stereographic projection

[¦ster·ē·ə¦graf·ik prə′jek·shən] (crystallography) A method of displaying the positions of the poles of a crystal in which poles are projected through the equatorial plane of the reference sphere by lines joining them with the south pole for poles in the upper hemisphere, and with the north pole for poles in the lower hemisphere. (mapping) A perspective conformal, azimuthal map projection in which points on the surface of a sphere or spheroid, such as the earth, are conceived as projected by radial lines from any point on the surface to a plane tangent to the antipode of the point of projection; circles project as circles through the point of tangency, except for great circles which project as straight lines; the principal navigational use of the projection is for charts of the polar regions. Also known as azimuthal orthomorphic projection. (mathematics) The projection of the Riemann sphere onto the euclidean plane performed by emanating rays from the north pole of the sphere through a point on the sphere.

Stereographic Projection

the correspondence between points on a sphere and a plane obtained as follows: from a certain point C—the center of the projection—on a sphere other points on the sphere are projected by radial lines onto a plane perpendicular to the radius OC and not passing through C (Figure 1). The plane usually passes through either the center of the sphere O or through C at the other end of the diameter CC. Each point M on the sphere that is distinct from C will correspond to a certain point M’ on the plane; with the exception of point C, to which no point on the plane corresponds, there will be a one-to-one correspondence between the points on the sphere and those on the plane.

Figure 1

Stereographic projections have two main characteristics. One is that circles on the sphere correspond to circles on the plane. In Figure 1, circle Γ corresponds to circle Γ’. Circles passing through the center of the projection correspond to straight lines on the plane (circles of infinitely large radius; in Figure 1, Ƴ and Ƴ’). The other characteristic is that the correspondence established by the stereographic projection is conformal; that is, the angles are conserved. For example, angle LMN on the sphere is equal to angle L’M’N’ on the plane.

Stereographic projections are perspective cartographic projections. They are frequently used in cartography because of all the equiangular projections they give the least variation of scale for circular areas. Stereographic projections are also used in such fields as astronomy and crystallography.

stereographic projection

Equatorial stereographic projection.Perspective projections in which a graticule of reduced earth is projected onto a plane tangential to reduced earth at a point from a point diametrically opposite to the point of tangency. In such a projection, the meridians and parallels are drawn on a flat sheet touching the reduced earth at a point opposite the source of light.
The scale expands away from the pole of projection and the angles and bearings are correctly projected. Areas expand and large areas get distorted. The great circle and the meridians are curves concave to the pole of projection.