44 



U. S. COAST AND GEODETIC SURVEY. 



must, of course, be considered as circles of infinite radii, 

 with centers at infinity. Anj circle either great or small 

 which passes through the pomt of projection will be pro- 

 jected into a straight line, since all of the projecting lines 

 will lie in the plane of the circle and will cut the mapping 

 plane in a straight line, which is formed by the intersection 

 of the plane of the circle with the mapping plane. 



Let us now take any other circle upon the sphere. Make 

 a great-circle section of the sphere containing the point of 

 projection and the pole of the given circle. This great 

 circle necessarily will also pass through the point that pro- 

 jects into the center of the map, i. e., the point antipodal to 



Fig. 11.— Proof that circles project into circles on stereographic projections. 



the point of projection. After this is done turn the great 

 circle section into the plane of the page. The plane of this 

 section ^will evidently be perpendicular to the plane of the 

 given circle, since the plane of any great circle containuig 

 the pole of the given circle would partake of this property. 

 In figure 11 let be the point of projection, KL the trace 

 of the mapping plane, BO the trace of the plane of the 

 circle, and let A be the point that projects into the center 

 of the map. The lines that project the circle under con- 

 sideration will evidently form an oblique cone that has the 

 given circle as a circular section. Any plane parallel to 

 the plane of this circle will also cut the cone m a circle. 



