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.

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.

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.

John P. Goode combined the sinusoidal and Mollweide (homolographic)
projections in his hybrid homolosine (*homolo*graphic +
*sinu*soidal) 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.

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.

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'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.

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.

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.

HEALPix, **H**ierarchical **E**qual **A**rea and
iso**L**atitude **P**ixelisation (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 3*H* 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 6*H* 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/MapProj/Normal/ProjPCyl/ProjPCH/projPCH.html — June 16, 2018

Copyright © 1996, 1997, 2006 Carlos A. Furuti