On determining the equations of stereographic projections in geodesy and map making

As a spherical surface, like the Earth, is not developable, the transposition of the spherical surface on a flat surface (as for maps, for instance) cannot be done without deformation, either of the distances, or of the angles. Most maps are drawn by stereographic projections, which keep intact the angles in the field. In the case of polar stereographic projection, the projection pole is one of the geographic poles, and the plane of projection is the equatorial plane of equation Z = 0 and a point on the sphere of coordinates M(theta, phi, r) will have the rectangular coordinates in the equatorial plane X and Y presented in (5). The meridians will have equations of the form (6), and the parallels circles of equations (7). In the case of the equatorial stereographic projection, the pole of projection is a point on the Equator, and the plane of projection is a plane which contains the axis of the poles, of equation Y = 0. A point on the sphere M(theta, phi, r) will have the coordinates in the projection plane X and Z presented in (11). The meridians will be the circular arc of equation (11), and the parallels will be circular arcs of equation (14).