1 Projective Geometry

Figure 1.2: Sextant is used to measure the angle between star/moon/sun and horizon. Imaging that you are located at the green point in Figure 1.1. It is possible to identify the location by knowing the projected location of the zenith—the point in the sky or celestial sphere directly above an observer—onto the earth surface given time. However, this is challenging because reliably identifying the zenith is often difficult. Instead, you can use other pre-charted stars. Consider a star A that is visible from my location. As the star is infinitely far away from the Earth, the lines of projections of the star A are parallel, i.e., the angles α between L and LA and between L and LA′ are preserved. Instead of measuring α, the angle 90◦ − α between LA′ and horizon can be reliably measured by Sextant as shown in Figure 1.2. Using α, the distance from my location to the projection of the star onto the Earth surface, eA is rα where r is the radius of the Earth. Any location at rα distance from eA can be my location, and therefore, there are infinite number of solutions. One more star can reduce the candidates to two, i.e., the intersection between two greate circles. Three star measurements can uniquely determine the global location. This celestial localization indicates that it is possible to know “where am I?” using points at infinity.