Orthographic projections are one of the map projections that cartographers use to create a representation of the Earth in a two-dimensional plane. With orthographic projections, the perspective point sits at an infinite distance, and the projection shows one of the hemispheres on the globe as it looks from outer space, making a great circle out of the horizon.
The history of the orthographic projection goes all the way back to ancient times, and its cartographic uses are documented as early as the second century BC, when Hipparchus used it to calculate the points for star rise and star set. The Romans used this projection to build sundials and figure out the correct position of the sun.
While the earliest maps of the world were drawn using the orthographic perspective, they at first appeared on woodcuts and were fairly crude in nature.
The formulas that are used to create a spherical orthographic projection come from trigonometry, and mathematicians write them in terms of latitude and longitude on the sphere itself. Latitudes that go past the map's range go through some trimming by calculating their distance from the middle of the projection. The purpose of this is to keep from plotting points on the opposing hemisphere.