Mercator: properties, uses, and limitations
The 1569 conformal cylindrical projection built for navigation: a constant compass bearing is a straight line, and the price is area inflating as sec squared of latitude - 91.5 times over at 84 degrees north.
Description
Gerardus Mercator published Nova et Aucta Orbis Terrae Descriptio in 1569. It solves one concrete navigational problem: draw a course of constant compass bearing - a rhumb line, or loxodrome - as a straight line, so a sailor can lay a ruler on the chart and read a heading off it.
To do that it has to be conformal: angles around any point on the map must match angles around the corresponding point on the ground. Parallel spacing accelerating polewards as sec(latitude) is exactly what cancels the east-west stretch that flattening the meridians introduces. That is not a flaw in the projection, it is the step that makes the straight bearing work.
Projection properties
- Family
- Conformal cylindrical
- Frame ratio
- 1.288:1
- Latitudes drawn
- 85.1° N to 70.0° S
- Pole line
- The pole cannot be drawn
- True-shape parallel
- Everywhere
- Area distortion doubles
- 45.0°
- Shape distortion doubles
- Never
- Held at 1.00 by definition
- Shape distortion
Meridians are evenly spaced vertical lines; parallels are horizontal, spaced further and further apart towards the poles. The graticule is orthogonal everywhere.
Conformal: shape distortion measures 1.00 at 0, 30, 45, 60, 72 and 84 degrees, without exception. Turn on the distortion-ellipse grid and every ellipse is a circle.
Area inflates as sec squared: 1.33 at 30 degrees, exactly 2 at 45, exactly 4 at 60, 10.5 at 72, 91.5 at 84, and 134 by 85.05.
Distance along the equator is true. Away from it, east-west and north-south are magnified by the same sec(latitude) together - which is what conformality means here, and why the area exponent is exactly the square of the length exponent.
Measured distortion
| Latitude | 0° | 30° | 45° | 60° | 72° | 84° |
|---|---|---|---|---|---|---|
| Area distortion | 1.00 | 1.33 | 2.00 | 4.00 | 10.5 | 91.5 |
| Shape distortion | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
Both rows are measured against this projection's own equator, taken as 1. They compare with each other and with nothing on another projection's page - every projection is normalised to itself.
Uses
Nautical charts. Nothing has replaced the straight rhumb line, and international charts are still mostly Mercator.
Large-scale mapping near the equator - Indonesia, equatorial Africa - where over a small extent the area error is negligible.
Web maps. Web Mercator (EPSG:3857) is the default coordinate system of nearly every online map, because conformality keeps buildings and roads unskewed at every zoom level.
Limitations
The poles cannot be drawn: sec squared diverges at 90. Web maps cut at 85.051129 degrees, about 85 degrees 3 minutes 4 seconds, where the area distortion measured here is 134 times.
At that cut-off Mercator's world is exactly square - measured here at a frame ratio of 1.000. The tool draws 85.1 N down to 70.0 S instead, because nothing in the catalogue reaches past 56 south.
Area is unusable. Greenland looks larger than Africa and covers a fourteenth of it. No map shaded by country area should sit on this base.
Parameters
How this site builds it (w and h are the canvas; sphere is the whole globe):
geoMercator().scale(w / 2π).translate([w / 2, scale * mercatorY(85.051129)])Web Mercator deserves its own note: it applies the spherical Mercator formulas to WGS 84 geodetic coordinates with a sphere radius of 6378137.0 metres. The result is not strictly conformal, and distances and directions drift away from the equator - but every major map service does it, so it became the de facto standard.
This site draws spherical Mercator, not Web Mercator's ellipsoidal approximation. At the accuracy this project claims the two are indistinguishable; for surveying or legal use they are not interchangeable.