A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Robin Stuart
Date: 2016 Dec 31, 07:47 -0800
If you are looking at the Wikipedia page https://en.wikipedia.org/wiki/Map_projection#Conformal what it states is "These are some conformal projections:" not the only ones. In fact there are an infinite number of mappings of the surface of a sphere to the plane that are conformal. Most of them would be pretty bizarre an not very useful for navigational purposes.
In case anyone is interested the whole fascinating topic is intimately connected to the theory of analytic functions of complex numbers. I touched on this in my talk at the Nav weekend in 2010 http://fer3.com/arc/m2.aspx/Materials-from-Navigation-Weekend-Talk-Stuart-jun-2010-g13201 which shows the surface of the sphere being mapped by stereographic projection onto the plane of complex numbers. The result is conformal. The subsequent application of any analytic function preserves the conformal property. In particular the Mercator projection is the logarithm of the stereographic projection,