Gnomonic projection

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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, onto any 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).^[1] More generally, a gnomonic projection can be taken of any n-dimensional hypersphere onto a hyperplane.

The projection is the n-dimensional generalization of the trigonometric tangent which maps from the circle to a straight line, and as with the tangent, every pair of antipodal points 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 points at infinity; often the projection is considered as a 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.^[2]

The gnomonic projection is 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 sundials 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 photographic lenses 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 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 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 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 vectors. The gnomonic projection of the hyperboloid of two sheets, treated as a model for the hyperbolic plane, is called the Beltrami–Klein model.