I'm not per say a GIS guru, I'm more of a user of the technology. On a recent project we were asked to analyze, provide, and submit all our analysis using a Double Stereo projection. What are the main advantages of using it vs. UTM, or any other projection.
As far as I can tell, "Double Stereographic" is identical to stereographic. Stereographic projections are favored for work around the poles because the meridians radiate in straight lines around the projected pole. The projection is conformal, implying the meridians also have the correct relative directions, too.
In the stereographic projection, all geodesics on the sphere are projected either to portions of lines or portions of circles in the plane. (This is because the formula for the projection is particularly simple.) A rotation of the sphere can be computed as a fractional linear transformation of the plane, which is the fundamental operation of inversive geometry. Thus, certain natural operations on the sphere are easily and directly computed and visualized in a stereographic projection.
In mathematics the stereographic projection is a fundamental tool: it creates a conformal one-to-one correspondence between the sphere and the plane with a "point at infinity". This unifies many aspects of complex analysis and algebraic geometry.