bigger on the map.be 8 times bigger on the map.

Robinson ProjectionRobinson Projection

Does not preserve size, area or shapeDoes not preserve size, area or shape

Robinson ProjectionRobinson Projection Frequently used.Frequently used. Distorts both size and shape, but not Distorts both size and shape, but not

too much.too much. The major benefit of the Robinson The major benefit of the Robinson

projection is that oceans are projection is that oceans are uninterrupted. This projection is useful uninterrupted. This projection is useful in depicting patterns of global in depicting patterns of global interactioninteraction..

Considered a compromise projectionConsidered a compromise projection

Goode’s ProjectionGoode’s Projection

Goode’s projection interrupts the Goode’s projection interrupts the oceans and tucks Australia and oceans and tucks Australia and New Zealand farther west than in New Zealand farther west than in reality. Therefore, land masses reality. Therefore, land masses appear relatively large compared appear relatively large compared to the oceans.to the oceans.

Minimized distortion in the shape Minimized distortion in the shape of the various land masses and the of the various land masses and the size of one land mass compared to size of one land mass compared to other land masses.other land masses.

Unprojected vs. Unprojected vs. LambertLambert

Peter’s ProjectionPeter’s Projection

The Peters Projection World Map is one of the most The Peters Projection World Map is one of the most stimulating, and controversial, images of the world. stimulating, and controversial, images of the world. When this map was first introduced by historian When this map was first introduced by historian and cartographer Dr. Arno Peters at a Press and cartographer Dr. Arno Peters at a Press Conference in Germany in 1974 it generated a Conference in Germany in 1974 it generated a firestorm of debate. The first English-version of the firestorm of debate. The first English-version of the map was published in 1983, and it continues to map was published in 1983, and it continues to have passionate fans as well as staunch detractors.have passionate fans as well as staunch detractors.The earth is round. The challenge of any world map The earth is round. The challenge of any world map is to represent a round earth on a flat surface. is to represent a round earth on a flat surface. There are literally thousands of map projections. There are literally thousands of map projections. Each has certain strengths and corresponding Each has certain strengths and corresponding weaknesses. Choosing among them is an exercise weaknesses. Choosing among them is an exercise in values clarification: you have to decide what's in values clarification: you have to decide what's important to you. That is generally determined by important to you. That is generally determined by the way you intend to use the map. The Peters the way you intend to use the map. The Peters Projection is an area accurate map. Projection is an area accurate map.

http://www.petersmap.com/page2.html http://www.petersmap.com/page2.html

Compare projections imageCompare projections image

A

DC

B

Equal Area Projection

ADC

B

On which map is the size of Greenland distorted the most?

Equal Area Projection

A

DC

B

Which map is best for navigation?

Equal Area Projection

A

DC

B

Which map best corrects most of the distortions associated with map projection?

Mercator ProjectionMercator Projection

What is bad about this projection?a. Shape b. Sizec. Distance d. direction

Equal Area Equal Area ProjectionProjection1.What is

bad about this projection?

a. Shapeb. Sizec. Distanced. direction2.What is best about this

projection?a. Shape b. Sizec. Distance d. direction

1.1. Map projections attempt to correct for errors Map projections attempt to correct for errors in in

a. transferabilitya. transferabilityb. relative size, distance, scale & proportionb. relative size, distance, scale & proportionc. relative size, distance, shape, & directionc. relative size, distance, shape, & directiond. distance, proximity, and topologyd. distance, proximity, and topologye. distance, shape, and lines of latitude and e. distance, shape, and lines of latitude and

longitudelongitude

2.2. The Mercator projection preservesThe Mercator projection preservesa. sizea. size b. areab. area c. shapec. shaped. scaled. scale e. distancee. distance

Goode’s ProjectionGoode’s Projection

What is bad about this projection?a. Shape b. Sizec. Distance d. direction