In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the pole or center of projection), onto a plane (the projection plane) perpendicular to the diameter through the point.

In the following section on projection categories, the earth is taken as a sphere in order to simplify the discussion. However, the Earth's actual shape is closer to an oblate ellipsoid . Whether spherical or ellipsoidal, the principles discussed hold without loss of generality.

In standard presentation, azimuthal projections map meridians as straight lines and parallels as complete, concentric circles. They are radially symmetrical. In any presentation (or aspect), they preserve directions from the center point.

Apr 3, 2025 · A map projection obtained by projecting points P on the surface of sphere from the sphere's north pole N to point P^' in a plane tangent to the south pole S (Coxeter 1969, p. 93). In such a projection, great circles are mapped to circles, and loxodromes become logarithmic spirals.

Oct 10, 2021 · Given a point P = (x, y) ≠ N P = (x, y) ≠ N on the unit circle, let s(P) s (P) denote the intersection of the line NP¯ ¯¯¯¯¯¯¯ N P ¯ with the x x -axis. See Figure 1.3.1. The map s:S1 ∖ {N} → R s: S 1 ∖ {N} → R given by this rule is called stereographic projection. Using similar triangles, it is easy to see that s(x, y) = x 1−y. s (x, y) = x 1 − y.

The maps of the sphere which are easiest to understand are the central projections. For these we choose a point called the center of the projection and an image plane, which is usually tangent to the sphere at some point.

Principle of stereographic projection. For stereographic projection, a line or a plane is imagined to be surrounded by a projection sphere (Fig. 1a). A plane intersects the sphere in a trace that is a great circle that bisects the sphere precisely. A line intersects the sphere in a point.

Stereographic projection gives us a more concrete way of identifying ̃C with a sphere, one which, moreover, yields a lot of geometrical insight. We shall take the Riemann sphere to be the usual unit sphere S f x; y; t 2 R3 : x2 y2 t2 1g, but we shall identify the real x; y …

From a technical standpoint, stereographic projection is distinguished by these two properties: It preserves the angle between paths. It takes circles on the sphere into circles on the plane, except that it takes circles through P into lines.

When the viewing point is at the top of a sphere that rests on the horizontal plane, central projection sends each point of the sphere to a unique point of the plane. This gives a mapping from the sphere to the plane that cartographers call stereographic projection.