Map Projections

## Cylindrical Projections

### Introduction

If the width of equatorial cylindrical maps is normalized by rescaling, then (since their spacing of meridians is identical) a projection's only defining feature is how its parallels are spaced. Here, a slice of the northern hemisphere is shown with degree and color gradient rulers (compare the azimuthal orthographic inset). Blue polar latitudes are necessarily compressed in the equal-area designs, and expanded by varied amounts in Miller's, Mercator's and the other perspective projections; only the equidistant Plate Carrée is linearly spaced as in the globe. Notice that the three leftmost projections are actually rescaled versions of the same principle, and that Mercator and central cylindrical maps stretch to the infinite.

In the equatorial — the most common, and frequently the only useful — aspect of all cylindrical projections:

• all coordinate lines are straight
• parallels (by convention horizontal) cross meridians always at right angles
• scale is constant along each parallel, so meridians are equally spaced
• all parallels have the same length; the same happens to meridians

Therefore,

• whole-world maps are always rectangular
• scale is identical in any pair of parallels equally far from the Equator
• scale differs considerably among parallels, reaching infinity at poles, which have zero length on the Earth but are as long as the Equator on a cylindrical map

On the transverse aspect, two opposite meridians lie over the Equator of the equatorial aspect; other properties don't hold.

Rolling a rectangular map and joining two opposite edges creates a tube, or a cylinder without end caps. In fact, some cylindrical projections are geometrically derived from closely fitting a tube around a sphere; the former may be secant or tangent, and as a result two parallels or the Equator, respectively, are standard lines.

All cylindrical projections are remarkably similar, being in fact only distinguished by parallel spacing. The very important, unique conformal cylindrical projection is named after Mercator and discussed elsewhere. There is a single equal-area cylindrical projection, in many rescaled versions.

As a group, cylindrical projections are more appropriate for mapping narrow strips centered on a standard parallel. Although useful for comparison of regions at similar latitudes, they are badly suited for world maps because of extreme polar distortion. Unfortunately, cylindrical maps are often employed in textbooks and other popular publications, perhaps due to poor research and their simple shape neatly fitting page frames.

#### Equidistant Cylindrical and Cassini Projections

Two particular cases of the equidistant cylindrical projection in the equatorial aspect differ only in the horizontal scale. On the first map, usually known as plate carrée, the standard parallel is 0°; on the second map, Gall's isographic projection has standard parallels at 45°N and S

Cassini's projection is a transverse aspect of the plate carrée, here with central meridians 70°E and 110°W.

The simplest of all map graticules belongs to the equatorial aspect of the equirrectangular projection, referred to by many names like equidistant cylindrical, plane chart, plain chart and rectangular. It is a cylindrical projection with standard meridians: all have constant scale, equal to the standard parallels's, therefore all parallels are equally spaced. It was credited to Erathostenes (ca. 200 b.c.) and to Marinus of Tyre (ca. 100). Its trivial construction made it widely used, even for navigation, until the Modern Age.

A special case of the equirrectangular projection is called Plate Carrée, or simple cylindrical: the Equator is a standard parallel, so it is twice as long as all meridians, making the map a 2 : 1 rectangle and the graticule's grid square. Fast, trivial equations led to its resurgence in rough computer-drawn maps, with early machines or real-time graphics. It is still commonly used in digitized textures (“skins”) of earthly and planetary features.

Another special case, James Gall's isographic projection (1885) has standard parallels at 45°N and 45°S, the same latitudes chosen for his equal-area orthographic design.

A transverse aspect of the equidistant cylindrical projection was proposed by César F. Cassini in 1745, and named after him. Cassini's grandfather, Jean Dominique (born Giovanni Domenico) was the most prominent member of a family of astronomers and cartographers. Several European countries used Cassini's projection in large-scale topographical maps until recently. Distortion is identical as in the normal aspect, thus a central meridian, the Equator, and three other meridians at multiples of 90° are all straight lines with equal, constant scale.

Seldom found, oblique equidistant cylindrical maps are useful for quickly calculating angular and linear distances from two points on the map to any other point.

#### Miller's Cylindrical Projection

Miller's cylindrical projection

The best-known (1942) of all projections published by Osborn M. Miller was mathematically conceived as a compromise to Mercator's, retaining its familiar shapes but with much smaller polar exaggeration. It applies a reduction factor of 0.8 to the latitudes before calculating Mercator's equations, and an inverse factor to the result. In consequence, the map can include the whole world. The projection is neither conformal nor area-preserving.

Miller created several other projections, including three other cylindrical designs; none was as popular as the one bearing his name.

www.progonos.com/furuti/MapProj/Normal/ProjCyl/projCyl.html — June 16, 2018