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Mercator Projection


MercatorProjection

The Mercator projection is a map projection that was widely used for navigation since loxodromes are straight lines (although great circles are curved). The following equations place the x-axis of the projection on the equator and the y-axis at longitude lambda_0, where lambda is the longitude and phi is the latitude.

x=lambda-lambda_0
(1)
y=ln[tan(1/4pi+1/2phi)]
(2)
=1/2ln((1+sinphi)/(1-sinphi))
(3)
=sinh^(-1)(tanphi)
(4)
=tanh^(-1)(sinphi)
(5)
=ln(tanphi+secphi).
(6)

The inverse formulas are

phi=2tan^(-1)(e^y)-1/2pi
(7)
=tan^(-1)(sinhy)
(8)
=gd(y)
(9)
lambda=x+lambda_0,
(10)

where gd(y) is the Gudermannian.

mobl

An oblique form of the Mercator projection is illustrated above. It has equations

x=tan^(-1)[(tanphicosphi_p+sinphi_psin(lambda-lambda_0))/(cos(lambda-lambda_0))]
(11)
y=1/2ln((1+A)/(1-A))
(12)
=tanh^(-1)A,
(13)

where

lambda_p=tan^(-1)((cosphi_1sinphi_2coslambda_1-sinphi_1cosphi_2coslambda_2)/(sinphi_1cosphi_2sinlambda_2-cosphi_1sinphi_2sinlambda_1)]
(14)
phi_p=tan^(-1)[-(cos(lambda_p-lambda_1))/(tanphi_1)]
(15)
A=sinphi_psinphi-cosphi_pcosphisin(lambda-lambda_0).
(16)

The inverse formulas are

phi=sin^(-1)(sinphi_ptanhy+(cosphi_psinx)/(coshy))
(17)
lambda=lambda_0+tan^(-1)((sinphi_psinx-cosphi_psinhy)/(cosx)).
(18)
mtra

There is also a transverse form of the Mercator projection, illustrated above (Deetz and Adams 1934, Snyder 1987). It is given by the equations

x=1/2ln((1+B)/(1-B))
(19)
=tanh^(-1)B
(20)
y=tan^(-1)[(tanphi)/(cos(lambda-lambda_0))]-phi_0
(21)
phi=sin^(-1)((sinD)/(coshx))
(22)
lambda=lambda_0+tan^(-1)((sinhx)/(cosD)),
(23)

where

B=cosphisin(lambda-lambda_0)
(24)
D=y+phi_0.
(25)

Finally, the "universal transverse Mercator projection" is a map projection which maps the sphere into 60 zones of 6 degrees each, with each zone mapped by a transverse Mercator projection with central meridian in the center of the zone. The zones extend from 80 degrees S to 84 degrees N (Dana).


See also

Gudermannian, Spherical Spiral

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References

Dana, P. H. "Map Projections." http://www.colorado.edu/geography/gcraft/notes/mapproj/mapproj_f.html.Deetz, C. H. and Adams, O. S. Elements of Map Projection with Applications to Map and Chart Construction, 4th ed. Washington, DC: U. S. Coast and Geodetic Survey Special Pub. 68, 1934.Pearson, F. Map Projections: Theory and Applications. Boca Raton, FL: CRC Press, p. 195, 1990.Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 38-75, 1987.

Referenced on Wolfram|Alpha

Mercator Projection

Cite this as:

Weisstein, Eric W. "Mercator Projection." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MercatorProjection.html

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