The stereographic projection is conformal, which means it preserves angles.
It is possible to modify the projection function, while retaining its conformal properties. The unmodified stereographic has circular lines of constant scale, which can be changed to follow more complex shapes.
More importantly, the variation in scale can also be minimised for a given region.
The projections above are available as d3.geo.modifiedStereographic in D3’s geo.projection plugin.