Concept

Convexity — where it appears

The property of a set that every straight line between two of its points stays inside it. It is what makes a polygon drawn through a set's boundary points a guaranteed under-estimate: a chord can only cut a corner, never leave the set and come back.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Also named here as polygon — the same set of essays touches all of them, so they are one junction rather than several.

The picture is kept, at four tolerances. One closed curve of 3001 vertices, simplified at four tolerances. Douglas–Peucker's promise holds in every panel: no discarded vertex is further than ε from the line drawn in its place, measured at 0.1158 against 0.128 in the last. The picture survives. The enclosed area does not: it falls by 5.43 per cent, and it falls rather than wandering, because cutting a corner takes area off and never puts it back.

A tolerance is a promise about the picture

Douglas–Peucker guarantees exactly one thing: no vertex it discarded is further than ε from the line drawn in its place. It says nothing about the enclosed area, nothing about which side of the boundary a point ends up on, and nothing about whether the curve still fails to cross itself — and all three are what the geometry is usually being asked.

applied · Generalise

What 16 bearings draw beside a river that bends. Everywhere within a 10 km walk of the centre, with a river 3 km to the north meandering ±2.0 km on a 4 km wavelength, crossable only at bridges every 2.5 km. The reachable ground is shaded — pale where it is reached directly and darker where the walk had to go by a bridge. The outline is the polygon 16 bearings draw. It misses 28.8% of the true ground, as a fan beside a straight river does, and it also CLAIMS 0.30% that is not there — ground cut off by a bend, which a chord between two neighbouring bearings passes straight over.

A bend in the barrier puts the drawn polygon over water

A straight river leaves the near side of a reach set convex, and that is what makes a fan of bearings fail in one direction only: every chord cuts a corner, nothing is claimed that is not there. A meander breaks both halves of that at once — a bearing can leave the set and come back, and a chord between two bearings can cross water. So the drawn polygon now over-claims as well as omitting. The question was whether the two partly cancel. They do not, and the reason is not their signs: the over-claim never reaches a fiftieth of the omission, because one scales with the whole set and the other only with the barrier.

paths · Reach

Named alongside it

The objects these essays reach for when they reach for this one.

Polygon Tolerance Area Barrier Containment Estimator Generalisation Measurement Non convexity Polyline Quadrature Reach set

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