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
