Editing Gnomonic projection (section)

{{Short description|Projection of a sphere through its center onto a plane}}
{{No footnotes|date=February 2019}}
[[File:Gnomonic projection SW.jpg|300px|thumb|Gnomonic projection of a portion of the north hemisphere centered on the geographic North Pole]]
[[File:Gnomonic with Tissot's Indicatrices of Distortion.svg|thumb|The gnomonic projection with [[Tissot's indicatrix]] of deformation]]

A '''gnomonic projection''', also known as a '''central projection''' or '''rectilinear projection''', is a [[perspective projection]] of a [[sphere]], with center of projection at the sphere's [[center (geometry)|center]], onto any [[plane (geometry)|plane]] not passing through the center, most commonly a [[tangent]] plane. Under gnomonic projection every [[great circle]] on the sphere is projected to a [[straight line]] in the plane (a great circle is a [[geodesic]] on the sphere, the shortest path between any two points, analogous to a straight line on the plane).<ref>{{cite journal |last1=Williams |first1=C.E. |last2=Ridd |first2=M.K. |year=1960 |title=Great Circles and the Gnomonic Projection |journal=The Professional Geographer |volume=12 |number=5 |pages=14–16 |doi=10.1111/j.0033-0124.1960.125_14.x }}</ref> More generally, a gnomonic projection can be taken of any [[n-sphere|{{mvar|n}}-dimensional hypersphere]] onto a [[hyperplane]].

The projection is the {{mvar|n}}-dimensional generalization of the [[tangent (trigonometry)|trigonometric tangent]] which maps from the [[circle]] to a straight line, and as with the tangent, every pair of [[antipodal point]]s on the sphere projects to a single point in the plane, while the points on the plane through the sphere's center and parallel to the image plane project to [[point at infinity|points at infinity]]; often the projection is considered as a [[Bijection|one-to-one correspondence]] between points in the hemisphere and points in the plane, in which case any finite part of the image plane represents a portion of the hemisphere.<ref name=Snyder>{{cite book |last=Snyder |first=John P. |title=Map Projections – A Working Manual |series=U.S. Geological Survey Professional Paper |volume=1395 |publisher=United States Government Printing Office |year=1987 |pages=164–168 |location=Washington, D.C. |url=https://archive.org/details/Snyder1987MapProjectionsAWorkingManual/page/n178 |doi=10.3133/pp1395 }}</ref>

The gnomonic projection is [[azimuthal projection|azimuthal]] (radially symmetric). No shape distortion occurs at the center of the projected image, but distortion increases rapidly away from it.

The gnomonic projection originated in [[astronomy]] for constructing [[sundial]]s and charting the [[celestial sphere]]. It is commonly used as a geographic [[map projection]], and can be convenient in [[navigation]] because great-circle courses are plotted as straight lines. [[Rectilinear lens|Rectilinear photographic lens]]es make a perspective projection of the world onto an image plane; this can be thought of as a gnomonic projection of the [[image sphere]] (an abstract sphere indicating the direction of each ray passing through a [[pinhole camera model|camera modeled as a pinhole]]). The gnomonic projection is used in [[crystallography]] for analyzing the orientations of lines and planes of crystal structures. It is used in [[structural geology]] for analyzing the orientations of fault planes. In [[computer graphics]] and computer representation of spherical data, [[cube mapping]] is the gnomonic projection of the image sphere onto six faces of a [[cube]].

In mathematics, the space of [[orientation (geometry)|orientations]] of undirected lines in [[3-dimensional space]] is called the [[real projective plane]], and is typically pictured either by the "projective sphere" or by its gnomonic projection. When the [[angle]] between lines is imposed as a [[distance function|measure of distance]], this space is called the [[elliptic plane]]. The gnomonic projection of the 3-sphere of [[unit quaternions]], points of which represent 3-dimensional rotations, results in [[Rodrigues vector]]s. The gnomonic projection of the [[hyperboloid of two sheets]], treated as a model for the [[hyperbolic plane]], is called the [[Beltrami–Klein model]].