The buffer that shrank on the way to the map

CartographyintermediateCartography

A contractor delivered the five-minute-walk coverage map for the cycle-hire scheme. It was built in a web-mapping tool, entirely in Web Mercator, and its coverage figure for Inner London is about a fifth lower than the one produced in British National Grid. Both teams insist they buffered by 400 metres. You have to explain, with numbers, why they are both telling the truth and only one of them is right.

Web Mercator (EPSG:3857) is conformal: shapes are right locally, but its metre is not a ground metre. At latitude φ the map is stretched by a scale factor

k = 1 / cos(φ)

so a distance of 400 "metres" in EPSG:3857 is only 400 / k metres on the ground.

Compute and report, using the docking-station snapshot:

scale_factor — k at the mean latitude of the stations, to 4 decimals
ground_radius_m — the ground distance a 400 m EPSG:3857 buffer really has at that latitude, to the nearest metre
covered_km2_mercator — the Inner London coverage you get if you buffer every station by ground_radius_m (dissolved, clipped to the 13 Inner London boroughs), km²
lost_pct — how much smaller that is than the correct 400 m coverage, as a percentage of the correct figure

The correct coverage is the same computation with a true 400 m buffer in EPSG:27700 (or a geodesic buffer); do both and compare.

What to submit

Submit: Enter the named answers in the Answers form.

Objectives

Derive the Web Mercator scale factor at London's latitude
Translate a projected buffer distance into its ground equivalent
Re-run the coverage analysis with the shrunken buffer and measure the shortfall

Constraints

These are where the teaching is. Read them twice.

Use the mean latitude of the stations for k, and use it for every station — the point is the magnitude, not a per-station correction
Coverage is dissolved before clipping and clipped to the 13 Inner London boroughs
Areas measured in EPSG:27700 or geodesically, so that the only variable is the buffer radius

Graded requirements

Web Mercator scale factor at the stations' mean latitude
Ground radius of a 400 m Mercator buffer
Inner London coverage with the shrunken buffer
Share of the true coverage lost

Buffers and distances computed in EPSG:3857 are the single most common silent error in web-built spatial analysis, and the size of the error grows with latitude.

Answers

Work the problem in whatever tool you like, then enter the answers here.

Web Mercator scale factor at the stations' mean latitude

scale_factor

Ground radius of a 400 m Mercator buffer

ground_radius_m

Inner London coverage with the shrunken buffer

km²

covered_km2_mercator

Share of the true coverage lost

lost_pct

Loading map

What will be checked

4 scored, 0 informational

Web Mercator scale factor at the stations' mean latitude
20%
Ground radius of a 400 m Mercator buffer
20%
Inner London coverage with the shrunken buffer
30%
Share of the true coverage lost
30%