BRieszk

/ˈbrisk/ is a selection of routines helpful in dealing with discrete measures converging to a continunous density. They will compute near-minimizers of varying quality for the Riesz energy-styled functionals. Several elementary underlying sets are built-in; general (well-behaved) implicit surfaces are supported. Proceed with care.

Below, for example, are 40,000 points on the tangle

distributed according to the absolute value of its Gaussian curvature.

Left: (approximate) Delaunay triangulation with faces color-coded by the value (not the absolute value) of curvature, blue-orange/low-high; right: (approximate) Voronoi diagram with faces colored by the number of edges, blue-yellow/few-many (hexagons are the majority). Here the approximation is understood in the sense that both the Delaunay and Voronoi are computed locally in the tangent plane, then transplanted back to the surface. Notice also that the goal is never to reach an exact local minimizer of the respective energy, rather to emulate certain features of such minimizers. Accordingly, no Hessians are computed and no careful line searches performed. In fact, it is rather remarkable that these points aren’t terrible on the energy scale; proving a theorem about this is left as an exercise to the reader for now.

The Github repo for BRieszk is here.

Tangle 1 Tangle 2

3dRBFnodes

Along the same lines, one can produce reasonable nodes for RBF-functions and use the latter for, say, geomodelling. A reasonably scalable implementation can be found below; it handles up to 3M nodes in 3d on an i5 CPU laptop. There is nonempty code overlap with BRieszk. A detailed discussion is contained in the paper.

The Github repo for 3dRBFnodes is here. Below are two pictures showing Andes, the one on the right is color-coded by elevation.

Andes 1 Andes 2