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.
Double Stereographic vs Stereographic.
Mathematically a stereographic projection is directly projecting the sphere to the plane. A double projection is projecting to another mathematical spheroid first (Guassian Spheroid). Both projection centers are located on opposite the tangent point of the plane (ie on the other side of the planet).
Why use either of these methods? The Canadian Province of New Brunswick uses a unique projection (NB Double Stereographic) because of the unique shape of NB (almost a square) this projection allows for minimal distortion around a circle with a scale factor of 1. Basically it is used in this case because NB straddles multiple UTM zones and it is convenient to use NB Double given the shape and population distribution in the province. It wouldn't be an ideal projection for a long skinny jurisdiction, as for say in Chile or Japan etc.