Ancient astronomers and cartographers realized the impossibility of stretching out the surface of a sphere on a plane without introducing distortions. To reduce the errors arising from this procedure, projection methods were introduced as a complement to the representations of the starry heavens by means of globes.
Cartographic projection, also called Lambert equivalent, for representing the northern terrestrial hemisphere on a plane tangent to the North Pole. The projection shows the northern hemisphere inscribed within a circumference corresponding to the equator, whose center represents the North Pole and whose surface carries a grid of radial lines (meridians) and concentric circles (parallels).
Projection of the celestial sphere on the plane intersecting the solstitial colure, with a projection point at infinity. In the orthogonal projection of both hemispheres, the meridians are shown by curved lines; the ecliptic and the parallels by straight lines. The oblique line of the ecliptic intersects the equator at the center (equinoctial point) and indicates the solstitial points on the circumference. The orthographic projection was introduced by Juan de Rojas Sarmiento.
Plane projection of the celestial sphere on the equatorial plane, using the South Pole as projection point. With this type of projection, one can show the entire northern hemisphere and a portion of the southern hemisphere down to the Tropic of Capricorn. Of the circles of the sphere, only the equatorial circle remains in its "true" position, as it coincides with the projection plane. The Tropic of Capricorn appears magnified and forms the outer circle of the representation, while the Tropic of Cancer seems shrunk. The ecliptic is shown as a eccentric circle, touching the two tropics in the solstitial points. This projection method, attributed to Ptolemy (2nd C. C.E.) or even to Eudoxus of Cnidus (350 B.C.E.), was developed to build the astrolabe.
Projection of the celestial sphere obtained on the plane intersecting the solstitial colure. The projection point is the intersection of the ecliptic and the equator (equinoctial point). With this type of projection both hemispheres are shown. Of the circles of the sphere, only the equator and the ecliptic appear as straight lines; all the others are displayed with increasingly curved shapes. This projection method was developed by the Muslim astronomer Al-Zarqali (Azarchel) (9th C. C.E.) to construct his universal astrolabe.