delaunay sweep circle initialization
I just realized that my question is irrelevant to making the algorithm work.
I mixed the radial and angular atributes of the points and mistakably thought them to need to be based at the same origin, but that's just confusing semantics.
Actually, the radial origin decides on the order of insertion of points and if set correctly, ensures no new point will apear inside an existing triangle, and the angular origin only helps to find the projection of a new point on the envelope (frontier / convex-hull in the case of [2]), so the angular origin indeed needs to be inside the seed envelope, but the radial origin need not. Moving the angular-origin inside the seed envelope will affect the actual progression of the algorithm, but will not affect correctness.

This makes things a lot easier - taking arbitrary point of the set, or somewhere in the plane and find the 2/3 points closest to it will work in general position, but it may result in a colinear triplet.
Following [2] and taking a point from the set, its nearest neighbour, and a 3rd point that creates the smallest circumcircle with them will always work as this is a delaunay triangle (nearest-neighbours are always delaunay edges, and the 2 triangles based on this edge have to create an empty circumcircle, so one of them has to be the smallest) which makes the circumcircle empty of other points, so using its origin as the radial origin is correct.

To do this for integers, we may take the second approach and round the result (checking for super-edge-cases), or take the first approach which is more straight forward for integer radial origin, and restart if we find a colinear triplet, or when the triangle is too obtuse to contain an integer point in its interior.
The last condition is only for making the algorithm fully integral (e.g for machines wihout Floating Point Arithmetics), but using precalculated float pseudo-angles [3] (avoiding use of slow atan2) and comparing them later in the binary search (tree), might be faster than using fully integral orientation test by demand.


pseudo-angles