February 16, 2013

Mapping the Earth with Complex Numbers

Conformal maps have their history in 18th century cartographic science, so I have updated my conformal map tool to be able to render the surface of the earth according to the conformal projection of your choice. The lower-right corner of the window has a ⊕ button that renders the earth as seen from space.

Below is the default (polar azumithal) projection. It places the equator on the unit circle, the south pole at infinity, and the prime meridian on the real line towards +1. Because the south pole is at infinity, there is a large distortion of sizes as you go south of the equator. However, the map is conformal, which means that locally, all angles are still accurate.

Here is a zoom-in on Greenwich, London at +0.35 on the polar projection. On the same projection, here is a zoom-in on Adelaide, Australia. Even though the southern hemisphere seems grossly large on the whole-plane rendering, it still looks correct locally.

The Mercator projection is the most well-known conformal projection, and here it can be rendered using e^iz. Other projections such as the classical azimuthal stereographic and the Lagrange projectections are also just other complex-valued functions.

Read about other conformal projections here.

Can you find functions that implement other interesting projections?

Posted by David at February 16, 2013 05:39 AM

Cool stuff!

Posted by: Alexandre at July 8, 2013 05:56 AM

(e^i1.37)(iz/11+0.7) is for India

Posted by: at July 14, 2018 08:19 PM

Amazing program! It's helping me a lot with some examples for Complex Analysis.
Thank u a lot!

Posted by: Sac-Nicté Salas at May 26, 2024 11:46 PM
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