Ivan Pribechttps://ivan-pi.github.io/tags/node-generation/Recent content on Ivan PribecHugo -- gohugo.ioen© 2025 — Ivan Pribec — All rights reserved.Fri, 04 Sep 2026 00:00:00 +0200Circle-Fitting Relaxationhttps://ivan-pi.github.io/gallery/circle-fit-relaxation/Fri, 04 Sep 2026 00:00:00 +0200https://ivan-pi.github.io/gallery/circle-fit-relaxation/<!-- Minimal working example: Poisson-disc sampling (Bridson) followed by circle-fitting relaxation: each node is attracted toward the centre of the least-squares (Kasa) circle through its k=6 nearest neighbours, blended with short-range repulsion. The residual of the same fit is the convergence monitor, shown as the colour of each point and as the mean readout. Hover to see one node's neighbours, fitted circle, and attraction; click to resample. Style after Jason Davies' Lloyd's Relaxation / Poisson-Disc pages. This is a Hugo content file: the front matter above is parsed, the rest is inserted verbatim by layouts/gallery/single.html. --> <div class="gallery-chart" id="chart"></div> <p class="gallery-readout"> <span>Iteration: <span id="iter">0</span></span> <span>mean &sigma;/h: <span id="sigma">&ndash;</span></span> <span>mean &psi;<sub>6</sub>: <span id="psi6">&ndash;</span></span> <span id="status"></span> </p>

Iteration: 0 mean σ/h: – mean ψ6: –

0.40 0 = circle fit, 1 = repulsion 8 repulsion exponent 6 neighbours, force and fit

Poisson-disc samples, relaxed by two forces. Click (or tap) the canvas to restart; hover to inspect one node.

The repulsion from the k nearest neighbours yj is $$f(x) = \sum_{j} \left( \frac{h}{|x - y_j|} \right)^{m} (x - y_j),$$ and each step blends it with the pull toward ĉ(x), the centre of the least-squares circle through the neighbours (Kåsa, 1976): $$\Delta x = \omega\, f + (1 - \omega)\, \big(\hat{c}(x) - x\big).$$

σ is the root-mean-square deviation of the neighbours from their fitted circle, in units of the spacing h: zero for a perfect ring. Each point is coloured by its own σ. ψ6 compares the directions to the six nearest neighbours with a perfect hexagon: 1 is hexagonal, near 0 is random. (At k = 3 a circle fits three points exactly, and σ stops meaning anything.)

The black squares are fixed. The first ten iterations run pure repulsion with a ramped-in step clamp; after that the sliders take over. A hard floor keeps every pair at least 0.7 h apart; the repulsive shift and the total step are clamped to 0.2 h, the step is damped by ½, and neighbour lists refresh every five iterations.

Inspired by the beautiful visualizations of Jason Davies.

]]>Advancing-Front Node Generatorshttps://ivan-pi.github.io/gallery/advancing-front-node-generators/Fri, 04 Sep 2026 00:00:00 +0200https://ivan-pi.github.io/gallery/advancing-front-node-generators/<!-- Faithful JavaScript ports of the two bulk node generators in the nodeplace library (Fortran/C): node_placing -- Fornberg & Flyer (2015) node_placing_heightfield -- van der Sande & Fornberg (2021), 2-D specialisation The ports preserve the seeding (cell-centred with jitter), the outward walk for the nearest outside-PDPs, the five cell-centred arc fractions, the splice semantics, the first-argmin tie rule, and the height-field direction-dependence correction (their section 2.3; inactive for constant radius). This is a Hugo content file: the front matter above is parsed, the rest is inserted verbatim by layouts/gallery/single.html. --> <div class="gallery-chart" id="chart"></div> <p class="gallery-readout"> <span>nodes: <span id="count">0</span></span> <span>front size: <span id="fsize">0</span></span> </p>

nodes: 0 front size: 0

60 both algorithms 1.0 both algorithms 5 Fornberg–Flyer 10 van der Sande–Fornberg 8 animation

Two bulk generators, ported line for line from the reference Fortran/C. Both sweep the box bottom to top, placing each node at the lowest admissible point and pushing an exclusion disc of radius r(x, y) into the front; the red line is the front. Click (or tap) the canvas to restart.

Fornberg–Flyer keep the front as an x-sorted list of potential dot positions (PDPs). The next node is the lowest PDP; it is accepted, and the PDPs inside its exclusion disc are replaced by a configurable number of cell-centred points on the exclusion arc (five by default), spliced between the nearest outside PDPs found by walking outward from the dot.

Van der Sande–Fornberg keep a height field on a background grid of grid-factor × ninit columns. The next node goes to the lowest column; each accepted dot raises the heights under its exclusion disc to the disc’s far hemisphere. When the radius varies, the node is first pushed off the discs of earlier nearby nodes — the direction-dependence correction of their section 2.3 (inactive for constant radius).

① ② ③ placed nodes nearest outside PDP current lowest point new front
Fornberg–Flyer, a snapshot from an actual run of the port. Dark dots are placed nodes; the front advances downward. ① the lowest PDP of the old front (grey dashed) is accepted as a node. ② the PDPs inside its exclusion disc are removed. ③ the angle between the directions to the nearest outside PDPs (dotted rays) is divided into five equal cells (ticks), and one point is placed at each cell’s centre — so the end gaps to the rays are half the dot-to-dot gap, keeping new PDPs clear of the surviving outside ones. The spliced result is the new front (red).
① ② old heights new heights follow the disc boundary background columns lowest column
Van der Sande–Fornberg. The height field (one value per background column) is the front. ① the node is placed at the lowest column of the old heights (grey dashed). ② the heights under its exclusion disc are raised to the disc’s far hemisphere, so the new heights (red) sample the disc boundary on the columns; elsewhere the old heights already lie outside the disc and are unchanged. The data is a snapshot from an actual run of the height-field port. Later nodes sit on the height curve, hence at least r away — for constant radius no node ever lands inside another’s disc.

References:

Fornberg, B., & Flyer, N. (2015). Fast generation of 2-D node distributions for mesh-free PDE discretizations. Computers & Mathematics with Applications, 69(7), 531–544. doi:10.1016/j.camwa.2015.01.009

van der Sande, K., & Fornberg, B. (2021). Fast variable density 3-D node generation. SIAM Journal on Scientific Computing, 43(1), A242–A257. doi:10.1137/20M1337016

Inspired by the visualisations of Jason Davies.

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