Function in_diametral_sphere 

pub fn in_diametral_sphere(
    a: Point3,
    b: Point3,
    c: Point3,
    d: Point3,
) -> Certified

Is d inside the diametral sphere of the triangle a, b, c?

The diametral sphere is the smallest sphere through a, b, c: its centre is the triangle’s circumcentre and its equator the circumcircle. For d coplanar with the triangle this is therefore the coplanar incircle test in 3D – the question a 3D Delaunay triangulation asks of a point lying exactly in the plane of a convex-hull face – and, unlike insphere, it needs no orientation: Sign::Positive means strictly inside, Sign::Negative strictly outside, Sign::Zero on the sphere.

The sign is that of

-( |n|^2 (w.w) - |u|^2 ((w x v).n) - |v|^2 ((u x w).n) )

with u = b - a, v = c - a, w = d - a and n = u x v: |n|^2 times the power of d with respect to the sphere, a degree-6 polynomial in the coordinates.

A degenerate triangle (collinear a, b, c) has no diametral sphere and returns Zero. A forward running-error filter decides almost every case; the exact expansion path decides the rest. The exact path needs every non-zero coordinate magnitude in [2^-100, 2^100], so that no product in the degree-6 determinant leaves binary64’s normal range; outside it, and for non-finite input, the result is Certified::Uncertain rather than a guess.