Zenithal Orthographic 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



In Orthographic projections, projections of parallels and meridians are created by projecting from a point at infinite distance away from the 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 the south of the circle at an infinite distance. As a result, the white rays radiating from that infinite point willl appear to go through the earth parallel to each other (as is the case of AB, CD and EF). The projection plane CE is situated tangentially to the north of the polar axis AB. Such an arrangement makes this projection a polar projection. Like other Zenithal projections, one can think of the orthographic projection as being created by the shadows casted upon the projection plane CE by the parallels and meridians.

With the red line segment GH representing an arbitrary parallel and <GOI & <HOJ equal to the angular distance away from the equator, shadows of parallels are located at the intersection of the light rays with the plane of projection such as C, B, and E. 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 and E as illustrated in Figure B. Mathematically, this distance can be calculated using circle geometry. As given in the beginning the polar axis and the equator are perpendicular to each other, therefore <GOI and <BOG are complementary angles. Hence, <BOG is also known as the co-latitude of <GOI. In other words, <BOG = 90° - <GOI. Geographically speaking, all latitudes are parallel to the equator and perpendicular to the polar axis. As such <GBO is a right angle and <BGO = <GOI (since they are alternate interior angles). Because BO is the diameter of the circle that is perpendicular to GH, this implies that GK = KH. In other words, GK is the radius used to draw the circles of the projected latitudes. By trigonometry, the following relationship can be constructed:

  • cos (<OGK) = GK/GO
  • Since <OGK = <GOI = angular distance of latitude and GO is the radius of the circle therefore GK = circle's radius * cos(angular distance of latitude)

In conclusion, the radius of the projected latitudes is equal to the circle's radius * cos (angular distance of the latitude). As such the radius of smaller latitudes becomes larger and can only project up to latitude 0 with its projected radius equal to that of the earth. This is because the cos funtion only have values ranging from 0 to 1 for any angle. The illustration in Figure C can be drawn by using the the intersection of the polar axis with the projection plane as the centre of all circular projected latitudes.

The meridians are drawn in a similar manner to all other meridians in the polar case in that they become equally spaced radius 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. Meridians become such straight lines because they are in reality semi-circles following the shape of the earth from the poles. As a result, when viewing from a birds eye view, it will be projected as the straight radius of the outer most latitude as seen in Figure C.