A space for lattice representation and clustering

Andrews, L.C.; Bernstein, H.J.; Sauter, N.K.

doi:10.1107/S2053273319002729

Figure 1
Example of a one-boundary virtual Cartesian point distance calculation. Only the VCP operation is shown, no reflections. Both points p₁ and p₂ are Selling reduced. The image is the three-dimensional, all-negative octant of the three $[{\bf S^6}]$ axes, s₁, s₃ and s₅; the reduction is done along the s₁ axis, and s₃ and s₅ are the two scalars that will be interchanged. The points are shown above or below the s₃/s₅ plane, with their projections onto that plane marked with a +. To compute the minimal distance between p₁ and p₂, begin by computing the Euclidean distance between the two. The s₁ reduction transforms p₁ into [p_1']

, but the metric changes when going from negative s₁ to positive s₁, so the simple Euclidean distance may not be minimal. To generate p₁(VCP) to which the distance may be shorter, project p₁ onto the s₃/s₅ plane, transform that projected point, and subtract s₁ from that point. The distance from p₁ to p₁(VCP) can now be used to decide whether it is shorter than the p₁ – p₂ Euclidean distance. The best distance for this case is the shorter of the distances between p₁ and p₂ as opposed to the distance between p₁(VCP) and p₂.

FOUNDATIONS
ADVANCES

ISSN: 2053-2733

Volume 75| Part 3| May 2019| Pages 593-599

https://doi.org/10.1107/S2053273319002729

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