map-projections.net: WVG-ESP

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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

 

Canters’ Optimisation of Wagner IX

Low-error optimization of Wagner IX, by Frank Canters

See below for a detailed description of the parameters
 

Projection Center/Tilt
 

Please leave the following four input field blank in any case!

Predefined Configurations
Image Options

Share this configuration

Share projection parameters only
Share parameters & image options

Values of the current configuration

Böhm’s Notation (adapted)

ix@67.131-86.562-206.7978-80.7486

Canters’ Notation

m1 = 0.7459
n  = 0.4809
k1 = 1.0226
k2 = 1.2664
p  = 0.5989

In Wagner’s Formula

m1 = 0.7459
n  = 0.4809
Cx = 1.707411
Cy = 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 m1 as 7/9 und n as 5/18.

d3 Implementation

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:

  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. a
    – Width of x-axis
    Min.: 20 / Max.: 100
    Reasonable values are probably higher than 50.
    Noted in percent: The lower the number, the narrower the projection.
Notes:
– 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).

Projection Center/Tilt

The map can be centered to any point on the earth’s surface. Oblique and transverse aspects are possible as well.

  1. Center Lat: Latitudinal projection center. Positive values represent a northern latitude, negative values represent a southern latitude.
    min: -90 / max: 90.
  2. Center Lon: Longitudinal projection center. Positive values represent a eastern longitude, negative values represent a western longitude.
    min: -180 / max: 180.
  3. 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.

  1. Continents: The land masses can be shown as grey silhouette, as outlines only, with a colored display of the countries – or not at all.
  2. Graticule: Meridians and Parallels can be shown at a spacing of 5, 10, 15, 18, 20, 30 or 45 degree – or not at all.
  3. 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
  4. Background projection: You can compare the rendered projection to one of 302 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.
  5. 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:

  1. 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. WVG-7), 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.
  2. Canters’ Notation: Parameter values as listed by Frank Canters in Small-scale Map Projection Design[3]:185 (Table 5.2).
  3. In Wagner’s Formula: Constants to insert into Wagner’s formula for Wagner VII/VIII, from Kartographische Netzentwürfe [1], see Umbeziffern: Notation.
  4. 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, map-projections.net.
 
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.

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

  1. ↑ Wagner, Karlheinz:
    Kartographische Netzentwürfe.
    Leipzig 1949.
  2. ↑ Dr. Rolf Böhm:
    Variationen von Weltkartennetzen der Wagner-Hammer-Aitoff-Entwurfsfamilie
    First publication in: Kartographische Nachrichten Nr. 1/2006. Kirschbaum: Bonn-Bad Godesberg.
    Quoted from www.boehmwanderkarten.de/archiv/pdf/boehm_kn_2_2006_2015_complete.pdf (german)
  3. ↑ a b c Canters, Frank:
    Small-scale Map Projection Design.
    London & New York 2002.
  4. ↑ 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 d3-geo – Seven Of Nine. The full source code is provided there.

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