Mercator projection

The Mercator projection ismap projection devised by Gerardus Mercator1569.

Like all map projections attemptingfitcurved surface ontoflat sheet,shape ofmap isdistortion oftrue layout ofEarth's surface. The Mercator projection wildly distorts area: Greenlandpresented as having roughlysame size as Africa, whenfact Africaapproximately 13 times larger than Greenland.

image:mercator.gif
Mercator projection: public domain Online Map Creation

Thisdonean important reason:projection isconformal map, that is,preserves angle. Any straight line onMercator map islineconstant bearing,loxodrome or rhumb line. This makesparticularly usefulnavigators even thoughplotted routeusually notgreat circle (shortest distance) route. Inerasailing ships,timetravel was subject toelementshencedistancetravel was not as important asdirectiontake especially since longitude could not be accurately determined.

To achieve this effect,Mercator projection stretches East-West distances by an increasing amount asdistance fromequator increases. The extreme casedistorted areaatpoless, wheretwo geographical points have become lines attopbottom ofmap.

The following equations determinexy coordinates ofpoint onMercator map from its latitude φlongitude λ (with λ₀ beinglongitude incentermap):

This isinverse ofGudermannian function.

Andinverse:

AlthoughMercator projectionstillcommon usenavigation, some critics argue thatno longer suitedrepresententire worldpublicationswall maps. The positionEuropeNorth Americamuch closer tocenter ofmap thanoughtbe,their increased size comparedAfricaseen as perpetuatingidea ofThird World's inferiority especially as itlargely relegated tolower part ofmap.

Furthermore, some Mercator maps omit most or allAntarctica. This haseffectplacing Europe atcenter ofmap. Mercator projectionsrarely usedatlasesdisplayworld. They continuebe usefulnavigation.

The Gall-Peters projection has been proposed as an alternativeaddress these concerns. This presentsvery different view ofworld:shapecountrieshighly distorted, especially away fromequator, but areapreserved. Nevertheless,1989 resolution by seven North American Geographical groups decriesuseall rectangular coordinate world maps includingGall-Peters.

Derivation ofProjection

The latitudelongitudeequivalent tospherical coordinates φθ, respectively. The radius R can be ignored. A straight line onMercator projection represents movement (e.g. ofship) onEarth atconstant anglerespectgeographic North.

Assume φ=0 (duesymmetry,following argumentsimilarany φ). Imagineplane tangent tosphere at point . Such plane can be described bypairunit basis vectors , where

Let us sayship moves at(clockwise) angle β away fromNorth (e_θ), thenship's movement is

where e_φ points towardsEast.

Now let us sayship moves at45 ° angle fromNorth (). Then, ifship movescertain distance ine_φ direction,will move an equal distance ine_θ direction.

However, movement ine_θ directionalonggeodesiccircumference

where R isEarth's radius.

Therefore ifship movesdistance ΔS_θ ine_θ direction, thisequivalentmoving an angle Δθ, where

(the Earth's radius isconversion factor).

Movement ine_φ directionalongcirclecircumference

Whenship movesdistance ΔS_φ ine_φ direction, thisequivalentmovement by an angle Δφ, where

Letting β=45°, then , so

Divide both sides by Rconvert differencesdifferentials,

.

Group togetherthetas,

Integrate,

And, since φdirectly proportionallengths onmap (x = φ), equation (1) means (since xy have uniform units ofsame length) that

Quod Erat Demonstrandum.

See also: cartography, oblique mercator projection, Nautical chart.

External link

• Resolution regardinguserectangular world maps: http://icg.harvard.edu/~maps/people/mercator.htm