Wagner Variations Generator
Custom map projections: Umbeziffern applied to projections with equally spaced parallels
Short info: see below – More Info: Wagner – Das Umbeziffern
– see also: WVG, based on Wagner VII
and the Interactive d3 implementation
See below for a detailed description of the parameters
Canters’ Optimisation of Wagner IX
Lowerror optimization of Wagner IX, by Frank Canters
Values of the current configuration
Böhm’s Notation (adapted)
ix@67.13186.562206.797880.7486
Canters’ Notation
m_{1} = 0.7459
n_{ } = 0.4809
k_{1} = 1.0226
k_{2} = 1.2664
p_{ } = 0.5989
In Wagner’s Formula
m_{1} = 0.7459
n_{ } = 0.4809
C_{x} = 1.707411
C_{y} = 1.340662
k_{ } = 1.0226
These values refer to the transcription of Wagner’s formula
in libproj4 by Gerald I. Evenden, v1.2, 2005.^{[4]}
Wagner himself wrote m_{1} as 7/9 und
n as 5/18.
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
the parameters. Change the values within the allowed range and hit the Render my projection button.
The five configuration parameters
Modify the projections using the following parameters:

Variant of:
Select the existing projection to be modified.
The parent projections of the Wagner variants are:
– Wagner IX: Equidistant azimuthal projection
– Wagner VI: Sinusoidal projection
– Wagner III: Apian II.

ψ_{1}
– 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.

λ_{1}
– 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.

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.

a
– Width of xaxis
Min.: 20 / Max.: 100
Reasonable values are probably higher than 50.
Noted in percent: The lower the number, the narrower the projection.
Projection Center/Tilt
The map can be centered to any point on the earth’s surface. Oblique and transverse aspects are possible as well.

Center Lat:
Latitudinal projection center. Positive values represent a northern latitude, negative values represent a southern latitude.
min: 90 / max: 90.

Center Lon:
Longitudinal projection center. Positive values represent a eastern longitude, negative values represent a western longitude.
min: 180 / max: 180.

Axial tilt:
Earth’s axial tilt. Positive values rotate counterclockwise, negative values rotate clockwise.
min: 180 / max: 180. A value of ±90 shows the transverse aspect.
Two examples of use:
– Wagner himself showed a modified example of Wagner VII, which was meant map the Atlantic only instead of the whole Earth.
He changed
the axial ration, centered to 20° North and 30° West, with an axial tilt of 90°,
but he explicitly stated that this is not necessarily the best solution for the area in question.
– Centered to 36° North, 45° East with a tilt of 37°
is a configuration that show all major land masses incl. Antarctica without interruptions.
Predefined Configurations
Instead of building your own projection, you can select an existing projection that can be generated using
the Wagner IX 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.
Image Options
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 150 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 260 others projections.
It’s a listing of all the cylindric, pseudocylindric and lenticular projections that are offered by mapprojections.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.
Download
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.
Inkscape,
LibreOffice Draw,
OpenOffice Draw,
and GIMP (all of them are available for Windows, macOS and Linux).
Commercial software (for Windows and macOS) include
PhotoLine,
Affinity Designer
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:

Böhm’s notation:
The representation of a Wagner variant as suggested by Dr. Böhm in his german article
Variationen von Weltkartennetzen
^{[2]}.
Böhm created this notation for variations of Wagner VII/VIII (s. WVG7),
I adapted it for Wagner IX.
The extended variant is my own suggestion, it only adds a prefix that indicates
which Wagner projection was modified.

Canters’ Notation:
Parameter values as listed by Frank Canters in
Smallscale Map Projection Design^{[3]:185}
(Table 5.2).

In Wagner’s Formula:
Constants to insert into Wagner’s formula for Wagner VII/VIII, from
Kartographische Netzentwürfe
^{[1]},
see Umbeziffern: Notation.

d3 Implementation:
Using the Customizable Wagner in d3 scripts, see below.
Terms of Use
SVG files:
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,
mapprojections.net.
Background projection images:
The background projection images are licensed under
CC BYSA 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.
Please also pay regard to the Legal Disclosure of this website.
Credits
Based on my original (clumsy and flawed) implementation of the
customizable Wagner IX, Peter Denner wrote the clean and extended
version that is presented here.
Thanks a lot, Peter!
This page utilizes a few scripts mentioned below. I’d like to thank the authors for their great work!
References

↑
Wagner, Karlheinz:
Kartographische Netzentwürfe.
Leipzig 1949.

↑
Dr. Rolf Böhm:
Variationen von Weltkartennetzen der WagnerHammerAitoffEntwurfsfamilie
First publication in: Kartographische Nachrichten Nr. 1/2006. Kirschbaum: BonnBad Godesberg.
Quoted from www.boehmwanderkarten.de/archiv/pdf/boehm_kn_2_2006_2015_complete.pdf
(german)

↑ ^{a b
c}
Canters, Frank:
Smallscale Map Projection Design.
London & New York 2002.

↑
libproj4, cartographic projection library, release 050319.
d3 Implementation
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 d3geo – Seven Of Nine.
The full source code is provided there.
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