Stereographic projection

Illustrative scheme of an azimutal stereo projection.

The stereographic projection is a graphic representation system in which the surface of a sphere is projected onto a plane by means of a set of lines that pass through a point, or focus. The projection plane is tangent to, or parallel to, the sphere, and the focus is the point on the sphere diametrically opposite the point of tangency of the plane with the sphere.

The surface it can represent is larger than a hemisphere. The most characteristic feature is that the scale increases as we move away from the center.

In its polar projection the meridians are straight lines, and the parallels are concentric circles. In the equatorial projection, only the equator and the central meridian are straight lines.

Astronomy

Stereographic projection of the celestial sphere with the most important elements.

There is another type of stereographic projection that is very useful for representing the celestial sphere. In this case, since the points to be projected are outside the sphere, the first step is to project them to the sphere, uniting them with its center. Once the external points are projected on the surface of the sphere, the procedure is analogous to that explained above. Usually, the plane is a horizontal one that contains the center of the sphere, and the focus of the projection is the nadir.

The characteristic of this projection is that it is more subjective, because it proposes the position of the observer P in the center and directly represents the azimuth and height of point A on the celestial sphere using local coordinates. It is used in architecture and engineering to represent the position of the Sun throughout the year, and to calculate sunlight and shadows that it produces.