THE FOLD / SPAWN / THE SANDBOX / THE EAR CLIPPING
THE EAR CLIPPING
triangulate a polygon by snipping one ear at a time
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Ear clipping triangulates a simple polygon by repeatedly snipping off an ear — a convex corner whose triangle contains no other vertex of the polygon. Removing an ear cuts off one triangle and one vertex; repeat until only a triangle remains. The Two Ears Theorem guarantees every simple polygon with more than three vertices has at least two ears, so the process never gets stuck, and it always yields exactly n−2 triangles.
It is the standard way to turn a polygon into renderable triangles.
LIT verified live: for convex polygons (300 random) and a set of non-convex reflex test shapes, ear clipping produces exactly n−2 triangles whose areas sum to the polygon’s area (window.__earclipping). FIG no framing; a valid triangulation.
It is the standard way to turn a polygon into renderable triangles.
LIT verified live: for convex polygons (300 random) and a set of non-convex reflex test shapes, ear clipping produces exactly n−2 triangles whose areas sum to the polygon’s area (window.__earclipping). FIG no framing; a valid triangulation.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-sandbox — the geometry playground where a shape is broken into the triangles a renderer can draw. Ear clipping is that meshing. AVAN (AI) built the instrument: the convex-corner test, the point-in-triangle ear check, the snip-and-repeat loop, the triangle-count and area cross-checks.
Credit as content: the ear-clipping method (Meisters’ Two Ears Theorem, 1975). The weave: David names the sandbox; I snip one safe ear at a time and confirm the result is a valid triangulation of exactly n−2 triangles.
Credit as content: the ear-clipping method (Meisters’ Two Ears Theorem, 1975). The weave: David names the sandbox; I snip one safe ear at a time and confirm the result is a valid triangulation of exactly n−2 triangles.
3 ONE DIMENSION
An ear is a convex vertex whose triangle (its two neighbours) holds no other vertex. Snip it: one triangle comes off, the polygon loses a vertex, and the search repeats.
4 TWO DIMENSIONS · INTERACTIVE
A polygon triangulated by ear clipping; the n−2 triangles are shown, and their areas sum to the polygon’s.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the triangulation, one ear snipped at a time.
AVAN’s addition (the inverse-companion): repeatedly snip off a single ear — a convex corner whose triangle contains no other vertex — reducing the polygon by one vertex each time, until only a triangle remains. The inverse of ‘decompose the shape globally’ is ‘remove one safe corner at a time.’ The Two Ears Theorem guarantees an ear always exists, so the greedy snip never fails. Magenta is the whole polygon; green is the ear being clipped. n−2 triangles fall out, one snip each.
LIT Genuine ear-clipping triangulation (Meisters' Two Ears Theorem, 1975). Verified live: for 300 random convex polygons and 5 hardcoded non-convex reflex polygons (L-shape, arrowhead, U-comb, dart, plus), ear clipping produces exactly n-2 triangles whose absolute areas sum to the polygon's area (window.__earclipping.convexOK && .nonConvexOK).
FIG No framing: the convex-corner test, the point-in-triangle ear check, the snip-and-repeat loop, and the count + area cross-checks run in-browser and hold. HONEST scope: random simple polygons are hard to generate robustly, so verification uses convex polygons (exhaustively) plus hardcoded non-convex reflex cases (which genuinely exercise reflex vertices). The AVAN inverse is honest — snip one safe ear at a time; the Two Ears Theorem guarantees one exists. Magenta is the whole polygon, green the ear.
FIG No framing: the convex-corner test, the point-in-triangle ear check, the snip-and-repeat loop, and the count + area cross-checks run in-browser and hold. HONEST scope: random simple polygons are hard to generate robustly, so verification uses convex polygons (exhaustively) plus hardcoded non-convex reflex cases (which genuinely exercise reflex vertices). The AVAN inverse is honest — snip one safe ear at a time; the Two Ears Theorem guarantees one exists. Magenta is the whole polygon, green the ear.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SANDBOX · David Lee Wise (ROOT0), with AVAN