Map Projections

### A Simple Projection plus Two Derived Works

#### Deducing the Equidistant Cylindrical Projection

 Equidistant cylindrical projection (Plate Carrée case) The effect of changing the standard latitude (parallels highlighted) of equidistant cylindrical maps

Suppose an arbitrary projection in whose equatorial aspect:

• All meridians are equally-spaced vertical standard 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 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 . Constant scale along any single parallel 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, French for a flat square surface), , and longitude and latitude are linearly mapped into and 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

 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:

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'.

#### 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:

• 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 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 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 and the auxiliary projection is given by:

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

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

 www.progonos.com/furuti    May 13, 2014