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delaunayWeights -- squared-norm weights inducing the Delaunay subdivision of a point set

Description

Returns the standard weight vector for the delaunaySubdivision of a point set: the $i$-th entry is $\|v_i\|^2$, the squared Euclidean norm of the $i$-th column of $A$. Lifting each point $v_i$ to $(v_i, \|v_i\|^2) \in \mathbb{R}^{d+1}$ and projecting the lower faces of the resulting convex hull back to $\mathbb{R}^d$ yields the Delaunay subdivision; passing these weights to regularSubdivision is the standard way to realise that construction.

i1 : A = transpose matrix {{0,0},{1,0},{2,0},{0,1},{1,1},{2,1}}

o1 = | 0 1 2 0 1 2 |
     | 0 0 0 1 1 1 |

              2       6
o1 : Matrix ZZ  <-- ZZ
i2 : delaunayWeights A

o2 = | 0 1 4 1 2 5 |

              1       6
o2 : Matrix ZZ  <-- ZZ

Caveat

Delaunay is intrinsically a Euclidean concept on a point set, so this function is meaningful only for point configurations -- not for vector configurations. No input-shape check is performed.

See also

Ways to use delaunayWeights:

  • delaunayWeights(Matrix)

For the programmer

The object delaunayWeights is a method function.


The source of this document is in /build/reproducible-path/macaulay2-1.26.06+ds/M2/Macaulay2/packages/Triangulations.m2:2677:0.