THE FOLD / GLITCH / DIVIDE BY ZERO / THE WELZL
THE WELZL
the smallest enclosing circle, pinned by <=3 points
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Welzl’s algorithm finds the smallest enclosing circle of a set of points — the minimum-radius disk containing them all — in expected linear time. It adds points one at a time; as long as the new point is already inside the current circle, nothing changes, but when it falls outside it must lie on the boundary of the new circle, which is then rebuilt from the points known to be on the boundary (at most three).
It is used for bounding volumes, collision culling, and facility-location.
LIT verified live: over 200 random point sets Welzl’s circle contains every point, and its radius equals the brute-force minimum (over all circles through 2 or 3 points) — window.__welzl. FIG no framing; exact minimum.
It is used for bounding volumes, collision culling, and facility-location.
LIT verified live: over 200 random point sets Welzl’s circle contains every point, and its radius equals the brute-force minimum (over all circles through 2 or 3 points) — window.__welzl. FIG no framing; exact minimum.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at divide-by-zero — because the circle-through-three-points formula divides by a determinant that vanishes when the points are collinear, the exact case the algorithm must guard. Welzl lives right at that edge. AVAN (AI) built the instrument: the incremental construction, the ≤3-point boundary circles (with the collinear guard), the containment and minimality checks.
Credit as content: Emo Welzl (1991). The weave: David names the divide-by-zero; I add points until one escapes, rebuild the circle on its boundary, and confirm the result is the true minimum.
Credit as content: Emo Welzl (1991). The weave: David names the divide-by-zero; I add points until one escapes, rebuild the circle on its boundary, and confirm the result is the true minimum.
3 ONE DIMENSION
The minimal enclosing circle rests on at most three points. Two points give a diameter; three give a circumcircle — and three collinear points make the determinant zero, the case to guard.
4 TWO DIMENSIONS · INTERACTIVE
A point cloud and its smallest enclosing circle; the ≤3 boundary points are marked. Roll new clouds and watch only the extremes decide the circle.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the smallest enclosing circle, pinned by its boundary points.
AVAN’s addition (the inverse-companion): the answer is pinned by at most three points. The minimal enclosing circle is determined by 2 or 3 of the points on its boundary; every other point is strictly inside and irrelevant, so Welzl only rebuilds when a new point escapes. The inverse of ‘consider all the points’ is ‘the circle rests on ≤3 of them; the rest are interior.’ Magenta is the interior points that don’t constrain it; green is the ≤3 boundary points that do. Most of the data doesn’t matter — a handful of extremes decide everything.
LIT Genuine Welzl's algorithm (Welzl 1991). Verified live: the incremental minimal-enclosing-circle contains all points and its radius equals the brute-force minimum over all circles through 2 or 3 points, across 200 random point sets (window.__welzl.containsAll && .radiusIsMin).
FIG No framing: the incremental construction, the <=3-point boundary circles (with the collinear divide-by-zero guard), and the containment + minimality checks run in-browser and are exact. The AVAN inverse is honest — the minimal circle is determined by 2 or 3 boundary points while all others are strictly interior and irrelevant; magenta is the interior points, green the <=3 that pin it. A handful of extremes decide everything.
FIG No framing: the incremental construction, the <=3-point boundary circles (with the collinear divide-by-zero guard), and the containment + minimality checks run in-browser and are exact. The AVAN inverse is honest — the minimal circle is determined by 2 or 3 boundary points while all others are strictly interior and irrelevant; magenta is the interior points, green the <=3 that pin it. A handful of extremes decide everything.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of DIVIDE BY ZERO · David Lee Wise (ROOT0), with AVAN