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Convex hull and Delaunay triangulation

(Inspired by Pole of inaccessibility of every country / KDO / Observable)

It’s pretty well known that by constructing the 3d convex hull of a set of points on a sphere, and then projecting the edges onto the surface of the sphere, one obtains the Delaunay triangulation of the points.

Under the assumption that the Earth is perfectly spherical, can that construction be used to find poles of inaccessibility? I’m thinking that it could just be a matter of intersecting each plane (each face of the convex hull of each countries boundary) with the sphere. Each resulting circle is guaranteed to be empty, if I’m not mistaken, so it’s just a case of picking the largest one (or the second largest, because all the largest circle will be exterior to the country’s border?)

I’m aware of GitHub - Fil/d3-geo-voronoi: Voronoi / Delaunay tessellations on the sphere but I don’t know if it offers a straightforward way to do this.

it works:

Using the hull is a good idea, as it allows to quickly eliminate many of the points without doing a full geoContains search (it also eliminates a bug with the Maldives which appears to have the wrong winding order in this topojson).

Note that it’s not mathematically exact, in the sense that the pole of inaccessibility should be based on lines, not points. But within the precision of the topojson it’s the best result we can give.