The Stereographic Projection
The plate of an astrolabe is a map of the sky on the flat plate of the astrolabe. Just as a map of the world is a projection of the spherical Earth on a flat sheet of paper, the astrolabe plate is a projection of the celestial sphere on an imaginary piece of paper placed through the flat plane of the equator. The projection used on the astrolabe is the stereographic projection.
The figure shows the principle of the stereographic projection. In the stereographic projection as used on the astrolabe, an imaginary line is drawn from the south celestial poleon to a point on the surface of the celestial sphere. The point where the line crosses the equator is marked on the instrument.
The stereograpic projection has two properties that make it ideal for astronomy:
- Circles on the celestial sphere are projected as circles on the projection.
- Angles on the celestial sphere are preserved on the projection.
Since most positions in the sky are measured by angles along circles, the stereographic projection is ideal for astronomical uses. All methods of astrolabe design use the fact that circles on the celestial sphere are preserved in the projection to determine the location and size of the circles on the astrolabe plate. The mathematics of the stereographic projection and details of both analytic and graphical methods of using the projection to draw an astrolabe plate are included in The Astrolabe.
Note that the stereographic projection shows the celestial sphere as seen from outside the sphere much as you would view a celestial globe.
The projection plane can be tangent to the celestial sphere or anywhere within it. Most star charts use the stereographic projection with the projection plane tangent to the sphere. It is also used in map making and other sciences such as crystallography.