Editing Mercator projection (section)

== Uses ==
[[File:Rhumb line vs great-circle arc.png|thumb|upright=1.3|A rhumb line (blue) compared to a great-circle arc (red) between Lisbon, Portugal, and Havana, Cuba. Top: orthographic projection. Bottom: Mercator projection.]]
Practically every marine chart in print is based on the Mercator projection due to its uniquely favorable properties for navigation. It is also commonly used by street map services hosted on the Internet, due to its uniquely favorable properties for local-area maps computed on demand.<ref>{{Cite web |title=Why Mercator for the Web? Isn't the Mercator bad? |url=https://www.mapthematics.com/forums/viewtopic.php?t=251 |access-date=2024-11-12 |website=www.mapthematics.com}}{{Self-published source|date=July 2020}}</ref> Mercator projections were also important in the mathematical development of [[plate tectonics]] in the 1960s.<ref>
{{cite book
| editor-last = Cox
| editor-first = Allan
| title = Plate Tectonics and Geomagnetic Reversals
| page = 46
| publisher = W.H. Freeman
| year = 1973
}}</ref>

===Marine navigation===
The Mercator projection was designed for use in marine [[navigation]] because of its unique property of representing any course of constant [[bearing (navigation)|bearing]] as a straight segment. Such a course, known as a [[rhumb line|rhumb]] (alternately called a rhumb line or loxodrome) is preferred in marine navigation because ships can sail in a constant compass direction. This reduces the difficult, error-prone course corrections that otherwise would be necessary when sailing a different course.

For small distances (compared to the radius of Earth), the difference between the rhumb and the [[great circle]] course is negligible. Even for longer distances, the simplicity of the constant bearing makes it attractive. As observed by Mercator, on such a course, the ship would not arrive by the shortest route, but it will surely arrive. Sailing a rhumb meant that all that the sailors had to do was keep a constant course as long as they knew where they were when they started, where they intended to be when they finished, and had a map in Mercator projection that correctly showed those two coordinates.{{sfn|Osborne|2013|pp=39–40}}

=== Web Mercator ===
{{Main|Web Mercator projection}}

Many major online street mapping services ([[Bing Maps]], [[Google Maps]], [[Mapbox]], [[MapQuest]], [[OpenStreetMap]], [[Yahoo! Maps]], and others) use a variant of the Mercator projection for their map images<ref>{{Cite journal |last1=Battersby |first1=Sarah E. |last2=Finn |first2=Michael P. |last3=Usery |first3=E. Lynn |last4=Yamamoto |first4=Kristina H. |date=June 1, 2014 |title=Implications of Web Mercator and Its Use in Online Mapping |url=http://dx.doi.org/10.3138/carto.49.2.2313 |journal=Cartographica: The International Journal for Geographic Information and Geovisualization |volume=49 |issue=2 |pages=85–101 |doi=10.3138/carto.49.2.2313 |issn=0317-7173}}</ref> called [[Web Mercator projection|Web Mercator]] or Google Web Mercator. Despite its obvious scale variation at the world level (small scales), the projection is well-suited as an interactive world map that can be zoomed seamlessly to local (large-scale) maps, where there is relatively little distortion due to the variant projection's near-[[conformal map projection|conformality]].

The major online street mapping services' tiling systems display most of the world at the lowest zoom level as a single square image, excluding the polar regions by truncation at latitudes of ''φ''<sub>max</sub>&nbsp;=&nbsp;±85.05113°. (See [[#Truncation and aspect ratio|below]].) Latitude values outside this range are mapped using a different relationship that does not diverge at&nbsp;''φ''&nbsp;=&nbsp;±90°.{{citation needed|date=February 2017}}

===Transverse Mercator===
{{Main|Transverse Mercator projection }}
A transverse Mercator projection tilts the cylinder axis so that it is perpendicular to Earth's axis. The tangent standard line then coincides with a meridian and its opposite meridian, giving a constant [[scale factor]] along those meridians and making the projection useful for mapping regions that are predominately north–south in extent. In its more complex ellipsoidal form, most national grid systems around the world use the transverse Mercator, as does the [[Universal Transverse Mercator coordinate system]].

===Oblique Mercator===
{{Main|Oblique Mercator projection}}
An oblique Mercator projection tilts the cylinder axis away from Earth's axis to an angle of one's choosing, so that its tangent or secant lines of contact are circles that are also tilted relative to Earth's parallels of latitude.<ref>{{cite web | url=https://mathworld.wolfram.com/MercatorProjection.html | title=Mercator Projection }}</ref> Practical uses for the oblique projection, such as national grid systems, use [[Oblique Mercator projection|ellipsoidal developments of the oblique Mercator]] in order to keep scale variation low along the surface projection of the cylinder's axis.

The animation below shows a continuous transformation between the normal Mercator projection and the transverse projection, running through the intermediate oblique states.  A rhumb line (blue) and great circle (red) from Tokyo, Japan, to Callao, Peru, change according to the projection's aspect. The rhumb between them is straight only on the normal projection. On one of the intermediate oblique projections, the great-circle path between the two cities is straight. In this plot the y-axis is always the projected axis of the cylinder. The sphere is rotated inside the cylinder to change aspect. Initially the axis is normal to the equator (projected as the thick green line) but ends transverse to it, when the projected equator extends to infinity in the vertical direction.

[[File:Mercator Polar Transverse.gif|thumb|500px|center|Continuous transformation from a normal Mercator projection to a transverse Mercator, with a rhumb line (blue) and great-circle line (red) from Tokyo to Callao.]]