In case of an azimuthal projection the surface of the sphere will be mapped to a tangent-plane whose boundary point is the break-through point (intersection) of the projection axis through the sphere surface. The azimuthal projection is a true projection.
The most obvious procedure follows intuitively from the outlook by projecting the surface of the sphere directly onto the tangent-plane via central projection (just like a light source).
Depending on the location of the projection center we differentiate the
For the gnomonic projection the projection centre Z is the centre of the sphere and for the stereographic azimuthal projection the opposite intersection of the projection axis with the sphere. The orthographic azimuthal projection is defined as a parallel projection (the projection centre Z lies in infinity).
The three mentioned projections are special perspective projections.
Another method results from the specification of differential-geometrical characteristics which the azimuthal projection should have. Besides conformality which is given for the stereographic azimuthal projection you also can construct equidistant and equal-area azimuthal projections which however can not be acquired directly from outlook (perspective) (see e.g. [3] under literature).