Zenithal Orthographic Equatorial 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 C

















Figure D

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. In the equatorial case, the light source comes from the west 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 in Figure A). The projection plane CE is situated tangentially to the east of the equator AB. 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.

Figure B

When viewed in this perspective, parallels are seen as the horizontal lines in Figure B, which is equivalent to the white lines in Figure C. In both diagrams, the distance between two consecutive latitudes vary. In fact, the parallels appear closer when further away from the equator. The distances that the projected parallels are away from the equator is as follows:

Given an arbitrary latitude DG in Figure A, it has the angular distance <BOG away from the equator AB. Since the equator AB goes through the centre of the earth and touches the tangential point B with the plane CE, <CBO is a right angle. Therefore a line segment GH with the same corresponding angle as <CBO (<GHO = <CBO) would make GH and CB parallel lines intersected by another two parallel lines DG and AB, implying that GH is equal to CB. Therefore the distance the projected parallels are away from the equator is constructed using the following trignometric relation:

  • sin (<GOB) = GH/GO
  • Since GO is the radius of the circle and <GOB is equivalent to the angular distance the latitude is away from the equator GH=circle's radius * sin (angular distance of latitude).

    Therefore, the distance projected parallels are away from the centre is equal to the circle's radius * sin (angular distance of latitude). As such, as the latitudes increase, the distance between consecutive parallels becomes smaller.

The meridians, on the other hand, are the vertical arcs seen in Figure C. The derivation of such arcs are related to the parallels in that the distribution of meridians along each parallels is equal to the distribution of the parallels along the central meridian. This distribution can be obtained by drawing a simple diagram as seen in Figure D. With the line segment on the bottom of the triangle equivalent in length and division as that along the upper half of the central meridian in Figure C, the radius of the latitudes are drawn parallel to the base of the triangle and lined up at the height, which intersects the base perpendicularly. Pick an arbitrary point along this vertical line (such as the point at the top of the triangle in Figure D) to form the triangle as seen. Arrange the latitude's radius starting with the smaller latitudes being closer to the base and touching the other two sides of the triangle. The intersections obtained by drawing lines from the top of the triangle to the divisions along the base would give proportional distribution of meridians along each corresponding parallel. Therefore such lines are representations of longitudes that increases in angular distance from right to left. As a result, the sloped side of the triangle functions like the central meridian. Location of meridians along the parallels can also be obtained mathematically. As stated before, the radius of the parallels are equal to the circle's radius * cos (angular distance of latitude) and since divisions along the base are taken from the upper polar axis, this implies that the division correspond to the relationship obtained above: circle's radius * sin (angular distance of latitude). However, because the lines are representation of longitudes, this means that the divisions are really circle's radius * sin (angular distance of longitude) and since the radius changes each time for each corresponding parallel the division of meridians along any parallel becomes:

earth's radius * cos (angular distance of latitude) * sin (angular distance of longitude).