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ARTICLE
Enhancing adaptive composite map projections: Wagner transformation
between the Lambert azimuthal and the transverse cylindrical equal-area
projections
Bernhard Jenny
a
*and Bojan Šavrič
b
*
a
Faculty of Information Technology, Monash University, Clayton, Australia;
b
Esri Inc., Redlands, CA, USA
ABSTRACT
The adaptive composite map projection technique changes the projection to minimize distortion
for the geographic area shown on a map. This article improves the transition between the
Lambert azimuthal projection and the transverse equal-area cylindrical projection that are used
by adaptive composite projections for portrait-format maps. Originally, a transverse Albers conic
projection was suggested for transforming between these two projections, resulting in graticules
that are not symmetric relative to the central meridian. We propose the alternative transverse
Wagner transformation between the two projections and provide equations and parameters for
the transition. The suggested technique results in a graticule that is symmetric relative to the
central meridian, and a map transformation that is visually continuous with changing map scale.
ARTICLE HISTORY
Received 28 April 2017
Accepted 10 September 2017
KEYWORDS
Map projection; dynamic
mapping; adaptive compo-
site map projection; adapta-
ble projection; Wagner
transformation; Umbeziffern;
web map
1. Introduction
Adaptive composite map projections combine several
projections to reduce map projection distortion on
interactive maps. The actual projection used is selected
according to the extent and latitude of the area visible
on the map as well as the map’s height-to-width ratio.
Multiple projections are combined and their para-
meters adjusted to create seamless transitions as the
user changes scale or re-centers the displayed area
(Jenny, 2012). The adaptive composite map projection
technique includes standard equal-area projections that
are recommended in the cartographic literature, and it
is based on established criteria for selecting projections
(Šavrič, Jenny, & Jenny, 2016; Snyder, 1987).
Adaptive composite map projections were origin-
ally developed as an alternative to the ubiquitous
web Mercator projection, which greatly inflates area
in polar regions (Battersby, Finn, Usery, &
Yamamoto, 2014). The technique can also be
included in geographical information system (GIS)
software to free users from the potentially confusing
task of selecting a distortion-reducing projection.
The selection can be automated, as the required
parameters –the geographic extent and the height-
to-width ratio of the map –can be determined for
web maps or GIS desktop applications. We therefore
believe that the adaptive composite map projection
technique has the potential to be useful to many
users and improve a large number of maps.
The projections used for adaptive composite maps
vary with the height-to-width format of a map. At larger
scales, landscape-format maps generally use conic projec-
tions, while square-format maps use azimuthal projec-
tions, and portrait-format maps use transverse cylindrical
projections (Snyder, 1987). Selecting a projection accord-
ing to the format of the mapreduces the overall distortion
caused by the map projection. Our contribution focuses
on equal-area portrait-format maps and aims at improv-
ing the transformation between the Lambert azimuthal
projection and the transverse equal-area cylindrical pro-
jection that are used for portrait-format continental-scale
and large-scale maps, respectively. Transverse cylindrical
projections are commonly used for portrait-format maps
because they result in less distortion than other types of
projections.
The current transformation between these two projec-
tions modifies a conic projection. It suffers from two
shortcomings: The shape of the graticule is not symmetric
relative to the central meridian, and the transformation
results in large angular and distance distortions.
The next section discusses existing transformations
in the adaptive composite map projections. The
CONTACT Bernhard Jenny bernie.jenny@monash.edu
*These authors contributed equally to this work.
CARTOGRAPHY AND GEOGRAPHIC INFORMATION SCIENCE, 2017
https://doi.org/10.1080/15230406.2017.1379036
© 2017 Cartography and Geographic Information Society
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following sections then propose a transverse variant of
the Wagner transformation to solve the shortcomings,
introduce parameters that result in a visually smooth
transition, and present results and conclusions. The
Appendix describes the method originally suggested
by Jenny (2012) for transforming between the
Lambert azimuthal and the transverse Lambert cylind-
rical equal-area projections, and documents its
shortcomings.
2. Transformations in adaptive composite map
projections
The user of a map with adaptive composite projections
can pan and zoom across the map at will. It is therefore
important to use a projection that results in a small
amount of distortion throughout the panning or zoom-
ing process. This can be achieved by continuously
adjusting the projection parameters to the currently
visible map extent such that distortion is minimized.
For example, the standard parallels of conic projections
can be positioned to minimize scale distortion (Šavrič
& Jenny, 2016).
Aside from minimizing distortion, it is equally
important to create visually seamless transitions
between projections when the user pans the map or
adjusts the scale. Abrupt and distracting changes of the
graticule geometry should be avoided. This can be
achieved by transforming projections that are conven-
tionally used in cartography to create intermediate
projections. The transformation parameters are con-
tinuously changed as the user enlarges or reduces
map scale, resulting in smoothly transitioning grati-
cules. Because users can stop panning or zooming as
they please, the transformed projections must always
result in a small amount of distortion.
So far, three conceptually and mathematically dis-
tinct transformations have been applied to adaptive
composite map projections. The first transformation
is blending between two projections. Spherical coordi-
nates are first transformed to Cartesian coordinates by
the two projections and then blended using a weight
that changes with map scale (Jenny & Šavrič,2017).
Blending does not create area-preserving projections
and therefore is primarily useful for transitioning
between a compromise world map projection and a
projection used at a continental scale.
A second transformation uses a conic developable
surface and adjusts the shape of the cone to transition
between the flat projective surface of an azimuthal
projection and a conic developable surface.
1
This
conic projection transformation is also useful for tran-
sitioning between azimuthal and cylindrical
projections. When transforming from an azimuthal
projection, the aperture of the cone (the maximum
angle between two lines connecting the apex and the
base of the cone) is set to π, resulting in an entirely
ablated cone that has the limiting shape of a flat plane.
This can be achieved by setting both standard parallels
of the conic projection to the point of tangency of the
azimuthal projection. The aperture is then steadily
reduced to form an increasingly pointed cone. When
the aperture reaches 0, the standard parallels of the
cone coincide with those of the cylinder; the cone has
become a cylinder, resulting in a cylindrical projection.
The conic projection transformation preserves the rela-
tive size of areas when applied to the equal-area conic
projection. It is used in adaptive composite projections
at large scales for landscape-format maps when the
user changes the central latitude of the map. The result
is a transformation between the azimuthal projection
(for poles), the conic projection (for medium latitudes),
and the cylindrical projection (for equatorial areas).
The conic projection transformation is also used for
landscape-format maps when the user changes map
scale between continental scales and larger scales.
Here, the Lambert azimuthal projection (used at con-
tinental scales) is transformed toward the conic, azi-
muthal, or cylindrical projections, which are used at
large scales.
The third transformation uses a method introduced
by Wagner (1932,1949,1962), which he referred to as
“Umbeziffern,”meaning renumbering. The Wagner
transformation generalizes the Aitoff transformation
(Siemon, 1937). Wagner presented three variations of
his renumbering technique for creating a new map
projection. For his area-preserving transformation,
Wagner first shrinks the geometry on a sphere by scaling
(or renumbering) each longitude and latitude value such
that the entire globe fits into a spherical segment. An
upper and a lower bounding parallel at !ϕBand a left
and a right bounding meridian at !λBdefine the extent
of the spherical segment, which is symmetric to the
central meridian and the equator. Wagner then selects
an existing equal-area projection and applies it to pro-
ject the spherical segment to Cartesian coordinates.
Because the previous renumbering shrank the longitude
and latitude values on the sphere, the area of the result-
ing map is too small, and an enlarging scaling factor is
applied to retain the correct area. Wagner finally adjusts
the height-to-width ratio of the graticule by multiplying
horizontal x-coordinates by a chosen factor, and divid-
ing y-coordinates by the same factor (Canters 2002). In
adaptive composite projections, the Wagner transfor-
mation is used to transform the Lambert azimuthal
projection used for continental-scale maps toward
2B. JENNY AND B. ŠAVRIČ
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various equal-area world map projections (Šavrič&
Jenny, 2014). The Wagner transformation preserves
the relative size of areas if the source projection is an
equal-area projection.
3. Transverse Wagner transformation
In this section, we modify the area preserving Wagner
transformation and apply it to Lambert’s azimuthal
projection for portrait format maps. It allows us to
smoothly transition from the Lambert azimuthal pro-
jection, which is used for continental maps, to the
transverse equal-area cylindrical projection, which is
used for larger scales. The transition to the transverse
cylindrical projection is achieved by creating a limiting
case of the transformation, where the bounding mer-
idian !λBand the bounding parallel !ϕBreach 0°.
We call this method the transverse Wagner
transformation.
The transverse Wagner transformation of the
Lambert azimuthal projection is created in two steps.
First, a spherical rotation is applied to longitude λand
latitude ϕcoordinates to rotate the true poles of the
Earth (Equation (1), after Snyder, 1987). The central
meridian λ0and central latitude ϕ0in Equation (1)
adjust the central point of the map.
tan λ0"ϕ0
!"
¼"sin ϕ
cos ϕ$cos λ"λ0
ðÞ (1)
sin ϕ0¼cos ϕ$sin λ"λ0
ðÞ
If the rotated longitude λ0is outside the valid range
between "πand þπ, then 2πis added to or sub-
tracted from the longitude λ0. Rotated longitude λ0and
latitude ϕ0coordinates are then projected to x=ycoor-
dinates with Equation (2).
x¼1
kffiffiffiffiffiffiffiffiffiffi
m$n
p$ffiffiffi
2
p$sin θ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þcos θcos nλ0
p(2)
y¼" k
ffiffiffiffiffiffiffiffiffiffi
m$n
p$ffiffiffiffi
2$
pcos θsin nλ0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þcos θcos nλ0
p
In Equation (2), xand yare the projected coordi-
nates, λ0is the rotated longitude, sin θ¼m$sin ϕ0,
where ϕ0is the rotated latitude, and the Wagner trans-
formation parameters are defined as m¼sin ϕB,
n¼λB=π, and k¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p$sin ϕB
2=sin λB
2
q, where λBand
ϕBare the bounding meridian and the bounding par-
allel respectively, and pis the ratio between the pro-
jected lengths of the central meridian and one quarter
of the equator.
Equation (3) is the inverse projection (including the
inverse spherical rotation) converting projected xand y
coordinates to longitude λand latitude ϕ, where
X¼x$k$ffiffiffiffiffiffiffiffiffiffi
m$n
p,Y¼"y$ffiffiffiffiffiffiffiffiffiffi
m$n
p=k,
Z¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1"X2þY2
ðÞ=4
p, and n,m, and kare the
Wagner transformation parameters as defined for
Equation (2).
λ0¼1
n$arctan Z$Y
2$Z2"1
$% (3)
ϕ0¼arcsin Z$X
m
$%
tan λ"λ0
ðÞ¼ sin ϕ0
cos ϕ0$cos λ0"ϕ0
!"
sin ϕ¼"cos ϕ0$sin λ0"ϕ0
!"
The bounding meridian λB, the bounding parallel
ϕB, and the ratio pdefine the characteristics of the
resulting graticule. For example, when the bounding
meridian λB¼π, the bounding parallel ϕB¼π
=
2, and
ratio p¼ffiffiffi
2
p, the transverse Wagner transformation
results in the parent projection, the Lambert azimuthal
equal-area. When the bounding meridian and bound-
ing parallel both equal 0°, the result is a transverse
cylindrical equal-area projection. The height-to-width
ratio of the cylindrical projection is adjusted with p.
When the ratio p¼π, the resulting projection is the
transverse Lambert cylindrical projection, which is
used for portrait-format maps at large scales.
The transition between the Lambert azimuthal and
the transverse Lambert cylindrical projections can be
accomplished by linearly blending the Wagner trans-
formation parameters using a weight wbetween 0 and
1. Equation (4) shows this linear blending. λB,ϕB, and
!
preplace λB,ϕB, and pin Equations (2) and (3).
λB¼π$w(4)
ϕB¼π
2$w
!
p¼π$1"wðÞþ
ffiffiffi
2
p$w
Figure 1 illustrates how the transverse Wagner
transformation creates a continuous transformation
between the Lambert azimuthal and the transverse
Lambert cylindrical projections using Equations (2)
and (4). We linearly interpolate between w¼1 for
the Lambert azimuthal (left) and w¼0 for the trans-
verse Lambert cylindrical projection (right).
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4. Minimizing change in distortion
The user of an adaptive composite projection can
adjust the scale and the central point of the map.
These dynamic parameters control the transitions
between various map projections. They also control
the transverse Wagner transformation between the
Lambert azimuthal and the transverse Lambert cylind-
rical projections. The spherical rotation of Equation (1)
is first applied to rotate the selected central point to the
center of the graticule. The change in map scale is then
linearly mapped to the weight w, which is then used
with Equation (4) to determine transformation para-
meters to create a transition as shown in Figure 1.
This linear interpolation of the weight wresults in a
visually inhomogeneous transition between the two map
projections, as illustrated in Figure 2.Comparingthefirst
two maps on the left, it can be observed that the graticule
is compressed vertically and stretched horizontally when
wis reduced from 1 to 0.75. The vertical compression is
indicated in the second map by two circled arrows start-
ing at parallels copied from the azimuthal map with w¼
1(redlines)andendingatparallelsforthemapwithw¼
0:75 (solid lines). The geometric change for wclose to 0.5
and 0.25 is small. When transitioning from the map with
w¼0:25 to the transverse Lambert cylindrical with
w¼0, the map is stretched vertically and compressed
horizontally. Two circled arrows on the rightmost map in
Figure 2 indicate the vertical stretching. A modest
amount of geometric change is expected as the under-
lying map projection is modified. However, the transi-
tional changes are very apparent and distracting when the
user changes the scale of the map.
Figure 1. The transverse Wagner transformation creates a continuous transition between the Lambert azimuthal (left) and the
transverse Lambert cylindrical (right). Only sections at the graticule centers are shown on the map. Scale is linearly mapped to w
(Equation (4)).
Figure 2. Transverse Wagner transformation with linearly interpolated transformation parameters. Red lines show the continental
outlines and graticule of the map immediately to the left.
4B. JENNY AND B. ŠAVRIČ
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To minimize this transitional change in projection
distortion, an alternative weight w0is needed for a
visually smoother transformation. We evaluated 10
different functions that map the linear weight values
w20;1½)to an alternative, nonlinear weight w0¼
fwðÞin the same value range, w020;1½). These func-
tions include the sine, cosine, tangent, and arctangent
functions, exponential and logarithmic functions, sim-
ple polynomial and root functions, and rational func-
tions. Table 1 lists the evaluated functions in two
groups. The first group includes functions where the
slope is increasing as wincreases, for example,
f1wðÞ¼w2. The second group includes functions
where the slope is decreasing as wincreases, for exam-
ple, f2wðÞ¼ ffiffiffiffi
w
p.
We applied each candidate function to the three trans-
formation parameters (the bounding meridian λB,the
bounding parallel ϕB,andtheratiop), and evaluated map
distortion change throughout the transverse Wagner trans-
formation. The evaluation was performed on a map with a
height-to-width ratio of 1.25. This ratio is the limit ratio
where Jenny (2012)switchesbetweensquareandportrait
formats. At a ratio of 1.25, the left and right edges of the
map are also the furthest from the central meridian, and as
such, distortion values are potentially larger. For the eva-
luation, we started with the Lambert azimuthal projection
(w¼1) and decreased the weight wby 0.05 in 20 steps to
transition to the Lambert cylindrical (w¼0). At each step,
we first generated a grid of 61 *77 ¼4697 regularly
arranged points over the area shown on the map. We
then inverse-projected the grid points using Equation (3),
decreased the weight wby 0.05, and computed the new grid
point locations using Equations (1) and (2) and the new
weight. For every grid point, we calculated the Euclidian
distance between the new position and the position at the
previous weight and finally summed all distances. The total
summed distances from all 20 steps were used to compare
all possible combinations of candidate functions. In total,
1331 different combinations were compared, which
included the 10 nonlinear functions of Table 1,aswellas
alinearblending.Finally,weperformedtheevaluationfor
the height-to-width map ratios of 1.5, 2, and 3, using grids
of 55 *83 ¼4565, 49 *97 ¼4753, and 39 *117 ¼
4653 points, respectively.
Equation (5) is the combination that resulted in the
smallest total distances for all tested height-to-width
ratios (1.25, 1.5, 2, and 3). In Equation (5) λB,ϕB,andp
are parameters for the transverse Wagner transformation
(Equation (2)) and w20;1½)is the weight as in Equation
(4). The weight wϕfor the bounding parallel ϕBis com-
puted with the cosine function from the first group in
Table 1 with wϕ¼1"cos π
2$w
!"
, while the weight wλfor
the bounding meridian λBuses the tangent function from
the second group: wλ¼4
π$arctan wðÞ. The ratio puses
the weight wϕfor the bounding parallel ϕB.
λB¼π$wλ(5)
ϕB¼π
2$wϕ
p¼π$1"wϕ
!"
þffiffiffi
2
p$wϕ
5. Results
Figure 3 shows a series of maps created with the transverse
Wagner transformation using transformation parameters
computed with Equation (5). The nonlinear parameteriza-
tion results in a visually continuous transformation.
Whereas the linearly interpolated transformation para-
meters result in an irregular and visually disturbing com-
pression and stretching pattern (Figure 2), the nonlinear
parameterization does not create any visually irregular
stretching or compression (Figure 3). The distortions
introduced with the nonlinear parameterization are so
small that it is impossible to visually detect differences
between the continental outlines of the current parameter-
ization (filled with gray in Figure 3)withtheparameteriza-
tion of the map to the left (red lines).
Table 2 presents summed distances between maps with
different values for the weight w.Wedecreasewby 0.25 as
Table 1. Ten nonlinear candidate weight functions
for finding a nonlinear parameterization of the trans-
verse Wagner transformation.
Group 1 Group 2
w2ffiffiffiffi
w
p
2w"1 log2wþ1
ðÞ
1"cos π
2$w
!" sin π
2$w
!"
tan π
4$w
!" 4
π$arctan wðÞ
w
2"w
2$w
wþ1
The slope of functions increases (Group 1) or decreases (Group
2) as wincreases.
Table 2. Summed distances between maps with different
weight wfor linear and nonlinear interpolations.
Weight w
change
Linear interpolation
(Equation (4), Figure 2)
Sum [m]
Non-linear interpolation
(Equation (5), Figure 3)
Sum [m]
1!0:75 348 175 825 27 082 672
0:75 !0:5 87 842 542 16 869 213
0:5!0:25 108 064 717 9 207 731
0:25 !0 267 506 544 2 728 324
20 steps with
Δw¼0:05
783 148 736 50 607 231
The first four rows show distances summed for map pairs with a difference
in wby 0.25. The last row shows summed distances for 20 maps that
differ in wby 0.05. All values calculated with a spherical radius of
6,371,009 m, with 61 *77 ¼4697 points arranged in a grid, and a
height-to-width map ratio of 1.25.
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in Figures 2 and 3to transition from the Lambert azi-
muthal projection to the transverse Lambert cylindrical
projection. We used 61 *77 ¼4697 points in a regular
grid with a height-to-width map ratio of 1.25 to compute
the summed distances. The first four rows in Table 2 show
that summed distances are considerably smaller when wis
interpolated with the nonlinear Equation (5) instead of the
linear Equation (4). The last row of Table 2 compares the
summed distances computed with 20 decrements of 0.05
applied to the weight w.Thedistancessummedoverall
grid points and all 20 maps of the nonlinear interpolation
are 15 times smaller than the corresponding summed
distances computed with linear interpolation.
6. Conclusion and future research
For portrait-format maps, the transverse Wagner transfor-
mation between the Lambert azimuthal and the transverse
Lambert cylindrical projections outperforms the transfor-
mation originally suggested by Jenny (2012)foradaptive
composite projections. The transverse Wagner transforma-
tion creates a graticule that is symmetric relative to the
central meridian, and is equal-area throughout the trans-
formation. The proposed nonlinear parameterization
results in a visually continuous and pleasing transforma-
tion with small geometric distortion. We conclude that the
originally suggested transformation should be replaced
with the transverse Wagner transformation for portrait-
format maps.
Asomewhatlessthanperfectconictransformation
remains in the adaptive composite projection for land-
scape-format maps for transitioning between the azimuthal
(for maps at continental scales) and the conic (for larger
scales) projections. Because the axis of this conic projection
aligns with the central axis of the map, the resulting grati-
cule is always symmetric relative to the central meridian.
However, distortion caused by the conic transformation is
comparatively large, because the conic and the azimuthal
projections are used with oblique aspects, a spherical rota-
tion, and a translation in order to avoid showing the empty
wedge created by the unfolding of the conic projection
(Jenny, 2012). Improving this transformation between the
Lambert azimuthal and the Albers conic projections is an
open challenge. Alternative projection transformations are
asubjectofrecentinterest.Forexample,thetransformable
Hufnagel projection family (Jenny, Šavrič,&Strebe,2017)
or Strebe’s(2016)adjustablepseudoconicequal-areapro-
jections could provide inspiration for finding a solution.
Note
1. The concept of a developable surface is useful for under-
standing map projections and the conic transformation
discussed here, but the majority of projections (including
the equal-area Albers conic projection) are not constructed
by projective geometry involving any developable surface.
Acknowledgments
The authors would like to thank Jane E. Darbyshire, Oregon
State University, for editing the text of this article, as well as
the anonymous reviewers for their valuable comments.
Disclosure statement
No potential conflict of interest was reported by the authors.
ORCID
Bernhard Jenny http://orcid.org/0000-0001-6101-6100
Bojan Šavričhttp://orcid.org/0000-0003-4080-6378
Figure 3. Transverse Wagner transformation with distortion-minimizing parameterization. Red lines show the continental outlines of
the map immediately to the left. Differences between consecutive maps are impossible to detect.
6B. JENNY AND B. ŠAVRIČ
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Appendix. Original conic transformation
When introducing adaptive composite map projections,
Jenny (2012) suggested a conic transformation for transition-
ing between the Lambert azimuthal and the transverse equal-
area cylindrical projections for portrait-format maps. The
cylindrical projection is used in transverse aspect for por-
trait-format maps, which requires a transformation with a
conic projection with a transverse aspect (that is, the main
axis of a transverse cone is in the equatorial plane). Figure 4
illustrates graticules created with this transverse conic trans-
formation. The red rectangles indicate the geographic area
shown on the map, as the user enlarges the scale of the
portrait-format map. In Figure 4, the cone’s apex is placed
on the right side, but it could also be placed on the left side,
which would create a mirrored graticule.
Figure 4. Transformation with a transverse Albers conic between the Lambert azimuthal (left) and the transverse Lambert cylindrical
(right). Red rectangles indicate the area shown on the map (below). Transverse Albers conic graticules are asymmetric relative to the
central meridian (three center maps).
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The transverse conic transformation creates two major
issues. First, the shape of the graticule is not symmetric
relative to the central meridian. This is visible, for example,
in the second map of Figure 4, where the meridians bulge
towards the left side. This asymmetric graticule pattern is
confusing, because it differs from the commonly used pattern
consisting of meridians symmetrically arranged around a
straight central meridian.
The second issue created by the transverse conic trans-
formation is related to map distortion. The angular and
distance distortions are large, because the center of the
azimuthal projection –which is the area with the smallest
distortion –cannot be used. This can be understood when
considering what happens when the conceptual cone of the
conic transformation is cut along a meridian to unfold it.
The center of the map cannot be used, because the cut
creates an empty wedge that would be disturbing when
shown on the map. This is illustrated by the second map in
Figure 4, where the red rectangle indicating the area shown
on the map is not placed at the geometric center of the
graticule, but instead translated horizontally in order to
avoid showing the area intersected by the empty wedge. To
compensate for this translation, the spherical coordinates are
rotated in the opposite direction before the projection is
applied. For a seamless transition between the azimuthal
(the first map on Figure 4) and the conic (the second map
on Figure 4) projections, a similar transformation is applied
to the azimuthal map. The result is a continuous transition
between the azimuthal and the conic. However, angular and
distance distortion are large.
8B. JENNY AND B. ŠAVRIČ
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