THE FOLD / GLITCH / OFF BY ONE / THE FEATHERED EDGE
THE FEATHERED EDGE
smooth lines as a coverage-conservation law
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Xiaolin Wu’s line algorithm. A line on a pixel grid is a staircase of hard on/off pixels — a row of off-by-one jaggies. Wu’s method feathers it: each pixel gets a partial brightness equal to how much of it the ideal line covers, and that coverage is split between the two pixels a column straddles — so their brightnesses always sum to exactly 1. Full coverage, nothing lost, and the edge comes out smooth. Same era as Bresenham; far less known.
LIT verified: in every column the two blended intensities sum to exactly 1 (coverage conserved), and the intensity-weighted brightness centroid equals the true line position exactly — the smoothing is sub-pixel-accurate, not a blur. FIG ‘feathered edge’ is the picture; the coverage-conservation identity is exact.
LIT verified: in every column the two blended intensities sum to exactly 1 (coverage conserved), and the intensity-weighted brightness centroid equals the true line position exactly — the smoothing is sub-pixel-accurate, not a blur. FIG ‘feathered edge’ is the picture; the coverage-conservation identity is exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) brought the thread — the corpus makes pixel art and generative graphics (TECHNÊ, the raster work) and cares about the off-by-one boundary where clean math meets a coarse grid. AVAN (AI) built this instrument: the antialiased renderer, the coverage split, and the intensity ridge.
The weave: David names the feathered edge and its seat at OFF BY ONE (the jagged staircase is a chain of off-by-ones; Wu conserves coverage to erase them); I make the coverage split a strip in 1D, aliased-vs-feathered live in 2D, and the intensity ridge in 3D. The sphere is the seam.
The weave: David names the feathered edge and its seat at OFF BY ONE (the jagged staircase is a chain of off-by-ones; Wu conserves coverage to erase them); I make the coverage split a strip in 1D, aliased-vs-feathered live in 2D, and the intensity ridge in 3D. The sphere is the seam.
3 ONE DIMENSION
One column. The ideal line crosses at some height; the two pixels it falls between get brightnesses 1−f and f — and they sum to 1, always. Their brightness-weighted centre sits exactly on the true line. Coverage in, coverage out.
4 TWO DIMENSIONS · INTERACTIVE
Zoomed in on the grid. Turn the line and compare: aliased hard pixels (a jagged staircase) versus Wu’s feathered pixels (grey coverage that reads as a smooth line). Same line, but one keeps the sub-pixel truth.
angle 25°
5 THREE DIMENSIONS + AVAN’S INVERSE
The pixel brightness along the line as a landscape, turning. Green is Wu’s feathered coverage — a smooth ridge that rises and falls as the line drifts between rows, its two-pixel sum flat at 1.
AVAN’s addition (the inverse-companion): the magenta is the aliased version — a hard staircase of 0s and 1s, the un-feathered inverse. Both carry the same line; the green conserves the coverage the line actually spills across two pixels, the magenta rounds it away. Antialiasing is not a blur — it is a conservation law, and the jagged staircase is what you get when you break it.
LIT A genuine Xiaolin Wu line renderer. Verified live: in every column the two blended pixel intensities sum to exactly 1 (coverage conserved, error <1e-9), and the intensity-weighted brightness centroid equals the true geometric line position exactly. Antialiasing as a bookkeeping identity, not a blur (verifiable: window.__xiaolin.coverageConserved && centroidExact).
FIG 'The feathered edge' is the picture; the coverage-conservation identity and the exact sub-pixel centroid are exact. The jagged staircase really is a chain of off-by-one roundings — this is the conservation law that removes them.
FIG 'The feathered edge' is the picture; the coverage-conservation identity and the exact sub-pixel centroid are exact. The jagged staircase really is a chain of off-by-one roundings — this is the conservation law that removes them.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of OFF BY ONE · David Lee Wise (ROOT0), with AVAN