366] 
487 
366. 
NOTE ON THE PROJECTION OF THE ELLIPSOID. 
[From the Philosophical Magazine, vol. xxx. (I860), pp. 50—52.] 
Consider an ellipsoid, situate any way whatever in regard to the eye and the 
plane of the picture; the apparent contour of the ellipsoid is an ellipse, the inter 
section of the plane of the picture by the tangent cone having the eye for vertex; 
this cone touches the ellipsoid along a plane curve (the intersection of the ellipsoid 
by the polar plane of the eye), which may be called the contour section; and the 
apparent contour is thus the projection of the contour section. Consider any other 
plane section; the projection thereof has double contact (real or imaginary) with the 
projection of the contour section: the common tangents are the intersections with the 
plane of the picture of the tangent planes of the tangent cone which pass through 
the pole of the section; or, what is the same thing, they are the tangents to the 
projection of the contour section, or to the projection of the section, from the point 
which is the projection of the pole of the section. The projection of the pole lies in 
the line which is the projection of the diameter conjugate to the plane of the section; 
and in particular, if the section is central, that is, if the plane thereof passes through 
the centre of the ellipsoid, then the pole is the point at infinity on the conjugate 
diameter; whence also if the eye be at an infinite distance, so that the projection is 
a projection by parallel rays, then the projection of the pole is the point at infinity 
on the projection of the conjugate diameter; and therefore the common tangents of 
the projections of the section and the contour section are in this case parallel to the 
projection of the diameter conjugate to the plane of the section. 
Suppose that the plane of the picture is parallel to a principal plane of the 
ellipsoid, and that the projection is by parallel rays; then if OA, OB, 00 are the 
projections of the semiaxes (OA, 00 will be at right angles to each other if the 
plane parallel to the plane of the picture is that of xz), the projections of the principal 
sections are the ellipses having for conjugate semidiameters OB, 0(7; 00, OA ; OA, OB