Equidistant cylindrical projection (Plate Carrée case) |
The effect of changing the standard latitude (parallels highlighted in blue) of equidistant cylindrical maps |
Suppose an arbitrary projection in whose equatorial aspect:
The rectangular result is the very simple cylindrical equidistant projection ("equidistant" only along meridians and one or two selected parallels), having a multitude of alternate names. It is neither conformal nor equal-area, and despite the resemblance to the stereographic cylindrical, it is not truly created by a perspective method.
Since scale is constant along all meridians, is simply ; two parallels are standard at , with circumference R. Constant scale along equal-length parallels means:
Different standard parallels affect only the map's width/height ratio. For the common special case of a standard Equator (usually known as the Plate carrée), , and latitude and longitude are directly mapped into and respectively, therefore rendering a world map into a 2:1 rectangle.
Boundary meridians of Winkel's first proposal and its foundation projections |
The cylindrical equidistant projection can be calculated quickily and presents a few interesting properties like immediate determination of angular and linear distances from two points. However, it creates extreme horizontal stretching along the poles. On the other hand, the sinusoidal projection is difficult to read at the polar regions due to high shearing. In both, scale is constant along each parallel, and all parallels are uniformly spaced.
The Winkel I projection is an arithmetic average of the sinusoidal and equidistant cylindrical projections. Neither equal-area nor conformal, it is defined as:
The Eckert V projection is a special case for . Winkel instead chose a standard parallel yielding a map with total area in scale with its width.
Consider the upper right quadrant of the map; the boundary meridian is given by
Winkel I map |
Because
If the area of an spherical Earth is ,
and therefore the standard parallels are about ±50°27"35'.
As the basis for his second proposal, Winkel used a pseudocylindrical projection in whose equatorial aspect:
Resembling Mollweide's but not equal-area, this auxiliary elliptical projection is essentially an extension of Apian's second globular design to a whole-world map. Consider the equation of an ellipse centered on the origin with horizontal major axis and minor axis :
In the northeastern quadrant, the boundary meridian is
Equally-spaced parallels imply
Winkel II map |
therefore . Since meridians are also equally spaced, horizontal scale is constant, so .
The Winkel II projection is a simple arithmetical average of this elliptical projection and the cylindrical equidistant: