Editing Map projection (section)

==Design and construction==
The creation of a map projection involves two steps:

# Selection of a model for the shape of the Earth or planetary body (usually choosing between a [[sphere]] or [[ellipsoid]]). Because the Earth's actual shape is irregular, information is lost in this step.
# Transformation of geographic coordinates ([[longitude]] and [[latitude]]) to [[Cartesian coordinate system|Cartesian]] (''x'',''y'') or [[Polar coordinate system|polar]] (''r'', ''θ'') plane coordinates. In large-scale maps, Cartesian coordinates normally have a simple relation to [[Easting and northing|eastings and northings]] defined as a grid superimposed on the projection. In small-scale maps, eastings and northings are not meaningful, and grids are not superimposed.
<!--# Reduction of the scale (it does not matter in what order the second and third steps are performed)-->

Some of the simplest map projections are literal projections, as obtained by placing a light source at some definite point relative to the globe and projecting its features onto a specified surface. Although most projections are not defined in this way, picturing the light source-globe model can be helpful in understanding the basic concept of a map projection.

===Choosing a projection surface===
[[File:Usgs map miller cylindrical.PNG|thumb|300px|A [[Miller cylindrical projection]] maps the globe onto a cylinder.]]
A surface that can be unfolded or unrolled into a plane or sheet without stretching, tearing or shrinking is called a ''[[developable surface]]''. The [[cylinder (geometry)|cylinder]], [[cone (geometry)|cone]] and the plane are all developable surfaces. The sphere and ellipsoid do not have developable surfaces, so any projection of them onto a plane will have to distort the image. (To compare, one cannot flatten an orange peel without tearing and warping it.)

One way of describing a projection is first to project from the Earth's surface to a developable surface such as a cylinder or cone, and then to unroll the surface into a plane. While the first step inevitably distorts some properties of the globe, the developable surface can then be unfolded without further distortion.

===Aspect of the projection{{anchor|Aspect}}===
[[File:Usgs map traverse mercator.PNG|thumb|300px|This [[transverse Mercator projection]] is mathematically the same as a standard Mercator, but oriented around a different axis.]]
Once a choice is made between projecting onto a cylinder, cone, or plane, the '''aspect''' of the shape must be specified. The aspect describes how the developable surface  is placed relative to the globe: it may be ''normal'' (such that the surface's axis of symmetry coincides with the Earth's axis), ''transverse'' (at right angles to the Earth's axis) or ''oblique'' (any angle in between).

===Notable lines===<!-- [[Central meridian (map projections)]] links to this section -->
{{comparison_of_cartography_surface_development.svg|300px}}
The developable surface may also be either ''[[tangent]]'' or ''[[secant line|secant]]'' to the sphere or ellipsoid. Tangent means the surface touches but does not slice through the globe; secant means the surface does slice through the globe. Moving the developable surface away from contact with the globe never preserves or optimizes metric properties, so that possibility is not discussed further here.

Tangent and secant lines (''standard lines'') are represented undistorted. If these lines are a parallel of latitude, as in conical projections, it is called a ''standard parallel''. The ''central meridian'' is the meridian to which the globe is rotated before projecting. The central meridian (usually written ''λ''{{sub|0}}) and a parallel of origin (usually written ''φ''{{sub|0}}) are often used to define the origin of the map projection.<ref>{{cite web
  | url       = http://www.geography.hunter.cuny.edu/~jochen/GTECH361/lectures/lecture04/concepts/Map%20coordinate%20systems/Projection%20parameters.htm
  | title     = Projection parameters
  | last      = Albrecht
  | first     = Jochen
  | publisher = City University of New York
  }}</ref><ref>{{cite web
  | url          = http://edndoc.esri.com/arcsde/9.2/concepts/geometry/coordref/coordsys/projected/mapprojections.htm
  | archive-url  = https://web.archive.org/web/20181128234249/edndoc.esri.com/arcsde/9.2/concepts/geometry/coordref/coordsys/projected/mapprojections.htm
  | archive-date = 28 November 2018
  | title        = Map projections
  | website      = ArcSDE Developer Help
  }}</ref>

===Scale===
{{further|Map scale factor}}
A [[globe]] is the only way to represent the Earth with constant [[scale (map)|scale]] throughout the entire map in all directions. A map cannot achieve that property for any area, no matter how small. It can, however, achieve constant scale along specific lines.

Some possible properties are:
* The scale depends on location, but not on direction. This is equivalent to preservation of angles, the defining characteristic of a [[conformal map]].
* Scale is constant along any parallel in the direction of the parallel. This applies for any cylindrical or pseudocylindrical projection in normal aspect.
* Combination of the above: the scale depends on latitude only, not on longitude or direction. This applies for the [[Mercator projection]] in normal aspect.
* Scale is constant along all straight lines radiating from a particular geographic location. This is the defining characteristic of an equidistant projection such as the [[azimuthal equidistant projection]]. There are also projections (Maurer's [[two-point equidistant projection]], Close) where true distances from ''two'' points are preserved.<ref name="SnyderFlattening"/>{{rp|234}}

===Choosing a model for the shape of the body===
Projection construction is also affected by how the shape of the Earth or planetary body is approximated. In the following section on projection categories, the earth is taken as a [[sphere]] in order to simplify the discussion. However, the Earth's actual shape is closer to an oblate [[ellipsoid]]. Whether spherical or ellipsoidal, the principles discussed hold without loss of generality.

Selecting a model for a shape of the Earth involves choosing between the advantages and disadvantages of a sphere versus an ellipsoid. Spherical models are useful for small-scale maps such as world atlases and globes, since the error at that scale is not usually noticeable or important enough to justify using the more complicated ellipsoid. The ellipsoidal model is commonly used to construct [[topographic map]]s and for other large- and medium-scale maps that need to accurately depict the land surface. [[Latitude#Auxiliary latitudes|Auxiliary latitudes]] are often employed in projecting the ellipsoid.

{{anchor|geoid}}A third model is the [[geoid]], a more complex and accurate representation of Earth's shape coincident with what [[mean sea level]] would be if there were no winds, tides, or land. Compared to the best fitting ellipsoid, a geoidal model would change the characterization of important properties such as distance, [[#Conformal|conformality]] and [[#Equal-area|equivalence]]. Therefore, in geoidal projections that preserve such properties, the mapped [[Geographic coordinate system#Geographic latitude and longitude|graticule]] would deviate from a mapped ellipsoid's graticule. Normally the geoid is not used as an [[Earth model]] for projections, however, because Earth's shape is very regular, with the [[undulation of the geoid]] amounting to less than 100&nbsp;m from the ellipsoidal model out of the 6.3&nbsp;million&nbsp;m [[Earth radius]]. For irregular planetary bodies such as [[asteroids]], however, sometimes models analogous to the geoid are used to project maps from.<ref>
{{Cite journal
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  | title   = Equal Area Map Projection for Irregularly Shaped Objects
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  | title   = Mapping Worlds with Irregular Shapes
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{{cite journal
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  | title   = Mathematical Basis for Non-spherical Celestial Bodies Maps
  | journal = Journal of Geospatial Engineering
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{{cite journal
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{{Cite book
  | doi     = 10.1007/978-1-4614-7762-4_6
  | chapter = CSNB Mapping Applied to Irregular Bodies
  | title   = Constant-Scale Natural Boundary Mapping to Reveal Global and Cosmic Processes
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  | series  = SpringerBriefs in Astronomy
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Other regular solids are sometimes used as generalizations for smaller bodies' geoidal equivalent. For example, [[Io (moon)|Io]] is better modeled by triaxial ellipsoid or prolated spheroid with small eccentricities. [[Haumea]]'s shape is a [[Jacobi ellipsoid]], with its major [[Axis of rotation|axis]] twice as long as its minor and with its middle axis one and half times as long as its minor.
See [[map projection of the triaxial ellipsoid]] for further information.