HomeSite MapAzimuthal Equal-Area/EquidistantHow Projections are CreatedAitoff, Hammer, Winkel TripelMap Projections

A Simple Projection plus Two Derived Works

Deducing the Equidistant Cylindrical Projection

Equidistant cylindrical projection
Equidistant cylindrical projection (Plate Carrée case)
Equidistant cylindrical projection
The effect of changing the standard latitude (parallels highlighted) 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, y is simply R phi; two parallels are standard at +-phi0, with circumference Rcos phi0. Constant scale along any single parallel means:

x=R lambda cos phi0, y=R phi

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, French for a flat square surface), cos phi0 = 1, and longitude and latitude are linearly mapped into x and y respectively, therefore rendering a world map into a 2:1 rectangle. The trivially fast computation has turned this format into a favorite for storing images intended for texturing spheres in computer graphics, and for panoramic scenes in digital photography, despite the range of detail density and shape distortion, both much higher at the poles than near the Equator.

Deducing the Winkel I and Eckert V Projections

Winkel I schematics
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 suffers from infinite 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:

x=R lambda (cos phi0 + cos phi)/2, y=R phi

The Eckert V projection is a special case for phi0 = 0. 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

x=pi R (cos phi0 + cos phi)/2
The quadrant's area is
S=int from 0 to pi R/2 pi R (cos phi0 + cos phi)/2 dy
Winkel I projection
Winkel I map

Because

y=phi R, 0<=y<=pi R /2
dy=R d phi, 0<=phi<= pi / 2
S=pi R^2/2 int from 0 to pi/2 (cos phi0+cos phi)dphi=pi R^2/2 (pi/2 cos phi0 + 1)

If the area of an spherical Earth is 4 pi R^2 = 4S,

piR^2=piR^2(pi/2 cos phi0+1)/2, 1=pi/2 cos phi0, phi0 = arccos2/pi

and therefore the standard parallels are about ±50°27"35'.

Deducing Apian's Extended Globular and Winkel II Projections

As the basis for his second proposal, Winkel used another auxiliary pseudocylindrical projection in whose equatorial aspect:

Resembling Mollweide's but not equal-area, this intermediary 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 a and minor axis a/2:

x^2/a^2+4y^2/a^2=1

In the northeastern quadrant, the boundary meridian is

xb=sqrt(a^2-4y^2)

Equally-spaced parallels imply

y=R phi
Winkel II projection
Winkel II map

therefore a=piR. Since meridians are also equally spaced, horizontal scale is constant, so x=lambda xb / pi and the auxiliary projection is given by:

x=lambda R sqrt(pi^2-4phi^2), y=R phi

The Winkel II projection is a simple arithmetical average of this elliptical projection and the cylindrical equidistant:

x=lambda R (cos phi0 +sqrt(pi^2-4phi^2)/pi)/2, y=R phi

Again, the result is neither conformal nor equal-area and the author preferred 50°27"35' N and S for standard latitudes.



HomeSite MapAzimuthal Equal-Area/EquidistantHow Projections are CreatedAitoff, Hammer, Winkel Tripel  www.progonos.com/furuti    May 13, 2014
Copyright © 1996, 1997 Carlos A. Furuti