THE FOLD / BOSS / THE WALL / THE HALF-PLANE INTERSECTION
THE HALF-PLANE INTERSECTION
a region carved by half-planes
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Half-plane intersection carves out the region that satisfies a set of linear inequalities. Each constraint a·x ≤ c keeps one side of a line — a half-plane — and the intersection of them all is a convex polygon (possibly empty or unbounded). You can build it by clipping: start with a big bounding box and slice it by each half-plane in turn, keeping only the inside. The result is exactly the feasible region of a linear program, and every point of it obeys every constraint at once.
LIT verified live: over thousands of random constraint sets, every vertex of the clipped region satisfies all the half-planes, and a point lies inside the region only if it satisfies every constraint (window.__half_plane). FIG no framing; the sequential clipping and the membership tests run in-browser.
LIT verified live: over thousands of random constraint sets, every vertex of the clipped region satisfies all the half-planes, and a point lies inside the region only if it satisfies every constraint (window.__half_plane). FIG no framing; the sequential clipping and the membership tests run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-wall — each half-plane is a wall, and the surviving region is what lies inside every wall at once. AVAN (AI) built the instrument: sequential polygon clipping against each half-plane, a vertex-feasibility check, and a membership-vs-constraints test.
Credit as content: convex polygon clipping (Sutherland–Hodgman, 1974) applied to half-plane intersection, the feasible-region primitive of computational geometry and linear programming. The weave: David names the walls; I confirm the clipped region’s vertices obey every constraint and that interior points satisfy them all.
Credit as content: convex polygon clipping (Sutherland–Hodgman, 1974) applied to half-plane intersection, the feasible-region primitive of computational geometry and linear programming. The weave: David names the walls; I confirm the clipped region’s vertices obey every constraint and that interior points satisfy them all.
3 ONE DIMENSION
A bounding box clipped by three half-planes, one at a time; each slice keeps the inside, and the survivor is their convex intersection.
4 TWO DIMENSIONS · INTERACTIVE
Random half-planes and their intersection polygon; every vertex satisfies all constraints, and sampled points are inside only when feasible.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the convex feasible region carved by the walls.
AVAN’s addition (the inverse-companion): don’t test points — carve the region. The inverse of ‘does this point satisfy every constraint?’ is ‘clip a box by each half-plane and the survivors are exactly the feasible points.’ Magenta is a point a wall rejects; green is the region inside them all. Carve, don’t test.
LIT Genuine half-plane intersection via convex polygon clipping (Sutherland–Hodgman, 1974), the feasible-region primitive of computational geometry / LP. Verified live: over 2000 random constraint sets, every result vertex satisfies all half-planes (window.__half_plane.verticesFeasible) and interior points satisfy every constraint (.membershipMatches).
FIG No framing: the sequential clipping and the membership tests both run in-browser. The AVAN inverse is honest — clipping a box by each half-plane so the survivors ARE the feasible points (rather than testing points against every constraint) is the constructive view of the feasible region; magenta is a point a wall rejects, green the region inside them all. Carve, don't test.
FIG No framing: the sequential clipping and the membership tests both run in-browser. The AVAN inverse is honest — clipping a box by each half-plane so the survivors ARE the feasible points (rather than testing points against every constraint) is the constructive view of the feasible region; magenta is a point a wall rejects, green the region inside them all. Carve, don't test.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE WALL · David Lee Wise (ROOT0), with AVAN