Editing Mercator projection (section)

== Properties ==
[[File:comparison_of_cylindrical_projections.svg|thumb|right|Comparison of tangent and secant forms of normal, oblique and transverse Mercator projections with standard parallels in red]]
The Mercator projection can be visualized as the result of wrapping a cylinder tightly around a sphere, with the two surfaces ''tangent'' to (touching) each other along a circle halfway between the poles of their common axis, and then [[conformal map|conformally]] unfolding the surface of the sphere outward onto the cylinder, meaning that at each point the projection uniformly [[scaling (geometry)|scales]] the image of a small portion of the spherical surface without otherwise distorting it, preserving angles between intersecting curves. Afterward, this cylinder is unrolled onto a flat plane to make a map. In this interpretation, the scale of the surface is preserved exactly along the circle where the cylinder touches the sphere, but increases nonlinearly for points further from the contact circle. However, by uniformly shrinking the resulting flat map, as a final step, any pair of [[parallel circles of a sphere|circles parallel]] to and equidistant from the contact circle can be chosen to have their scale preserved, called the ''standard parallels''; then the region between chosen circles will have its scale smaller than on the sphere, reaching a minimum at the contact circle. This is sometimes visualized as a projection onto a cylinder which is ''secant'' to (cuts) the sphere, though this picture is misleading insofar as the standard parallels are not spaced the same distance apart on the map as the shortest distance between them through the interior of the sphere.<ref>{{cite journal |last=Lapaine |first=Miljenko |title=A problem in 'Basic Cartography' |journal=International Journal of Cartography |volume=10 |number=1 |year=2024 |pages=118–131 |doi=10.1080/23729333.2022.2157106 |bibcode=2024IJCar..10..118L }} {{pb}} {{cite journal |title=Secant Cylinders Are Evil – A Case Study on the Standard Lines of the Universal Transverse Mercator and Universal Polar Stereographic Projections |last=Kerkovits |first=Krisztián |year=2024 |journal=  ISPRS International Journal of Geo-Information|volume=13 |number=2 |page=56 |doi=10.3390/ijgi13020056 |doi-access=free |bibcode=2024IJGI...13...56K }}</ref>

The original and most common [[aspect (cartography)|aspect]] of the Mercator projection for maps of Earth is the normal aspect, for which the axis of the cylinder is Earth's [[axis of rotation]] which passes through the North and South poles, and the contact circle is Earth's [[equator]]. As for all [[map projection#Cylindrical|cylindrical projections]] in normal aspect, [[circle of latitude|circles of latitude]] and [[meridian (geography)|meridians of longitude]] are straight and perpendicular to each other on the map, forming a grid of rectangles. While circles of latitude on Earth are smaller the closer they are to the poles, they are stretched in an East–West direction to have uniform length on any cylindrical map projection. Among cylindrical projections, the Mercator projection is the unique projection which balances this East–West stretching by a precisely corresponding North–South stretching, so that at every location the scale is locally uniform and angles are preserved.

The Mercator projection in normal aspect maps trajectories of constant [[bearing (navigation)|bearing]] (called ''[[rhumb lines]]'' or ''loxodromes'') on a sphere to straight lines on the map, and is thus uniquely suited to [[marine navigation]]: courses and bearings are measured using a [[compass rose]] or protractor, and the corresponding directions are easily transferred from point to point, on the map, e.g. with the help of a [[parallel ruler]].

Because the linear scale of a Mercator map in normal aspect increases with latitude, it distorts the size of geographical objects far from the equator and conveys a distorted perception of the overall geometry of the planet. At latitudes greater than 70° north or south, the Mercator projection is practically unusable,{{whom|date=July 2024}} because the [[linear scale]] becomes infinitely large at the poles. A Mercator map can therefore never fully show the [[polar regions of Earth|polar areas]] (but see [[Mercator projection#Uses|Uses]] below for applications of the oblique and transverse Mercator projections).

The Mercator projection is often compared to and confused with the [[central cylindrical projection]], which is the result of projecting points from the sphere onto a tangent cylinder along straight radial lines, as if from a light source placed at Earth's center.<ref>{{cite journal |last1=Frederick Rickey |first1=V. |last2=Tuchinsky |first2=Philip M. |title=An application of geography to mathematics: history of the integral of the secant |journal=Mathematics Magazine |date=May 1980 |volume=53 |issue=3 |page=164 |doi=10.2307/2690106 |jstor=2690106 |url=https://www.jstor.org/stable/2690106 |access-date=18 August 2022}}</ref> Both have extreme distortion far from the equator and cannot show the poles. However, they are different projections and have different properties.