The Euler Spiral Map Projection

A Euler spiral is a curve whose curvature changes linearly with its curve length. A Euler spiral can therefore be used to create a map projection by projecting a curved globe onto a flat spiral. The interesting point for cartographers is that the more spirals used in a Euler spiral map projection the less distortion there is.

Now for the math. By cutting a sphere along a spiral with width 1 / N and flattening out the resulting shape we create a Euler spiral when N tends to the infinity. In other words we can create a map projection whose distortion tends to zero as N tends to the infinity.

If this sounds a little confusing then it might help to play with an interactive Euler spiral map. This interactive Euler Spiral Map allows you to adjust the number of spirals used in the projection by changing the thickness of the spirals. By reducing the thickness of the spirals you can increase the number of spirals used in the map projection. The more spirals you create then the less distortion in the projection.

Unfortunately for cartographers a Euler spiral map projection is not very useful for navigating with. If you are still confused then this excellent Numberphile video explains the projection far more clearer than I can: