Cartesian transforms and ellipsoids

Point transforms are free functions over Cartesian points: translation, rotation about an axis through a centre (Rodrigues), mirroring through a plane, inversion through a centre, alignment of one vector to another, and plane fitting from at least three non-collinear points.

The finite cell duplicates every atom whose fractional coordinate is within the tolerance of a face, edge or corner, using a coordinate-wise duplicate test (not periodic distance, which would merge 0 and 1).

Displacement ellipsoids diagonalize the symmetric Uij tensor with Jacobi rotations, sort the eigenvalues, take square roots for the semi-axis lengths and scale by sqrt(chi2_3(p)), the square root of the chi-square quantile for three degrees of freedom: 1.538, 2.500, 2.796 and 3.368 at p = 0.5, 0.9, 0.95 and 0.99, linearly interpolated between them and clamped to the end values outside [0.5, 0.99]. A non-positive-semidefinite tensor is rejected.