Zenithal Stereographical Polar Projection


Lines of Latitude
Lines of Longitude
Prime Meridian
Facts on Earth
Zenithal Projections
Gnomonic Polar
Gnomonic Equatorial
Stereographic Polar
Stereographic Equatorial
Orthographic Polar
Orthographic Equatorial
Simple Conic Projection
Cylindrical Equal-area Projection





Figure A

Figure B

Figure C


Stereographic projections are created by projecting from a point at one end of the diameter in a circle to a projection plane which sit tangential to the other end of the diameter. This situation is clearly depicted in Figure A where the light source comes from point A with white rays radiating out of it. Geometrically, the light source sits on the bottom end of diameter AB. In terms of geography, the light source sits on the south pole of the earth with the projection plane CI on the north pole, thus making the projections on this plane a polar projection. Visually, one can think of the stereographic projection as being created by the shadows casted upon the projection plane CI by the parallels and meridians.

With the yellow lines representing parallels and the angle the blue lines makes withe the equator representing angular distance away from the equator EF, shadows of parallels are located at the intersection of the light rays with the plane of projection such as C, B, D, E, F, G, H, and I. As noted in the picture, these rays all goes through the same intersection point made by the parallel with the circle. When looking upon this plane, latitudes will appear as circles whose radius is equal to the distance the polar axis AB is away from such intersections as C, B, D, E, F, G, H, and I as illustrated in Figure B. Mathematically, this distance can be calculated using circle geometry. Referring to Figure A, if an arbitrary latitude JK is chosen, its angular distance is <EOJ. Because the polar axis and the equator are perpendicular to each other, <EOJ and <BOJ are complementary angles. As such <BOJ is also known as a co-latitude. In other words, <BOJ = 90° - <EOJ. A similar situation occurs on the supplementary side of the latitude. Thus, <FOK=angular distance of latitude and <KOB = co-latitude. Therefore, <KOB=<BOJ and <EOJ=<FOK. Utilizing a fact from circle geometry, there is a central angle <JOK subtended by the arc JK. Since <JAK is also subtended by the same arc JK, but touching the side of the circle, the relationship between <JOK and <JAK is as follows: <JAK= 1/2 <JOK. Because JK is perpendicular to the polar axis BA which goes through the center of the circle, BA bisects JK and similarly <JOK and <JAK. This is because <JAK and <JOK are opposite angles to the bisected latitude. The relationship <JAK = 1/2 <JOK becomes <JAB+<KAB=1/2 (<JOB+<KOB). Since <KAB = <JAB and <JOB = <KOB, 2 <JAB = 1/2 (2 <JOB) therefore <JAB=1/2<JOB. With AB equivalent to the diameter of the earth or 2 radius and the plane CI tangential to the circle, therefore, <CBO is a right angle. Thus the following relationship can be formed: tan (<JAB) = JB (radius of the latitude) / circle's diameter. In other words, radius of the latitude = 2 circle's radius * tan(1/2<JOB). In conclusion, radius of latitude = radii * tan(co-latitude). Since tangent reaches infinity at 90°, the polar stereographic projection cannot reveal an entire hemisphere. Another interesting fact about the stereographic polar projection is that smaller latitudes are farther away from the center point of the projection (which represents the north pole) at the point where the polar axis intersects the projection plane.

By visually imagining the meridians from a birds eye view, one can imagine that the meridians, which is the semi circle following the shape of the earth from the poles, will be projected as the straight radius of the outer most latitude as seen in Figure C. These equally spaced radius are rotated around the center with the desired angular interval. Therefore, the number of meridians with angular distance n° apart from each other would have 360°/n° number of meridians.