Cartographic Projection
every map is wrong, and by exactly how much
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 4 essays across 3 fields — each of them below, with the objects they name alongside it.

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 34.1% of the true ground, as a fan beside a straight river does, and it also CLAIMS 0.14% 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 that: the near side is no longer convex, and a chord between two bearings can cross water. So the drawn polygon now over-claims as well as omitting — by a seventh of a per cent, at every bend measured, and more bearings do not reduce it. The question was whether the two errors partly cancel. They do not, and the reason is not their signs: the over-claim never reaches a hundredth of the omission, because one scales with the whole set and the other only with the river.

paths · Reach Beside a bending river, the fan's chords cross water; a route drawn from the river's own outline never does. Everywhere within a 10 km walk of the centre, shaded pale, with a river 3 km north meandering ±2.0 km on a 4 km wavelength and bridges every 2.5 km. Left: the polygon 16 bearings draw, with the ground it claims and should not in solid ink — chords spanning the bends. Averaged over where the first bearing falls, it misses 35.0% of the true ground and claims 0.15% that is not there. Right: every point reachable by a route with at most one crossing — straight out, or straight to a bridge and straight on — worked out from the river's outline and the bridges alone, with no fan and no shortest path. It misses 6.4% and claims nothing.

A reach set drawn from the river's outline is a bound again

Beside a bending river, a fan of bearings drawn with straight chords claims a strip of water, and the drawing stops being a lower bound. The fan's own numbers cannot repair it: what a bearing records about the river is where the bridges are, not where the bends are, and the best a rule built from them does is cut the over-claim by seven eighths while giving up twenty times as much ground. The river's outline repairs it exactly. Cutting each bearing at the first bank costs three and a half times the over-claim it removes; keeping the chords wherever a route over one bridge reaches costs one and a half. Drawn from the outline and the bridges with no fan at all, that route set misses 6.4 per cent of the reachable ground where the fan misses 35.

paths · Reach Two solved maps and the line between them keep what the whole family of exponents keeps. Area inside a band, per cent, against the band as a fraction of each region's least worst departure, from 0.3 to 1. Heavy: the best map on the line between the p = 1 map and the least worst map, found by a search along the line alone. Thin: the best of twelve exponents from 1 to 32, each solved separately. Dashed: the rule taking p = 1 below 0.9 and the least worst case above it; shaded, what that rule gives up. On the square the line trails the best exponent by at most 0.00 points and beats it by up to 1.3; the rule gives up 0.9. On the ellipse the line trails by at most 1.1 and the rule gives up 5.9.

Two maps and the line between them do the work of every exponent

A conformal map's log scale is linear in its coefficients, so every blend of the map that minimises the mean departure and the least worst map is itself a conformal map, available the moment the two are solved. Searched along that one line for each band, the blend keeps within 0.17 points of the best of twelve exponents on every compact region and beats all twelve by up to 1.35 on the square; it trails by 1.5 at most on the thinnest ellipse and 0.74 on four regions with a bay, where the rule of switching between the two ends gave up as much as 7.1. The family's middle members lie up to two thirds of their own departure off the line. The line does not reproduce them; it does their work.

choosing · Condition

Named alongside it

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

PolygonToleranceVerificationBarrierReach setShortest pathAreaContainmentEstimatorGeneralisationInterpolationLeast-squares

All concepts