10 S. L. Penfield — Stereographic Projection. 



through the cone will be an ellipse, but it is not necessary for 

 the present discussion to demonstrate this point. A line join- 

 ing the center of h V and JS, passes through^?, bisects the angle 

 of the cone at S, and may be regarded as an axis of the cone. 

 Let it now be imagined that the cone b V S revolves about the 

 axis Sjp. Since the section b V is an ellipse, and an ellipse is 

 a figure of binary symmetry, the surface of the cone during 

 the first quarter of the revolution will depart somewhat from 

 its original position, and then on continued turning it will 

 approach more and more to its original position until a revolu- 

 tion of 180° has been accomplished, when the correspondence 

 of the conical surfaces, will be complete. After the revolution 

 of 180°, the points a and b of the cone in its original position 

 will be transferred to a' and b', respectively, and the section 

 a' b' must be a circle, because a b was a circle. Figure 5, A, 

 illustrates a most important feature of the stereographic pro- 

 jection ; namely, that no matter where p is located, if the pole 

 of the projection is at S, all circular sections corresponding to 

 a'b' are horizontal, and parallel to the plane of the equator ; 

 hence, the lines of projection running from. S to the circle a b 

 or a' V intersect the plane of the equator in a circle. Proof 

 of this feature of the stereographic projection is very simple. 

 The angles /3 and /3\ figure 5, A, are equal, because they 

 belong to the same cone before and after a revolution of 180° 

 about the axis S p. Draw the line a c parallel to b' a', and $" 

 will equal /3' because of the construction. The angle ft (on 

 the circumference of the circle at b) is measured by half the 

 arc Sa, and the angle fi" is measured by half the arc#c/ 

 therefore, since y£ and ft" are equal, the arcs Sa and Sc are 

 equal. This being true a c must be a chord, at right angles to a 

 line joining the north and south poles, and hence horizontal, or 

 parallel with the trace of the equator X Y. What holds good 

 for a circle cutting the meridian at the points a and b, figure 5, 

 A, holds good for a circle in any possible position on the sphere ; 

 hence,, all circles on the sphere will appear in stereographic 

 projection as circles on the pla?ie of the projection. 



Figure 5, A, further illustrates an interesting feature of 

 conic sections ; namely, that through an oblique cone circular 

 sections are possible in two directions, — parallel to a b and a' V . 

 All other sections are ellipses. 



The construction of a small circle in stereographic projec- 

 tion corresponding to the problem of which, so to speak, an 

 elevation has just been given in figure 5, A, becomes a very 

 simple matter, and is illustrated in figure 5, B, case 7. The 

 divided circle here corresponds to the equator, and the north 

 pole N is in the center. A north and south meridian would 

 appear in the projection as a diameter of the circle, and the 



