The diagram FIG1.JPG to the right shows in 3 dimensions the Circle of Position corresponding to a given observation of a celestial body's altitude. An observer anywhere on a circle centered at GP will measure the same angle from horizon to celestial body B. This circle is a plane figure, and can be visualized as the result of slicing the sphere (Earth) with a flat plane. Geometry in a plane is much easier than on the surface of a sphere, hence there will be numerous mathematical simplifications in understanding, as well as in deriving algorithms for computer programs.

From one measurement of a single celestial body, an observer's position is limited to being somewhere on a circle of position. A second measurement gives a second circle of position, and in order to be on both circles, the observer must be at one of the two circles' intersection points.

The diagram F1.JPG to the right shows two circles of position, each resulting from observation of a different celestial object. The two circles intersect in two points, one of which must be the observer's position.

The GP's may be conveniently represented as vectors from the Earth's center to the GP point on the surface. By doing so, a set of simple vector formulas can be used to describe the planes of the two circles of position, and to construct an equation for the straight line produced by the intersection of these two planes. This line is important, since it intersects the Earth's surface at two points, one of which is the observer's position.

The diagram FIG5A.JPG above...

The diagram FIG6A.JPG above...