Angle-triple projection

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Angle-triple projection of the earth

Distortions of the angle-triple projection illustrated with the Tissot indicatrix

The Winkel-Tripel-Projektion is a map network design published in 1921 by Oswald Winkel (1874–1953) for the entire surface of the earth. It represents a compromise between area and angle fidelity and is therefore one of the most widely used world map projections. Compared to the similar Robinson projection , it achieves less distortion, but gives up the positional accuracy, so that curved circles of latitude arise. Compared to equal area projections such as the Mollweide projection or the Eckert IV projection , it avoids their relatively strong shape and angle distortions, but does not achieve complete area accuracy. The projection is therefore only useful for general and thematic world maps.

Projection formula

The projection formula is the arithmetic mean of the rectangular flat map and the Aitov projection :

${\displaystyle x={\frac {1}{2}}\left[\lambda \cos(\phi _{1})+{\frac {2\cos(\phi )\sin \left({\frac {\lambda }{2}}\right)}{\mathrm {si} (\alpha )}}\right]}$

${\displaystyle y={\frac {1}{2}}\left[\phi +{\frac {\sin(\phi )}{\mathrm {si} (\alpha )}}\right]}$

• ${\displaystyle \lambda }$ is the longitude (relative to the central meridian)
• ${\displaystyle \phi }$ is the latitude
• ${\displaystyle \alpha :=\arccos \left(\cos(\phi )\cos \left({\tfrac {\lambda }{2}}\right)\right)}$
• ${\displaystyle \phi _{1}}$is the latitude of the standard parallels of the flat map. Winkel chose for his projection . ${\displaystyle \phi _{1}=\arccos \left({\tfrac {2}{\pi }}\right)}$
• ${\displaystyle \mathrm {si} }$is the non-normalized cardinal sinus .

At the same time, Winkel presented two other map network designs. These are known as angle I (the arithmetic mean of the flat map and the sinusoidal projection ) and angle II . The triple projection is therefore also called angle III .

use

The projection has been used by the National Geographic Society for world maps since 1998 .

literature

• Günter Hake: Cartography I . de Gruyter, 1982, ISBN 3-11-008455-4 .
• Kurt Stüwe: Introduction to the geodynamics of the lithosphere . Springer, 2000, ISBN 978-3-540-67516-7 , pp. 28 ( limited preview in Google Book search).

Web links

Commons : Winkel-Tripel-Projection  - collection of images, videos and audio files

• Winkel's Triple Projection . In: Three Modifications for Azimuthal Projections. Carlos A. Furuti, September 21, 2002.
• Winkel Tripel Projection . In: Modified Azimuthal Projections. Carlos A. Furuti, December 22, 2002.
• Description of the inverse solution

Individual evidence

1. ^ Oswald Winkel: New combinations of graticules. In: Petermann's communications. 67, 1921, pp. 248-252.
2. ↑ ^{a b} Winkel triple projection — Help | ArcGIS for Desktop. Retrieved June 17, 2020 . 
3. ↑ deducing the angle I and Eckert V Projections . In: A Simple Projection plus Two Derived Works. Carlos A. Furuti, September 21, 2002.
4. ↑ deducing the angle Projection II . In: A Simple Projection plus Two Derived Works. Carlos A. Furuti, September 21, 2002.