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Stereographic projection of the unit sphere from the north pole onto the plane z = 0, shown here in cross section. The unit sphere S 2 in three-dimensional space R 3 is the set of points (x, y, z) such that x 2 + y 2 + z 2 = 1.
The maturation of complex analysis led to general techniques for conformal mapping, where points of a flat surface are handled as numbers on the complex plane.While working at the United States Coast and Geodetic Survey, the American philosopher Charles Sanders Peirce published his projection in 1879, [2] having been inspired by H. A. Schwarz's 1869 conformal transformation of a circle onto a ...
Can display nearly the entire sphere's surface on a finite circle. The sphere's full surface requires an infinite map. Other azimuthal projections are not true perspective projections: Azimuthal equidistant: r(d) = cd; it is used by amateur radio operators to know the direction to point their antennas toward a point and see the distance to it.
They are written in terms of longitude (λ) and latitude (φ) on the sphere. Define the radius of the sphere R and the center point (and origin) of the projection (λ 0, φ 0). The equations for the orthographic projection onto the (x, y) tangent plane reduce to the following: [1]
Gnomonic projection of a portion of the north hemisphere centered on the geographic North Pole 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, onto any plane not passing through the center, most commonly ...
It sends the point (0, 0, −1) to (0, 0), the equator z = 0 to the circle of radius √ 2 centered at (0, 0), and the lower hemisphere z < 0 to the open disk contained in that circle. The projection is a diffeomorphism (a bijection that is infinitely differentiable in both directions) between the sphere (minus (0, 0, 1)) and the open disk of ...
For example, lines traced from the eye point at 45° to the picture plane intersect the latter along a circle whose radius is the distance of the eye point from the plane, thus tracing that circle aids the construction of all the vanishing points of 45° lines; in particular, the intersection of that circle with the horizon line consists of two ...
Projects the globe onto eight octants (Reuleaux triangles) with no meridians and no parallels. 1909 Cahill's butterfly map: Polyhedral Compromise Bernard Joseph Stanislaus Cahill: Projects the globe onto an octahedron with symmetrical components and contiguous landmasses that may be displayed in various arrangements. 1975 Cahill–Keyes projection