Map Projections

### Some Hybrid Pseudocylindrical Projections

Several authors have attempted to combine advantages of two or more existing projections. Values can be mathematically averaged, or different pieces of the map may be separately projected along lines of similar scale. This latter approach is made convenient in pseudocylindrical projections due to their straight parallels with constant scale; therefore different map "slices" can be projected separately, then fused, i.e., "stitched" together, possibly after rescaling.

 Nell's projection

#### Nell's Pseudocylindrical Projection

Not counting the obsolete trapezoidal, the first known flat-polar pseudocylindrical projection was published by Adam Nell in 1890; actually it is the limiting case of Nell's pseudoconic projection for the ellipsoid, with the Equator as a reference parallel. Its x-coordinates are similar to an average of a cylindrical projection with the sinusoidal, but using an auxiliary angle; the whole map is equal-area.

 Nell-Hammer projection

#### The Nell-Hammer projection

Published in 1900 by Hammer, a flat-polar projection derived from Nell's suggestions directly averages x-coordinates of Lambert's equal-area cylindrical and the sinusoidal projection, with y-coordinates adjusted to preserve areas.

#### Goode's Homolosine

 Goode's homolosine map, in the almost never used uninterrupted form. In contrast, the interrupted version was very popular.

John P. Goode combined the sinusoidal and Mollweide (homolographic) projections in his hybrid homolosine (homolographic + sinusoidal) projection of 1923-25: three horizontal stripes are joined at the two parallels with the same length in the two base projections - approximately 40°44'12"N and 40°44'12"S. Latitudes higher than the boundary parallels are represented using Mollweide's projection, and the remaining area in the central stripe by the sinusoidal.

The meridians are broken at the joint and the result is not appreciably better than either original methods used alone; however, horizontal scale is preserved in nearly 65% of the map and the polar caps are reasonably legible while preserving the sinusoidal's constant meridian scale at the tropical band. This projection, especially designed for interruption, was for long quite popular in atlases.

#### Boggs Eumorphic

 Uninterrupted Boggs eumorphic map

Created by S.W. Boggs, the eumorphic (Greek for "well-shaped") projection of 1929 is another hybrid. However, instead of discretely joining separate bands, it defines its y-coordinates as the arithmetic average of corresponding sinusoidal and Mollweide coordinates. The x-coordinates are calculated for an equal-area map, usually presented in interrupted form.

#### The Sinu-Mollweide Projection

 Sinu-Mollweide projection in its plain oblique form. The interrupted version is more interesting

Another fused pseudocylindrical design, Allen K. Philbrick's Sinu-Mollweide projection (1953) shares the base projections and fusion latitude of Goode's homolosine; however,

• instead of three bands, there are only two, using the Mollweide projection above 40°44'12"S and the sinusoidal below it
• the preferred aspect is not equatorial but oblique, with 55°N 20°E at the intersection of the original Equator and the central meridian, which retains the North Pole
In this simple form, this projection has been published in commercial maps, but like the homolosine's, its author favored an interrupted version.

#### Eckert V

Eckert's fifth proposal (1906) can be built as an arithmetic mean of the sinusoidal and plate carrée projections (actually the y-coordinates are the same in both). As a result, poles are mapped to straight lines with half the Equator's length. All the meridians but the central are sinusoids. Although the map does not preserve areas, it deceptively resembles the much more popular equal-area Eckert VI projection.

#### Winkel I

 Winkel's first projection with 50°27"35'N and S as standard parallels

The first projection published by O.Winkel in 1921 is a generalization of Eckert's V using the equidistant cylindrical projection with any two opposite parallels standard, not necessarily the Equator (therefore only the horizontal scale is changed from the special case). Winkel preferred 50°27"35' N and S, which makes the total area proportional to the equatorial circumference.

 Winkel II map

#### Winkel II

Also published in 1921, this projection averages the equidistant cylindrical and a 2:1 elliptical projection similar to Mollweide's, but with equally-spaced parallels and therefore not equal-area (some sources maintain Mollweide's itself is the base projection).

The resulting map, also neither conformal nor equal-area, is constructed much like Winkel's first projection.

World maps in the HEALPix projection
 Although the main purpose of HEALPix is not cartographical, using its projection for world maps shows some interesting properties. This HEALPix map with H = 2 has facet edges marked in green. Each facet can be further hierarchically subdivided into smaller squares.
HEALPix map with H = 4. Compared with the map above, facets are still square, but the overall aspect ratio differs.
HEALPix map with H = 6, rescaled in order to show facets as equilateral triangles. Each facet may be subdivided into four triangles.

#### The HEALPix Projection

HEALPix, Hierarchical Equal Area and isoLatitude Pixelisation (Górski and others, 1999), is a collection of standards and resources for efficient storage and processing of large sets of data for astronomical and cosmological research. At discrete points ("pixels") covering a conceptual celestial sphere surrounding the Earth, satellite probes detect incoming radiation, like gamma rays and the cosmic microwave background; the measured values are saved on a raster grid for further analysis.

HEALPix defines a family of hybrid interrupted pseudocylindrical projections mapping from the celestial sphere to the plane. A HEALPix map comprises H lobes; in each lobe in the normal aspect, the equatorial band is mapped to a square using Lambert's equal-area cylindrical projection; the polar areas are mapped to two right isosceles triangles using a rescaled, interrupted form of Collignon's projection. The boundary parallels, approximately 41°48'37" N and S, are chosen in order to make the triangular regions cover 1/3 of the total area.

For a special case, with H = 2 and dispensing with the equatorial band, the result is an interrupted Collignon map in two squares. In general, changing H affects the unscaled aspect ratio (much like in variations of Lambert's projection) and the parallel of least shape distortion; the most common case, H = 4, can be trivially folded into a cube, like polyhedral projections.

The whole map may be divided into 3H identical facets, each a square with vertical and horizontal diagonals; a single facet is split along the boundary opposite the central meridian, but this can be fixed moving one half to the opposite side of the map, leaving all facets whole in a herringbone lay-out. Further, each facet may be recursively divided into 4 smaller squares. This hierarchical organization allows data processing in different levels of detail. In the last level, pixels are also squares with horizontal and vertical diagonals; this unorthodox orientation may be fixed by the artifice of rotating the entire map by 45 degrees. Since the overall result is equal-area, raster computation yields consistent results. The pseudocylindrical property provides uniform pixel distribution along parallels. Another favorable feature for large data sets is the easily predictable number and location of neighbor pixels.

Variants of HEALPix projections involve further rescaling. For instance, with H = 6, stretching makes the triangles equilateral; this makes for 6H triangular facets, and each may be subdivided into 4 smaller triangles; pixels are likewise triangular. For H = 3, an inverse stretching creates 3 hexagonal facets, which may be subdivided after some artifices for sharing segments between neighbor facets.

 www.progonos.com/furuti    September 17, 2012
Copyright © 1996, 1997, 2006 Carlos A. Furuti