# stereographic projection

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## stereographic projection

[¦ster·ē·ə¦graf·ik prə′jek·shən]## 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.

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

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.