A Simple Projection plus Two Derived Works
Suppose an arbitrary projection in whose equatorial aspect:
- All meridians are standard
equally-spaced vertical lines
- All parallels are horizontal, equally-spaced, equally long
lines
The rectangular result is the very simple cylindrical equidistant
projection (equidistant only along meridians and two
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 meridians, y is simply
R
; two parallels are standard
at ±0, with circumference R
cos 0. Constant scale along
equal-length parallels means:
Different standard parallels only change the map's
width/height ratio. For the common special case of standard
Equator (usually known as the plate carrée),
cos 0 = 1, and latitude and longitude are
directly mapped into y and x respectively,
therefore a world map is a 2:1 rectangle.
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. Both have constant scale along
equally-spaced parallels.
The Winkel I
is an arithmetic average of the two preceding projections. It is
neither equal-area nor conformal and is defined as: 
- x = R
(cos 0 +
cos ) / 2
- y = R
The Eckert
V projection is a special case for 0 =
0. Winkel instead chose a standard parallel yielding a map
with area in scale with its width.
For the upper right quadrant of the map, the right edge is 
or about
±50°27"35'. 
For his second proposal, Winkel used an arbitrary projection in
whose equatorial aspect:
- parallels are equally-spaced horizontal lines
- meridians are equally-spaced elliptical arcs
- the whole map fits a 2:1 ellipse
Resembling Mollweide's (but
not equal-area), this auxiliary elliptical projection is
essentially an extension of Apian's
second globular
design, defined as:
The Winkel II projection is a simple arithmetical average
of this elliptical projection and the cylindrical equidistant; it
is neither conformal nor equal-area.
Again, ±50°27"35' is chosen as the standard
parallels.
The underlying elliptical projection is also (erroneously)
mentioned to be an unmodified Mollweide ellipse, with
non-uniformly spaced parallels.
 |  |  |  |  |  | | www.progonos.com/furuti September 21, 2002 |