Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

QUADRIC SURFACES

549

x2 y2 7C2
+xb —1k —c-, z= k.
a2 b2 C2
If c2 < c2, these equations represent, in the plane z = k, an
ellipse whose center is on the axis of z and whose axes lie in
the (z, x)- and (y, z)-planes and have the lengths
2a i- 2b 1- -.
As k increases from 0 toward c as its limit, this ellipse, rising
frpm the section by the (x, y)-plane, grows continuously
smaller and shrinks finally to a point,-the point (0, 0, c).
Similarly, if k decreases from 0 toward -c as its limit.
The surface generated by the
Fig. 2 (or Fig. 6 of Ch. XXII *)
shows it in its entirety. The surface is evidently symmetric in the
origin, 0, and in the coordinate
axes and coordinate planes. 0 is
called the center of the ellipsoid; FIG. 2
the coordinate axes, the axes of the
ellipsoid; and the coordinate planes, the principal planes of the
ellipsoid. The sections by the principal planes are known as
the principal sections.
The dimensions of the ellipsoid, measured along the axes,
are 2a, 2b,'2c. These numbers, in the order of their magnitude, are known as the major axis, mean axis, and minor axis
of the ellipsoid.
Here, and throughout the chapter, we speak of a (plane)
section of a surface only when the plane in question meets the
surface in a curved line. Sections by parallel planes we shall
call parallel sections.
* Figs. 6-10 of Ch. XXII, drawn originally to represent quadric surfaces of revolution, picture equally well the corresponding general quadric surfaces studied in this chapter. One has merely to imagine that a
different ratio of foreshortening along the axis of x has been chosen.