Presented by Karlheinz Wagner in 1941 using the name Flächentreuer Entwurf aus der flächentreuen Azimutalprojektion durch Umbeziffern entwickelt (Equal-area projection, derived
from the equal-area azimuthal projection by Umbeziffern).
See below for a detailed description of the parameters
The projection is equivalent, there’s no areal inflation anywhere.
What’s this all about?
Karlheinz Wagner’s method of map projection transformation – called das Umbeziffern – consists of
selecting a part of an existing projection and mapping the entire surface of the earth onto this part.
The resulting new projection is determined by five configuration parameters.
You can find a detailed description of this process in the article
Umbeziffern – The Wagner Transformation Method.
On this page you can experiment with this mechanism by modifying
four of the five parameters. Change the values within the allowed range and hit the Render my projection button.
The five configuration parameters
This example is using the equatorial aspect of the azimuthal equal-area projection as parent projection,
as the well-known Wagner VII and Wagner VIII do. Modify using the following parameters:
ψ1 – the bounding parallel of the parent projection.
Min. value: 1 / max. value: 90
In effect, lower values increase, higher values decrease the pole line length.
At 90°, the poles become points. At 1°, the pole line is (almost) as long as the equator.
λ1 – the bounding longitude of the parent projection.
min.: 1 / max.: 180
Lower values generate a less pronounced, higher values generate a more pronounced curvature
of the parallels.
At 1°, the parallels become (almost) straight lines. Note: A value of 180° will generate a faulty image.
In that case, try a value < 180, even 179.99° will do.
φ1 – reference parallel for the areal distortion.
Determines the latitude that shows the amount of areal inflation noted in #4.
You can’t modify this parameter, the reason is given in the article
Umbeziffern – The Wagner Transformation Method.
S60 – amount of areal distortion.
min.: 0, max.: 99.999
Noted in percent, a value of 200 corresponds to a areal inflation with a factor of 1.2 at 60° N/S.
A value of 0 will result in an equal-area projection.
p – axial ratio.
min.: 1 / max.: 9999 Reasonable values range (depending on parameters 1 to 4) roughly between 150 and 250.
Noted in percent: The ratio of central meridian to equator. At 200, the equator is twice as long
as the central meridian.
Lower values increase the height of the projection, higher values increase the width.
– There is a form validation which prevents sending values that don’t meet the min/max values noted above.
However, you can try to enter values like e.g. 0.01 for ψ1.
In that case, hit the »Ignore validation & render« button but be aware that a faulty image might be rendered, or no image at all.
– Depending on your browser and the regional settings of your system, the decimal marker can be a dot, a comma, or either of them.
If you run into problem using a dot, try a comma (and vice versa).
The map can be centered to any point on the earth’s surface. Oblique and transverse aspects are possible as well.
Latitudinal projection center. Positive values represent a northern latitude, negative values represent a southern latitude.
min: -90 / max: 90.
Longitudinal projection center. Positive values represent a eastern longitude, negative values represent a western longitude.
min: -180 / max: 180.
Earth’s axial tilt. Positive values rotate counterclockwise, negative values rotate clockwise.
min: -180 / max: 180. A value of ±90 shows the transverse aspect.
Please note that the values of areal inflation are calculated for the equatorial aspect only, i.e. if you enter
a nonzero number for the Lat or axial tilt values, the areal inflation values do not refer to the map image
that is currently displayed.
Instead of building your own projection, you can select an existing projection that can be generated using
the Wagner VII formula. The corresponding configuration parameters will be filled into the form.
This might contribute to a better understanding pf the parameters and is a great start for your own experiments. See below for a detailed description of the various implementations of
Canters’ optimization of Wagner IX.
These option don’t modify the projection in itself but only the image that is generated.
Continents: The land masses can be shown as grey silhouette, as outlines only,
with a colored display of the countries – or not at all.
Graticule: Meridians and Parallels can be shown at a spacing of
5, 10, 15, 18, 20, 30 or 45 degree – or not at all.
Map scale factor: This has nothing to do with the nominal scale that you’ll often find
on printed maps. It’s just an internal factor of the script that renders the projection. The default value of 162 was chosen because with this value, all predefined projections fit into the given boundary.
Higher values increase, lower valued decrease the size of the projection.
Min.: 30 / max.: 300
Background projection: You can compare the rendered projection to one of 281 others projections.
It’s a listing of all the cylindric, pseudocylindric and lenticular projections that are offered by map-projections.net.
The background projection can’t be scaled. To match rendered and background projection in size, you just have to play around
with the map scale factor until you succeed. Sorry.
Tissot Indicatrix: Visualizes the distortions of the current configuration.
The implementation that is used here might not be absolutely accurate, but it’s close enough for a
reasonable evaluation. Read more about Tissot’s indicatrix here.
You can download the generated projection to view or edit it in other applications.
The full projection will be saved to your hard disk, even if here on this page it isn’t shown fully because it
exceeds the bounding frame.
The projection will be saved as SVG file. SVG means Scalable Vector Graphics, and guess what, it is exactly that. ;-)
Which makes is a good choice for line drawings like they are used here.
There are numerous applications to open and edit SVG files, many of them are free, e.g.
and GIMP (all of them are available for Windows, macOS and Linux).
Commercial software (for Windows and macOS) include
and Adobe Illustrator.
The background projection you may have selected will not be part of the SVG file.
But you can download it separately (as PNG file) using the link Download background projection which in this case
will be visible beneath the projection image.
Values of the current configuration
For the current configuration, the following values are displayed:
The representation of a Wagner variant as suggested by Dr. Böhm in his german article
Variationen von Weltkartennetzen.
The extended variant is my own suggestion, it only adds a prefix that indicates
which Wagner projection was modified.
Parameter values as listed by Frank Canters in
Small-scale Map Projection Design:185
In Wagner’s Formula:
Constants to insert into Wagner’s formula for Wagner VII/VIII, from
see Umbeziffern: Notation.
Using the Customizable Wagner in d3 scripts, see below.
Values to insert into the generalized Wagner which is available in the
map projection software Geocart (version 3.2 and up).
A list of values of the areal inflation at latitudes in steps of 10 degrees (plus 85°
to show the enormous increase near the poles). Displayed as factor (e.g. 1.200)
and in percent (20).
Of course, this list is dismissed when you’ve rendered an equivalent projection.
All generated SVG images that are generated on this
page are in the public domain. You may use the images in
any manner, including modifying the content and design,
electronic dissemination, and offset printing. The
author of this site, Tobias Jung, renounces all financial claim to the
data and invites you to use it for personal, educational,
and commercial purposes.
No permission is needed to use the SVG files.
Crediting the author is unnecessary. However, if you wish
to cite the data source, simply use: Tobias Jung,
Background projection images:
The background projection images are licensed under
CC BY-SA 4.0.
For more information please refer to the link Download background projection which is
visible beneath the projection image in case you have selected a background projection.
The author provides this page as a free piece of service
and is not responsible for any problems relating to
accuracy, content, design, and how it is used.
Canters, Frank: Small-scale Map Projection Design.
London & New York 2002.
The projection image above is generated by scripts using D3.js scripts.
However, if you want to know how to do it, please don’t look at this page’s source code!
I’m still using the first quick & dirty version here. Sometime soon, it’ll be replaced by the
newer and cleaner implementation: See
Customizable Wagner Projections using d3-geo – Seven Of Nine.
The full source code is provided there.