Map projection
In cartography, a map projection is any of a broad set of transformations employed to represent the curved two-dimensional surface of a globe on a plane.
Rather, any mathematical function that transforms coordinates from the curved surface distinctly and smoothly to the plane is a projection.
The Earth and other large celestial bodies are generally better modeled as oblate spheroids, whereas small objects such as asteroids often have irregular shapes.
[8] Therefore, more generally, a map projection is any method of flattening a continuous curved surface onto a plane.
Because the Earth's curved surface is not isometric to a plane, preservation of shapes inevitably requires a variable scale and, consequently, non-proportional presentation of areas.
Similarly, an area-preserving projection can not be conformal, resulting in shapes and bearings distorted in most places of the map.
Data sets are geographic information; their collection depends on the chosen datum (model) of the Earth.
Carl Friedrich Gauss's Theorema Egregium proved that a sphere's surface cannot be represented on a plane without distortion.
[7]: 147–149 [10]: 24 By spacing the ellipses regularly along the meridians and parallels, the network of indicatrices shows how distortion varies across the map.
[13] Rather than the original (enlarged) infinitesimal circle as in Tissot's indicatrix, some visual methods project finite shapes that span a part of the map.
[16] Another way to visualize local distortion is through grayscale or color gradations whose shade represents the magnitude of the angular deformation or areal inflation.
[17] To measure distortion globally across areas instead of at just a single point necessarily involves choosing priorities to reach a compromise.
Some schemes use distance distortion as a proxy for the combination of angular deformation and areal inflation; such methods arbitrarily choose what paths to measure and how to weight them in order to yield a single result.
Moving the developable surface away from contact with the globe never preserves or optimizes metric properties, so that possibility is not discussed further here.
Some possible properties are: Projection construction is also affected by how the shape of the Earth or planetary body is approximated.
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.
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, conformality and equivalence.
The most common projection surfaces are cylindrical (e.g., Mercator), conic (e.g., Albers), and planar (e.g., stereographic).
Some of the more common categories are: Because the sphere is not a developable surface, it is impossible to construct a map projection that is both equal-area and conformal.
[29]Lee's objection refers to the way the terms cylindrical, conic, and planar (azimuthal) have been abstracted in the field of map projections.
If maps were projected as in light shining through a globe onto a developable surface, then the spacing of parallels would follow a very limited set of possibilities.
Where the light source emanates along the line described in this last constraint is what yields the differences between the various "natural" cylindrical projections.
The resulting conic map has low distortion in scale, shape, and area near those standard parallels.
These are some conformal projections: Equal-area maps preserve area measure, generally distorting shapes in order to do so.
Great circles are displayed as straight lines: Direction to a fixed location B (the bearing at the starting location A of the shortest route) corresponds to the direction on the map from A to B: Compromise projections give up the idea of perfectly preserving metric properties, seeking instead to strike a balance between distortions, or to simply make things look right.
For smaller-scale maps, such as those spanning continents or the entire world, many projections are in common use according to their fitness for the purpose, such as Winkel tripel, Robinson and Mollweide.
Due to distortions inherent in any map of the world, the choice of projection becomes largely one of aesthetics.
In 1989 and 1990, after some internal debate, seven North American geographic organizations adopted a resolution recommending against using any rectangular projection (including Mercator and Gall–Peters) for reference maps of the world.
