A bevy of area-preserving transforms
for map projection designers
Daniel “daan” Strebea
aMapthematics LLC, Seattle, WA, USA
Accepted February 17, 2018 for publication in
Cartography and Geographic Information Science
assigned 16 March, 2018
published online 5 April, 2018
This Author’s Manuscript has the same content as the publisher’s Version of
Sometimes map projection designers need to create equal-area projections to best
ﬁll the projections’ purposes. However, unlike for conformal projections, few trans-
formations have been described that can be applied to equal-area projections to
develop new equal-area projections. Here, I survey area-preserving transformations,
giving examples of their applications and proposing an eﬃcient way of deploying an
equal-area system for raster-based Web mapping. Together, these transformations
provide a toolbox for the map projection designer working in the area-preserving
Map projection; equal-area projection; area-preserving transformation;
area-preserving homotopy; Strebe 1995 projection
CONTACT Daniel “daan” Strebe. Email: firstname.lastname@example.org
It is easy to construct a new conformal projection: Find an existing conformal projec-
tion and apply any complex analytic function to it. Voil`a, new conformal projection!
In practice, ﬁnding the right function for the purpose usually takes some work, but
nevertheless, the toolbox is rich and powerful. Not so for equal-area projections. A few
techniques have been exploited over the centuries, even fewer explicitly described, and
in general the domain remains lightly explored. No area-preserving analog to complex
analysis is known to exist.
This work doesn’t solve such a big problem. Instead, it collects together in one place
a body of techniques map projection designers can use to eﬃciently generate new equal-
area map projections. Some are obvious; some are simple but clever; some are a little
more involved. Their inventions range from centuries ago to being introduced in this
paper. I give examples, with particular attention to a novel replacement scheme for
the Web Mercator.
Since the advent of digital computers, several iterative methods for minimizing
distortion across a chosen region have appeared in the literature. Iterative systems are
speciﬁcally outside the purview of this work. They are adequately described by their
authors in any case. I invite the interested reader to peruse, for example, Dyer and
Snyder (1989) or Canters (2002, pp. 115-244).
For simplicity, I discuss the manifold to be projected as a sphere, but the techniques
I describe generally do not depend on this speciﬁcity. The techniques fall into two
categories: plane-to-plane transformations and “sphere-to-sphere” transformations –
but in quotes because the manifold need not be a sphere at all. Some of the techniques
combine both kinds of transformations or use the plane as an intermediary.
In the case of plane-to-plane transformations, the presumption is that the manifold
has been projected to the plane already in a way that preserves areas. Hence, the
consideration is purely planar, and regardless of what manifold was designated as
the original surface before projecting, the only question is whether the plane-to-plane
transformation itself preserves area measure.
Terminology is not well standardized for equal-area projections. More in line with
the ﬁeld of diﬀerential geometry than with mathematical cartography, the title of this
paper uses the term area-preserving transformation so as to be readily understandable
to the widest audience. However, I do not use that term in the body of this text. To
avoid the awkwardness of noun forms of equal-area, I will sometimes use equivalence.
The term authalic also appears in the literature, but is redundant to my needs here
and will not be used.
•λrefers to a geodetic longitude;
•ϕrefers to a geodetic latitude;
•θrefers to a counterclockwise angle from positive xaxis on the cartesian plane;
•ρrefers to a distance from origin on the cartesian plane;
•0added to a variable name denotes a variable transformed from the variable of
the same name that lacks the prime symbol.
As noted by Strebe (2016), the equal-area property for a mapping from sphere to plane
can be deﬁned either strictly or loosely: strictly in that
∂ϕ =R2cos ϕ, (2.1)
(with Rbeing the radius of the generating globe) or loosely in that
∂ϕ =scos ϕ, (2.2)
with sbeing any non-zero, ﬁnite, real value. The rationale for the strict case is that
the projection maps regions on the globe to the regions on the plane such that the
area measure of any mapped region remains the same as the area measure of the
unmapped region. The rationale for the looser case is that relative areas are preserved
throughout the map, regardless of how their area measure might be scaled with respect
to the generating globe. Simple, isotropic scaling is an area-preserving transformation
by the looser condition.
On its own, scaling is not terribly interesting. Combined with other techniques,
however, it becomes powerful, as we shall see in subsequent sections.
2.2. Aﬃne transformation
Also noted by Strebe (2016), aﬃne transformation preserves areas. That is, given x
and yas the planar coordinates of an equal-area map projection,
c d x
x0, y0is also equal-area, provided the matrix is not singular and a, b, c, d are constant.
When b= 0, c = 0, a =d, the aﬃne transformation degenerates to the simple, isotropic
scaling described in Section 2.1. With b= 0, c = 0, a 6=d, the aﬃne transformation
describes a scaling such that the xand ydirections scale diﬀerently. This can be
useful, particularly when d= 1/a. This last case preserves overall area of the map,
and can adjust the proportions and distortion characteristics of the map to better suit
the purpose. A common example is based on the cylindric equal-area presented by
Lambert (Tobler 1972), with primitive formulae
y= sin (ϕ).(2.4)
In its original form, the projection has no distortion along the equator, and “correct”
scale there. By scaling the height by secϕ0and the width by cos ϕ0, a chosen latitude ϕ0
can be made to have no distortion and to represent nominal scale. No less than seven
variants of Lambert’s original that use this technique have been formally described
independently by nine others, including:
•Gall (1885) (ϕ0= 45◦, “Gall orthographic,” now usually “Gall–Peters,” pre-
sented in 1855)
•Smyth (1870) (ϕ0= arccos p2/π, “Smyth’s equal-surface”)
•Behrmann (1910) (ϕ0= 30◦)
•Craster (1929) (identical to Smyth equal-surface)
•Balthasart (1935) (ϕ0= 50◦)
•Edwards (1953) (ϕ0= 37◦240, “Trystan Edwards”),
•Peters (1983) (identical to Gall orthographic, ﬁrst public mention in 1967)
•Tobler and Chen (1986) (ϕ0= arccos p1/π, “Tobler’s world in a square”)
•Abramms (2006) (ϕ0= 37◦300, “Hobo–Dyer,” commissioned in 2002)
Each of these projections can be described as an aﬃne transformation on Lambert’s
0 sec ϕ0·Lcea.(2.5)
Other examples of this technique can be found in pseudocylindric projections when
the simplest form of the generating formulae does not yield the desired latitude to be
free of distortion along the central meridian. Boggs (1929), for example, stretches x
to slightly more than twice the primitive formulae’s results, and compresses yby the
Aﬃne transformation relates to the scaling constant sreported in Equation (2.2).
c d (2.6)
such that Ais not singular and a, b, c, d are constant when applying Ato the projec-
s= det A,(2.7)
with det being the determinant.
I found no instance of aﬃne transformation with shear components exploited in
cartographic maps before Strebe (2017). As will be seen in Section 2.9, general aﬃne
transformations can serve useful purposes in conjunction with other transformations.
However, perhaps the value of aﬃne transformation on its own has been underappre-
ciated, as discussed next.
Consider an arbitrary, non-conformal projection P. As described by Tissot (1881),
the distortion undergone by a geographical point pwhen projected by Pcan be de-
scribed as the projection of an inﬁnitesimal circle around pinto an inﬁnitesimal ellipse
on the plane. The area of this projected Tissot ellipse, as a ratio to the original circle
from the sphere, gives the amount of ﬂation (or, areal inﬂation or deﬂation, as per
Battersby, Strebe and Finn (2017)) at the projected point.
Angles also undergo changes when projected. Anchored at the center of the unpro-
jected circle, we can construct an unlimited number of orthogonal axes, each rotated
from the rest. When projecting, there will always be some orthogonal axis from the
sphere that remains orthogonal on the plane and therefore remains undeformed. Con-
versely, some axis originally orthogonal must undergo greater deformation than any
other when projected. Likewise, axes will exist for every angle in between undeformed
and maximally deformed. It is the greatest deformation that is used to characterize
angular deformation at the point. The ratio of the major and minor axes of the Tissot
ellipse can be used to compute that maximal angular deformation.
Laskowski (1989) describes the relationship of a projection’s Jacobian matrix Jto
the projection’s local distortion:
where Nis the meridional radius of curvature and Mis the radius of curvature for
the parallel. For the sphere, both Nand Mare 1, and the relationship reduces to
0 1 (2.9)
after inverting the right-side matrix. In this context, the Jacobian matrix is given as
The signiﬁcance of this description is that Tdescribes the aﬃne transformation applied
to the inﬁnitesimal circle from the sphere that results in the Tissot ellipse on the plane.
I will make use of that fact after some preparatory remarks.
Let us say we have a detailed base map prepared for use in a service deployed on
the World Wide Web. We wish to accept arbitrary data sets to represent and overlay
onto the base map in order to augment the map with information customized for a
user’s needs. The bulk of the rendering work and detail needed for the complete map
has already gone into the base map, and we do not wish to render the base map anew
for every custom map because we want to conserve computational resources and to
reduce delivery time.
The scenario turns out to be common: Practically every large-scale mapping service
on the Web serves up “tiles” rendered in advance and upon which user-speciﬁed data
gets overlaid and displayed. The tiles may be raster, as in Google Maps (Rasmussen
2011), or vector, as in Mapbox (2017), but either way, they have been prepared in
advance on a speciﬁc map projection. That speciﬁc projection for the major commer-
cial services is the “Web Mercator,” which is the spherical Mercator projection. At
small scales, its mathematical usage is unremarkable, but its portrayal of the world
is controversial. At large scales, its portrayal of local areas is unremarkable, but its
mathematical usage is controversial. As described by Battersby, Finn, Usery and Ya-
mamoto (2014), small-scale controversy stems from the usual criticisms of Mercator:
It shows gross area disproportion across the map. Large scale controversy stems from
using geodetic coordinates as surveyed against an ellipsoidal model, but projected
using the spherical Mercator. Technically, this practice makes the map slightly non-
conformal, and also contrary to practice in any other context such that the US National
Geospatial-Intelligence Agency felt compelled to issue an injunction against its use for
Department of Defense work (US NGA 2014).
On the other hand, Web Mercator brings considerable beneﬁts to the online map-
ping scenario: At local scales, any place in the world away from the poles gets treated
without perceptible distortion, and therefore “fairly.” North is always up, so orienta-
tion is consistent and familiar. Tiles can be rendered and stored in advance, saving
time and enormous computational resources when serving up tiles. Because Mercator
is locally correct everywhere, adjusting the scale bar is the only change needed for
rendering when panning north-south, and no change at all is needed east-west.
By contrast, using any other projection would eliminate at least one of those bene-
ﬁts. In particular, if the mapping system wished to serve up an equal-area projection
instead, most of the world would be horribly distorted even at local scales. In or-
der to correct that, the mapping system would have to customize the projection’s
standard parallels, at the very least, in order to serve up tiles fair to everyone – at
huge expense for rendering if panning north-south. But most mapmakers would not
be pleased with simple, rectangular equal-area projections; they would want a pseu-
docylindric or something more elaborate, and in that case, the central meridian would
also need to be adjusted and the map rendered accordingly. This means panning in
any direction requires continuous rerendering throughout the operation.
Figure 1. Wagner VII, 15◦and 4◦graticules, aﬃne
transformation to beneﬁt North American Paciﬁc
Now, consider any non-conformal pro-
jection. Because the projection is not
conformal, it distorts angles across most
of the map, which is why it normally can-
not be used eﬃciently for Web map ser-
vices that provide usual pan-and-zoom
functionality. However, if we consider the
local distortion Tfrom Equation (2.8)
as an aﬃne transformation, we can undo
that distortion for any particular point
of interest psimply by applying T−1(p)
to the entire map. This would distort the
rest of the map, of course, but there are
two reasons why this is still reasonable:
(1) When zoomed in, we do not even
display those portions of the map that
would be heavily distorted; and (2) Why
were the undistorted parts of the original
map special anyway? We have changed
how distortion is distributed, but have
not necessarily worsened the overall dis-
tortion. Figure 1 shows this treatment
for the Wagner VII projection.
But, why would we do this? It turns
out that aﬃne transformations are ex-
ceedingly eﬃcient on modern computing
hardware due to the ubiquity of graph-
ics processing units (GPUs) in desktop
and mobile computers. As per Sørensen (2012) and endless other sources, aﬃne trans-
formations, being matrix-vector multiplications, are essentially what GPUs are made
for. By oﬄoading aﬃne transformation from the central processing unit to the GPU,
speed can be increased by several orders of magnitude. Therefore, even though a Web
mapping service might construct static tiles on a particular projection in a particu-
lar aspect, it need not be bound to the original distortion characteristics. Instead, it
could vary the locus of low distortion according to the progression of pan and zoom
by applying, to the entire displayed portion of the map, the aﬃne transformation that
would undo distortion at the point of most interest.
Aﬃne transformation used this way would introduce problems of its own. Marks
whose shapes are not intended to be projected, such as labels or dots for cities, could
not be reasonably rendered on the base maps because of the abuse they would suﬀer
when distorted by the aﬃne transformation. Instead, they would have to be rendered
and applied after the transformation. Furthermore, in order to avoid serious aliasing
when transforming, the original rendering would need to happen at several times the
target resolution, and then be scaled down (decimated) after transforming, implying
more calculation. Aside from increased computational costs, subtleties such as label
placement then get pushed to run-time, where they cannot be corrected by human
intervention. Still, freeing the Web mapping service from the conﬁnes of Mercator
while retaining most of the beneﬁts of using it suggests value in this novel technique.
The need for emancipation is particularly urgent for visualizations of statistical in-
formation at small scales, where Mercator shows up frequently but wholly inaptly.
According to Gartner Inc. (2017), the business intelligence and analytics market is led
by Tableau, Microsoft’s Power BI, and Qlik with their visualization-based approaches
to displaying business data. Displaying data on maps is important to all three plat-
forms, as per Gartner’s description of capabilities they deem critical for their rankings.
Meanwhile, as of this writing, Tableau’s only “natively” supported projection is the
Web Mercator (Dominguez 2016); Power BI’s is also Web Mercator; and Qlik’s was
until 2014, when plate carr´ee (“Unit” projection, in Qlik parlance) was added (Mu˜noz
2015). Unfortunately, plate carr´ee is not much better than Mercator over most of the
world that is relevant to business. All of these systems have ways to show maps in other
projections, but doing so requires abandoning the system’s background map tiles. (In
Power BI’s case, I found no reliable source describing the native base map projection,
and so relied on my own investigations.)
2.3. The Bonne transform
The Bonne projection appeared in rudimentary form in the early 1500s, and was likely
deﬁned precisely by the late 1600s. It is an equal-area projection consisting of arcs of
concentric circles to represent latitude, with each arc subtending the angle needed to
give it proportionally correct length for the parallel it represents. One parallel ϕ1must
be chosen to have no distortion. Meridians intersect a given parallel at constant inter-
vals. The projection is symmetric east-west. As ϕ1approaches the equator, the Bonne
approaches the sinusoidal projection. As ϕ1approaches 90◦, the Bonne approaches the
Another way to think about the projection is as an area-preserving transformation of
the sinusoidal projection. To review, the sinusoidal is a pseudocylindric projection, and,
therefore, in equatorial aspect its parallels are straight lines. Each parallel has correct
scale in the direction of the parallel for its entire length. To transform the projection
into the Bonne, each parallel is bent into the arc of a circle without stretching it,
such that each arc’s circle is concentric to the others and therefore all have a common
center. The radius of the circle for any parallel’s arc ultimately is determined by the
latitude chosen for ϕ1. If ϕ1is a low latitude, the common center will be far from the
map, parallels will curve gently, and the arc subtended by each parallel will be a small
fraction of its circle. If ϕ1is the equator, the common center will be oﬀ at inﬁnity
and the arcs will be straight segments. If ϕ1is at a high latitude, the common center
will be close to the map, parallels will curve rapidly, and the arc subtended by each
parallel will be a large fraction of its circle.
Figure 2. 15◦graticules and centered at 11◦E.
Top: Minimum-error pointed-pole equal-area projection
(Snyder, 1985, p. 128).
Bottom: Aﬃne scaling to correct center scale, and
Bonne transform applied with ϕ1= 22◦
Thought about this way, the Bonne
projection is a planar transform. What
makes it unusual is that its range need
not be conﬁned to the projection itself
because we only need use longer arcs
in order to extend the range, up un-
til the point where the arc wraps back
upon itself. With this extended range,
the transform can be applied to arbitrary
other projections even without scaling
ﬁrst. Therefore, the Bonne transform is a
parameterized, equivalent planar trans-
form, while the Bonne projection is an
instance of that transform applied to the
I used the Bonne transform to create
a series of equal-area projections based
on manipulations of existing projections.
The projections are bilaterally symmet-
ric, equal-area, and have curved paral-
lels in equatorial aspect. Each can be pa-
rameterized by a “latitude of curvature,”
which is the ϕ1of the Bonne transform
but without the meaning of a standard
parallel in the resulting projection. First,
I apply an aﬃne transformation to the original projection to give it correct scale at
the center. Then I interrupt the base projection along the equator by applying the
Bonne transform independently to both northern and southern hemispheres, greatly
reducing distortion in each hemisphere (Figure 2). As an interruption scheme, this is
similar to some cordiform projections from the 16th century, such as those of Oronce
Fine, 1531, “Nova, Et Integra Universi Orbis Descriptio,” or Gerard Mercator, 1538,
untitled double cordiform map as copied by Antonio Salamanca, c. 1550. However,
whether the motivation was the same or not is unclear. Those earlier forms were not
equal-area and arose out of geometric construction rather than planar transformation.
As found in Geocart 1.2 (1992), a commercial map projection software package I
authored, originally I did not split the equator all the way to the central meridian.
Instead, after centering the map at 11◦E to prevent separating the Chukchi peninsula,
I interrupted along the equator from the left edge eastward 90◦through the Paciﬁc,
and split the same distance inward from the east edge. The central portion of the
map remained unchanged, while the eastern and western outer wings of the northern
hemisphere curled upward from the cusps of the interruptions, and likewise downward
for the southern. By means of the partial interruption, the map avoided slicing major
I achieved the partial interruption by applying the left half of the Bonne transform
to the leftmost 90/360 of the map, and the right half of the Bonne transform to the
rightmost 90/360 of the map. A vertical line struck at the inner limit of each equatorial
interruption became the “central meridian” for each of the half-Bonne transforms.
This procedure yielded a smooth, equal-area map. However, as a piecewise function,
the projection could not have continuous derivatives and therefore its distortion char-
acteristics must change abruptly. I abandoned partial interruption for this projection
series starting with Geocart 2.0 (1994). Nevertheless, this application of the Bonne
transform to only parts of a map emphasizes its nature as a “transform,” more than
a “projection,” and also illustrates that planar transforms can be applied piecewise
to portions of the full equal-area map if convenient. This ﬂexibility is unavailable to
conformal maps because a transformation of a conformal map that yields a continuous
mapping necessarily aﬀects the entire map.
2.4. Directional path oﬀset
Figure 3. 15◦graticules and centered at 11◦E. Appli-
cation of directional path oﬀset to close interruption
Aﬃne transformation in the most gen-
eral sense can be thought of as a combi-
nation of scaling independently in both
directions, rotating around the origin,
and shearing. Shearing is the operation
that turns a rectangle into a rhombus.
As an aﬃne operation, shearing, too,
preserves area. A conceptual model for
shearing’s area preservation will be help-
ful. Considering a rectangle we wish to
shear such that it leans rightward, we
can break the rectangle up into an in-
ﬁnite number of slivers as rectangles of
the same width and inﬁnitesimal height.
We leave in place the sliver at the base
of the original rectangle. We slide all the
slivers above it an inﬁnitesimal distance
to the right. Then we hold the sliver next
to the base sliver ﬁxed, and slide every-
thing above it again by the same amount.
Then we hold the next sliver up ﬁxed,
and repeat, all the way up to the top
sliver. It should be clear that area has
not changed, since we kept each sliver the same length and (inﬁnitesimal) height, and
opened no extra space between them.
Notice that the concept for area preservation holds regardless of whether the shift
amount is constant or otherwise. In other words, we can deform that rectangle into
other shapes besides rhombi while preserving area. We must be careful not to open up
space between the slivers, or skew them with respect to each other; we may only slide
them against each other, all in parallel. If we honor those conditions, we can deﬁne an
arbitrary path for that left edge. The same argument holds for pressuring the original
rectangle from any constant direction, not just against one of its edges. Applying this
principle to the projection of Figure 2, we arrive at Figure 3.
I used this technique in a National Geographic animation (Strebe, Gamache, Vessels
and T´oth (2012)) in order to progressively close up the interruptions in the closing
sequence featuring a Mollweide projection. The projection remains equal-area through-
out. I also exploited this technique in the 1992 series mentioned in Section 2.3. In their
original form wherein the equatorial interruption was only partial, the option to close
up the interruptions or leave the map interrupted was available. When I abandoned
partial interruption in favor of the full interruption, I eliminated the option to show
the projections as interrupted from Geocart, which now always uses the directional
path oﬀset to close them up. The projections still appear in the extant Geocart 3.2
(2018) in the closed-up form, but were never formally described. They are shown here
as Figure 4. The meridians of those projections are kinked at the equator as a conse-
quence of the manipulation, a typical, undesirable side eﬀect of directional path oﬀsets
when used to close interruptions.
Figure 4. 30◦graticules and centered at 11◦E; Bonne ϕ1in parentheses.
Top: Strebe-sinusoidal (“cartouche”) (25◦4401600), Strebe-Hammer (15◦), Strebe-Kavraiskiy V (16◦270).
Bottom: Strebe-Mollweide (16◦), Strebe-Snyder ﬂat-pole (20◦), Strebe-Snyder pointed-pole (22◦)
2.5. Meridian duplication
Aitoﬀ (1892) introduced a brilliant little device that may have been the ﬁrst promi-
nent sphere-to-sphere transformation for cartographic maps. His invention was to halve
the longitudinal value of every location on the sphere, squeezing the sphere into one
hemisphere. Then, after projecting the hemisphere onto the plane as an equatorial
azimuthal equidistant, the method stretches the map horizontally 2:1 in order to com-
pensate for the squeezing on the sphere. By these means he created the Aitoﬀ projec-
tion, a simple, elliptical, pseudoazimuthal projection that reduces shearing compared
to the similar pseudocylindric Apian II projection by gently curving the parallels.
Aitoﬀ published the ﬁrst map on the projection in Atlas de g´eographie moderne in
Aitoﬀ’s invention soon inspired Hammer (1892) to pull the same trick on the Lam-
bert azimuthal equal-area projection. Hammer’s insight was that Aitoﬀ’s sphere-to-
sphere transform preserves areas because the squeezing on the sphere maintains a
constant relationship between the diﬀerential properties deﬁning the area metric on
the sphere. Therefore, projecting the squeezed sphere onto the plane via an equal-area
projection must result in an equal-area projection. Scaling, as noted in Section 2.1,
also preserves area, and so the resulting projection must be equivalent. The Hammer
projection has seen much use as a curved-parallel alternative to the pseudocylindric
Mollweide in much the same way that the Aitoﬀ is a curved-parallel alternative to
the Apian II. Startlingly, despite the Hammer projection’s favorable properties, it is
computationally much cheaper than the Mollweide because it requires no iteration.
Figure 5. Strebe’s heart projection,
ϕ1= 85◦, “meridian duplication fac-
tor” = 6/5, or n=5/6
Hammer chose n=1/2to multiply longitudes by in
his formulation, but over the years a few other values
for nwere proposed, such as Ros´en’s 7/8or Eckert-
Greifendorﬀ’s 1/4. Briesemeister proposed an oblique
case of the Hammer with change in aspect ratio (Sny-
der, 1993, pp. 236-240).
The Bonne projection, especially its specialization
in the form of Werner (ϕ1= 90◦), frequently appears
as a novelty to show the world in a heart, with equiv-
alence as a bonus. Dissatisﬁed with the shape’s aes-
thetics, I devised a variant by 1992 and incorporated
it into Geocart 1.2 as simply the “heart” projection.
It has not been otherwise described. I observed that
“meridian duplication,” as I call Aitoﬀ’s innovation,
would suﬃce to modify the Bonne into a more or less
idealized heart shape. The projection permits conﬁg-
uration via ϕ1of the Bonne and the meridian dupli-
cation factor, set by default to 85◦and 6/5, respectively (Figure 5). My convention for
the meridian duplication factor is the reciprocal of ngiven above in Hammer’s formu-
lation and below in Wagner’s, and happens to coincide with that of Snyder (1993, p.
Kronenfeld (2010) also uses meridian duplication in developing a simple sphere-to-
sphere transformation. In Kronenfeld’s case, longitudes are expanded with the intent
of projecting a fraction of the globe onto a larger section of the globe while preserving
areas. Kronenfeld refers to this as “longitudinal expansion factor.” This procedure
must discard regions of the globe; otherwise parts would wrap and overlap. With nas
above, αas the desired magniﬁcation, λ0as the “reference meridian,” and ϕ1as the
“reference parallel” (together constituting the “origin,” in a sense), the transformed
spherical coordinates (ϕ0, λ0) are given as
n(sin ϕ−sin ϕ1) + sin ϕ1, λ0=λ0+n(λ0−λ).(2.11)
How (or whether) the transformed spherical coordinates then get projected to the plane
is up to the projection designer. I note that Kronenfeld’s transform is the equivalent of
excerpting a rectangle from a cylindrical equal-area projection, scaling it, translating
it on the plane to center it at projected (ϕ1, λ0), and deprojecting back to the sphere
via the inverse of another cylindrical equal-area projection. Each of these steps is noted
individually in this paper.
2.6. Das Umbeziﬀern
Wagner (1932) generalized Aitoﬀ’s notion to reassign not only longitudinal values,
but latitudinal values as well. He called the procedure Umbeziﬀern, or “renumbering,”
referring to the reassignment or relabeling of the longitude and latitude values. The
theory was developed in depth by Karl Siemon over a series of papers in 1936–8 and
then deployed by Wagner over the course of his life in the development of many texts
and projections, starting in 1941 (Canters, 2002, pp. 119-124). Wagner and Siemon’s
explications went beyond just equal-area projections to accommodate several con-
straints, but what concerns us here is the equivalence transformation, which Canters
refers to as Wagner’s second transformation method. Presuming that
•0◦N, 0◦E projects to x= 0, y = 0;
•f1denotes the base projection’s generating function for x;
•f2denotes the base projection’s generating function for y;
•x0, y0are the coordinates produced by Wagner’s second transformation;
•nis the fraction to multiply longitudes by;
•kis a desired scaling (such as to eliminate distortion at selected parallel at central
•mis a free parameter
√mnf1(u, v), y0=1
where v=nλ, sin u=msin ϕ.
Besides those of Wagner himself, projections using Wagner’s method were devised by
B¨ohm (2006). ˇ
Savriˇc and Jenny (2014) present an adaptable pseudocylindric projection
from Wagner’s method, and Jenny and ˇ
Savriˇc (2017) use it to transition between
Lambert azimuthal equal-area and a transverse cylindric equal-area in order to improve
the adaptive projection system Jenny (2012) describes.
The diﬀerential forms that identify a projection as equivalent are not conducive to
generating new projections. Given suitable constraints and boundary conditions, it
can be done, but in the general case requires nested integrals with the implication of
enormous computational cost. Most of the techniques described in this work are, in
essence, ways to avoid dealing directly with the inﬁnitesimals.
Van Leeuwen and Strebe (2006) describe a mechanism for approaching the two-
dimensional problem of area preservation one dimension at a time. While we only
applied the method in the context of polyhedral faces, it is generally applicable. Our
work proves that you can reach the diﬀerential qualiﬁcations of an equal-area projec-
tion by slicing up the space in one direction (or dimension) such that each slice has the
correct area, and then dicing up the slices in a diﬀerent direction (or dimension) such
that the second dimension’s dices also preserve areas. The two directions need not be
orthogonal–and cannot be everywhere–but the results will still retain equivalence. The
paper generalizes an earlier projection by Snyder (1992) and provides a theory for it.
Any equal-area projection can be thought of in these terms. For example, ﬁnding
the position of a latitude in the well-known Mollweide projection can be thought of
as slicing the ellipse with a straight, horizontal parallel placed so that the proportion
of the global area higher in latitude is correct with respect to the global area lower in
latitude, and then dicing up the same space with meridians whose spacing is constant
along the parallel. The dice condition of constant spacing is forced by how we chose
More as a way of thinking about equal-area projections than a “technique,” slice-
and-dice has no general formulae to resort to, and in many cases provides no shortcut
to a solution. Van Leeuwen and I used it in the context of our work because the great-
and small-circle partitions we explored sliced and diced the space in computationally
eﬃcient ways. I do not explore the technique further here but instead refer the reader
to the 2006 paper for proof of the concept and applications.
2.8. Substitute deprojection
As noted in ˇ
Savriˇc, Jenny, White and Strebe (2015) in describing the “Strebe trans-
formation,” an equivalent mapping from the sphere back onto the sphere can be cre-
ated by projecting from sphere to plane while preserving areas and then deprojecting
back onto the sphere by means of some other equal-area projection’s inverse. This is
a sphere-to-sphere mapping mediated by projection onto the plane. The result may
then be projected again to the plane by means of yet another equal-area projection.
A reason to do this is to draw upon the vast corpus of extant equal-area projections.
After the Bonne transform experiments that resulted in the projections of Figure 4,
I considered ways to eliminate the unsatisfactory discontinuity at the equator without
changing the other characteristics. Noting that Mollweide and Hammer share the same
projection space and are both equal-area, I conceived of a vector space that transforms
Mollweide to Hammer, with the intent of applying that vector space to other equal-
area projections. This readily generalizes to the observation that any portion of an
equal-area projection Acould be treated as if it were a portion of any other equal-area
projection B, thence deprojected back onto the sphere via B’s inverse B0, and ﬁnally
projected back to the plane by yet another projection Cin order to arrive at a ﬁnal
projection D. Aﬃne transform Xis permissable between Aand B0if needed for A’s
domain to ﬁt within B’s conﬁnes or even just to change how much of Bimpacts A.
Aﬃne transformation at that stage normally would be counteracted by its inverse after
projecting via C, and so the full sequence would look like this:
Ultimately, I chose Eckert IV as A, Mollweide as B, and Hammer as C. For XI chose
with s= 1.35 recommended.
Collected, this is the formulation of the Strebe 1995 projection (Figure 6):
scos ϕpsin λp, y =sD sin ϕp
1 + cos ϕpcos λp
sin ϕp=2 arcsin √2ye
xe=sλ(1 + cos θ)
√4π+π2, ye= 2 √πsin θ
s√4 + π,
where θis solved iteratively:
θ+ sin θcos θ+ 2 sin θ=1
2(4 + π) sin ϕ.
Figure 6. Strebe 1995 projection, 15◦graticule and
centered at 11◦E, s= 1.35
As quoted by Raposo (2013), my goals
for the projection design were to pre-
serve area; maintain bilateral symmetry;
and push as much distortion as feasi-
ble into the oceans and away from the
land masses without resorting to inter-
ruptions. Geographer Marina Islas, who
used the projection for a map tattooed
on her upper back, appreciated it for its
organic shape and Afro-centric presen-
tation (Zimmer, 2011, pp. 90-91), (New
York Times, 2011, image in online re-
Savriˇc et al. (2015) used substitute deprojection in constructing novel map projec-
tions for a study of map reader preferences with regard to map projection aesthetics.
One of their hypotheses was that map readers would prefer projections whose outer
corners are softer, or rounder. To test the hypothesis, they developed a Wagner VII
with rounded corners and a Miller with rounded corners. (Though the substitute de-
projection they used does preserve areas, that particular beneﬁt was lost on the Miller
projection, which is not equivalent.)
2.9. Strebe’s homotopy
Strebe (2017) describes an eﬃcient method for synthesizing a continuum of equal-area
projections between any two chosen equal-area projections. A parameterized contin-
uum between two projections is known as a homotopy in algebraic topology and related
ﬁelds. The need for such a continuum arose in the context of Jenny’s eﬀorts to im-
prove his adaptive composite projection system of 2012, where no good transition from
Lambert azimuthal equal-area to a transverse cylindric equal-area projection had been
found. While the system I devised then solved the immediate need, its applicability is
far more general: It can produce a continuum between two arbitrary projections with
few restrictions, whether equal-area, conformal, mixed, compromise, or otherwise. If
both the initial and terminal projections are equal-area, the result will be equal-area.
If both are conformal, the result will be conformal. The general description is the same
where Cis the “weighted average” projection; Ais the initial projection; A0is its
inverse; Bis the terminal projection; and 0 ≤k≤1 is the weighting. (As k→0, this
formulation should be taken as a limit.) This process includes a substitute deprojection
in the form of A0(k·A).
In the context of equal-area projections, Equation 2.15 may not suﬃce because the
point on the plane P=k·Aas k→0 might not be undistorted for the given A. If
A(P) is distorted, then Formula 2.15 will not result in Aas k→0 because A0(P) back
Figure 7. 15◦graticules on Bonne-Albers homotopies. Left: Strebe homotopy, k=1/2, ϕ1= 29◦300, ϕ2=
45◦300, ϕ3= 37◦402400; Right: adaptable equal-area pseudoconic, k= 0.47, ϕ1= 10◦, ϕ2=ϕ3= 40◦.
onto the sphere would undo the original distortion. A simple, failsafe way to attain A
in that situation is to apply an aﬃne transformation Mato the result to reassert the
original distortion. The strength of Mamust wane as kprogresses away from 0. One
formulation of Ma, not unique, is
Na=k·I+ (1 −k)·Ta(P)
where Tameans the Tissot transformation at A(P) from Equation 2.8, and Iis the
Likewise, if B(P) is distorted, then Equation 2.15 will not suﬃce because B(A0[k·
A(P)]) asserts B’s distortion even when k= 0, where we expect C=A. We could
correct that by reversing the distortion of B(P) by applying an aﬃne transformation
Mb. One formulation, not unique, is
Nb=k·I+ (1 −k)·T−1
The general expression for equal-area homotopies then becomes
Homotopies generated by this method are asymmetric; that is, reversing Aand B
and replacing kwith 1 −kin the formulation will not result in the same C. In my
2017 work, I demonstrate both directions of homotopy between Albers and Lambert
azimuthal equal-area for the full sphere. The method worked well despite the very
diﬀerent topologies of a conic and an azimuthal projection. Figure 7 gives another
example, comparing my homotopy method to my adaptable equal-area pseudoconic
projection (itself a homotopy) that hybridizes Albers and Bonne (Strebe 2016). Despite
the markedly diﬀerent methodologies, practically identical results can be obtained
because of the number of degrees of freedom available in parameterization.
Strebe’s homotopy uses substitute deprojection at the level of inﬁnitesimals. In that
sense it is both a generalization of, and a specialization of, substitute deprojection.
Generalization, because it extends the concept into an area-preserving calculus; spe-
cialization, in that only two of the three projections are independent of each other, as
well as because it imposes a sharply deﬁned goal onto the procedure: homotopy. Sub-
stitute deprojection on its own, meanwhile, imposes no goal other than the synthesis
of new equal-area projections out of existing ones.
2.10. Axial yanking
I developed this technique as a generalization of scaling. If we pick an axis for the
projection (often the central meridian or the equator on an equatorial aspect with
bilateral symmetry), we can choose a function fto apply to the partial derivatives
of the projection in the direction of that axis. Considering a given point Palong the
axis, insofar as we apply the reciprocal of that evaluated function fto every point
perpendicular to the axis from P, we will have arrived at a new equal-area projection.
In a sense, we scale each parallel line (geometrically parallel, not parallels of latitude)
by a diﬀerent amount.
Let me illustrate with the Hammer projection. Reviewing its generating functions
(after Snyder, 1989, p. 232):
D=√2 (1 + cosϕcos nλ)−1
ncos ϕsin nλ, y =Dsin ϕ, (2.19)
2is the meridian duplication factor Hammer chose. Partial derivatives
relevant later are:
2zcos ϕcos ϕcos2nλ + cos ϕ+ 2 cos nλ(2.20)
2zcos2ϕcos nλ + 2 cos ϕ+ cos nλ(2.21)
z= (1 + cos ϕcos nλ)−3
The Hammer projection has no distortion at the center, but is distorted everywhere
else, including along the central meridian and along the equator. Let us suppose we
want a similar projection but with an undistorted central meridian. To achieve this,
we can “yank” the vertical axis to be undistorted by letting fdiscussed above be
the reciprocal of Equation 2.21. However, we do not need to evaluate that explicitly
because we chose fsuch that ywould be ϕalong the central meridian. That is, ∂y/∂ϕ
will be 1 along the central meridian.
Here is how to express that. Given Has the Hammer projection deﬁnition, for every
[xc, yc] = H(ϕc, λc) projected coordinate, ﬁnd ϕcm such that [0, yc] = H(ϕcm,0), and
then force y=ϕcm. Next, in order to preserve equivalence, multiply xcby ∂
∂ϕ H(ϕcm ,0)
because we implicitly divided the inﬁnitesimal height of the entire line ycby that value
when we set y=ϕcm.
Finding yc=H(ϕcm,0) means needing the inverse for the yvalue of H, but only
along the central meridian such that λ= 0:
yc=√2 sin ϕcm (1 + cos ϕcm)−1
Figure 8. Hamusoidal, n=1/2, 30◦graticules, 10◦increments in angular deformation. Left to right:
Hammer; vertical axis undistorted; horizontal axis undistorted; both axes undistorted
Solving for ϕcm,
ϕcm = sgn (yc) arccos 1−1
At λ= 0, simplifying Equation 2.21,
∂ϕ H(ϕ, 0) = √2
2pcos ϕ+ 1 (2.24)
which, when ϕ=ϕcm as per Equation 2.23, is
∂ϕ H(ϕcm,0) = 1
Consolidated, Hammer with vertical axis yanked to eliminate the axis distortion is,
c, y = sgn (yc) arccos 1−1
with [xc, yc] as [x, y] from 2.19.
Using the same concepts, we could, instead, yank the horizontal axis to have no
distortion. This makes use of Equation 2.20 instead of 2.21, but follows a substantially
similar derivation. Abbreviating,
c, y =yc·1
But, why stop there? Because the vertical operation does not aﬀect the horizontal
axis, and vice versa, we can apply both in succession to render both the vertical and
horizontal axes without distortion. Not surprisingly, this last projection is much like
the sinusoidal but with curved parallels. Which gets applied ﬁrst does matter, but the
diﬀerence is small across the full allowable range of n. Whether one axis or both get
straightened, I call this family of projections “hamusoidal.” It appears as Figure 8.
Noting that we can parameterize Hammer with a choice of nand still yank the
horizontal axis to constant scale, we can pair a northern hemisphere having one pa-
rameterization with a southern hemisphere having a diﬀerent parameterization while
retaining continuity across the equator. Doing so, I arrived at the whimsical equal-area
“kiss” projection of Figure 9.
Figure 9. “Kiss” pro jection, hamusoidal north n=
0.85, south n= 0.65. 15◦graticule, 10◦increments in
We eliminated distortion along an axis
in these examples, yielding a side eﬀect
of computational simplicity. However, as
noted at the top of this subsection, what
we are really doing is applying some
function to the partial derivatives along
the axis. The resulting axis is the integral
of those partials. If we want something
other than constant scale along an axis,
we would chose fto be something other
than the reciprocal of the partial deriva-
tive. I do not explore that further here
other than to note the obvious ﬂexibility
and easy integrability of polynomials as
a distortion function.
2.11. Radial oozing
An azimuthal projection in north polar
aspect has the basic form
2, ρ =g(ϕ),whence
x=ρcos θ, y =ρsin θ. (2.28)
If we generalize the system such that ρbecomes a function of both ϕand λ, and such
that θbecomes a function of λ, we have
θ=f(λ), ρ =g(ϕ, λ).(2.29)
The consequence of this relaxation is that meridians remain straight, but angles be-
tween them vary, and parallels are no longer described by circles.
We wish to constrain this system to be equal-area. As given by Strebe (2016), the
general condition for equivalence in polar coordinates from a spherical model is
∂ϕ = cos ϕ. (2.30)
In this case, ∂θ/∂ϕ = 0 because θis a function solely of λ, and therefore
∂ϕ = cos ϕ. (2.31)
This is not fully constrained; conic and pseudoconic equal-area projections also satisfy
this diﬀerential equation because ∂ρ/∂λ = 0 in those cases. What we wish to do here is
to squish a Lambert azimuthal equal-area projection so that it oozes into some shape
other than circular. A way to do this that meets the criteria set out above is to require
that all parallels retain the proportion of their spacing along the meridians, while
meridians themselves vary in length. This implies that ρis the same as Lambert’s,
but shrunk or expanded by some function hthat depends only on λor θ, with ∂ρ/∂ϕ
inheriting the same dependency. Lambert’s ρis given by 2 sin(π
into Equation 2.31,
−h2 sin( π
∂ϕ hh2 sin( π
2)i= cos ϕ, (2.32)
∂λ =h−2(λ) or ∂λ
∂θ =h2(θ) (2.33)
(depending on whichever is convenient for the projection designer’s speciﬁcation for
the projection), and therefore
θ=Zh−2(λ) dλor λ=Zh2(θ) dθ. (2.34)
If we specify our projection via the shape we want for the outer boundary, presum-
ably we express the shape in terms of ρaround the full sweep of θ, or in cartesian
coordinates to cast into those polar coordinates. We know how to proportion the par-
allels along any meridian implied by ρ(θ); all that is left is to determine the meridional
spread thus implied. This is available from Equation 2.34. You might have noticed that
this turns ρinto a mixed-up function of ϕand θfrom ϕand λ, but because θhas no
dependence on ϕand is invertible with λ, nature will ignore the transgression.
As an example, let us deform the Lambert azimuthal equal-area projection into
an equal-area square. We will deﬁne the perimeter in terms of ρand θ, where ρ=
h(θ)2 sin π
2. The perimeter is not a continuous function because of the corners,
so instead we will treat only the 0th octant θ= [0,π
4) here, with the remaining even
octants being a rotation of the 0th octant and the odd octants being a reﬂection plus
rotation of the 0th octant. The area of a sphere of unit radius is 4π, so each side of a
square of that area has length 2√π. Simple trigonometry yields
h(θ) = √π
and therefore, by Equation 2.34,
4tan θ, whereby
θ= arctan 4
πλand, as noted before,
ρ= 2h(θ) sin π
=√πsec θsin π
x=ρcos θ, y =ρsin θ. (2.36)
Figure 10. “quasiazimuthal equal-area square” pro-
jection, 15◦graticule, 10◦increments in angular defor-
I show the result in Figure 10. Many
years ago, in a personal communication
from Waldo Tobler, I received a plot-
ted projection that is apparently iden-
tical. However, nothing about the con-
text remains. Tobler (2008, 31) notes, “It
is relatively easy to ﬁt equal area maps
into regular n-sided polygons,” and illus-
trates the pentagonal case, but does not
Using these same principles, I de-
veloped the whimsical “quasiazimuthal
equal-area apple,” shown in Figure 11,
appearing in Geocart 1.2 (1992). The
outer boundary is described piecewise
as conic sections, the piecewise nature
of which accounts for the discontinuities
in the distortion diagram. The bound-
ary description is too involved to present
I use “quasiazimuthal” to mean a
projection which, in polar aspect, has
straight meridians without constant an-
gular separation between them. Recog-
nizing that a region boundary on an
equal-area projection says nothing about
the region’s interior, it follows that an
equal-area projection in some particular
shape, such as an ellipse or a square or
an apple, is not unique. Qualifying the
description with “quasiazimuthal” spec-
iﬁes which among an unlimited number
of projections it is.
Lastly, we observe that a plane-to-plane form of the equal-area condition in polar
where the primes mean the transformed coordinates, and without primes mean the
original polar coordinates on the plane. We are no longer concerned with the original
manifold; insofar as it got projected by some equivalent means onto the plane, we can
set our origin wherever we like and use Equation 2.37 to establish that the result truly
preserves areas. As applied to planar radial oozing, we prohibit a dependency of θ0on
ρ, simplifying the condition to
∂ρ =ρ. (2.38)
As in the spherical case, we posit some function h(θ) or h(θ0) to warp ρinto ρ0. We
are no longer concerned about the spacing of parallels; this is merely linearly stretching
ρinto ρ0, and so ρ0=hρ and ∂ρ0/∂ρ =h, and Equation 2.38 simpliﬁes to
∂θ = 1 or h2(θ0)∂θ0
∂θ = 1
∂θ =h−2(θ) or ∂θ0
h2(θ)or θ=Zh2(θ0) dθ0.
Figure 11. “quasiazimuthal equal-area apple” projec-
tion, 15◦graticule, 10◦increments in angular deforma-
Applying this to the example of equiv-
alently and radially oozing a unit circle
into a square with sides of length √π, we
θ0= arctan 4θ
2ρsec θ. (2.39)
And ﬁnally, I note that radial oozing is
analogous to axial yanking. In radial ooz-
ing, the angle subtending meridians is
speciﬁed by formula, just as is the linear
spacing along the axis caused by yank-
ing. The radial scaling is thereby deter-
mined by the derivative of the radial for-
mulation, just as the perpendicular scal-
ing is determined by the derivative of the
I have gathered and described here 11
distinct methods for generating equal-
area map projections. Three methods are
new to the literature: axial yanking, di-
rectional path oﬀset, and radial oozing.
I have identiﬁed the Bonne projection
methodology as a transform in its own
right. I have described novel uses for
aﬃne transformation, a method which
seems not to have literature devoted to
it in its most general forms. I have re-
ported several novel projections as ex-
amples of using these techniques, and de-
scribed the methods behind some projec-
tions found in the Geocart software but never described formally. Further, I proposed
a way that aﬃne transformation could be used to render locally correct maps for any
region without having to rerender the underlying raster tiles based on a single projec-
tion. This technique would open up Web mapping to the use of any projection without
discarding the beneﬁts of the ubiquitous Web Mercator.
By consolidating these methods into one monograph, I hope to help map projection
designers develop and explore the domain of equal-area projections. I ﬁnd the theory
of equivalent projections lacking when compared to conformal projection theory. As
a description of techniques, the present work does not directly advance theory, but I
nevertheless hope it will inspire ideas and research directed toward a comprehensive
understanding of area-preserving transforms.
I am grateful for the attention given to this paper by the several anonymous reviewers,
as well as by my less anonymous wife, Tiﬀany Lillie. It is a better paper for their eﬀorts.
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