Cylindrical equal-area projection


In cartography, the cylindrical equal-area projection is a family of cylindrical, equal-area map projections.

Cylindrical projections

The term "normal cylindrical projection" is used to refer to any projection in which meridians are mapped to equally spaced vertical lines and circles of latitude are mapped to horizontal lines.
The mapping of meridians to vertical lines can be visualized by imagining a cylinder wrapped around the Earth and then projecting onto the cylinder, and subsequently unfolding the cylinder.
By the geometry of their construction, cylindrical projections stretch distances east-west. The amount of stretch is the same at any chosen latitude on all cylindrical projections, and is given by the secant of the latitude as a multiple of the equator's scale. The various cylindrical projections are distinguished from each other solely by their north-south stretching :
The only cylindrical projections that preserve area have a north-south compression precisely the reciprocal of east-west stretching : equal-area cylindrical. This divides north-south distances by a factor equal to the secant of the latitude, preserving area but heavily distorting shapes.
Any particular cylindrical equal-area map has a pair of identical latitudes of opposite sign at which the east–west scale matches the north–south scale.

Description

Formulae

All cylindrical equal-area projections use the formula:
where λ is the longitude, λ is the central meridian, φ is the latitude, and φ is the standard latitude, all expressed in radians.
Some cartographers prefer to work in degrees, rather than radians, and use the equivalent formula:

Simplified formula

Stripping out unit conversion and uniform scaling, the formulae may be written:
Hence the sphere is mapped onto a stretched vertical cylinder. The stretch factor S is what distinguishes the variations of cylindric equal-area projection.

Discussion

The various specializations of the cylindric equal-area projection differ only in the ratio of the vertical to horizontal axis. This ratio determines the standard parallel of the projection, which is the parallel at which there is no distortion and along which distances match the stated scale. There are always two standard parallels on the cylindric equal-area projection, each at the same distance north and south of the equator. The standard parallels of the Gall–Peters are 45° N and 45° S. Several other specializations of the equal-area cylindric have been described, promoted, or otherwise named.
ProjectionImageCreator Standard parallels north and southWidth-to-height aspect ratio
cylindrical equal-area
φ02
LambertJohann Heinrich Lambert Equator ≈ 3.142
BehrmannWalter Behrmann 30° ≈ 2.356
Smyth equal-surface
= Craster rectangular		Charles Piazzi Smyth ≈ 37°04′17″2
Trystan EdwardsTrystan Edwards 37°24′≈ 1.983
Hobo–DyerMick Dyer 37°30′≈ 1.977
Gall–Peters
= Gall orthographic
= Peters		James Gall
Promoted by Arno Peters as his own invention		45° ≈ 1.571
BalthasartM. Balthasart 50°≈ 1.298
Tobler's world in a squareWaldo Tobler ≈ 55°39′14″1

History

The invention of the Lambert cylindrical equal-area projection is attributed to the Swiss mathematician Johann Heinrich Lambert in 1772. Variations of it appeared over the years by inventors who stretched the height of the Lambert and compressed the width commensurately in various ratios. See [|Named specializations] table.
The Tobler hyperelliptical projection, first described by Tobler in 1973, is a further generalization of the cylindrical equal-area family.
The HEALPix projection is an equal-area hybrid combination of: the Lambert cylindrical equal-area projection, for the equatorial regions of the sphere; and an interrupted Collignon projection, for the polar regions.