In the hyperbolic paraboloid (figs. 62 and 63) the sections by the planes of zx, zy are the parabolas z = x²/2a, z = − y²/2b, having the opposite axes Oz, Oz′, and the section by a plane z = γ parallel to that of xy is the hyperbola γ = x²/2a − y²/2b, which has its transverse axis parallel to Ox or Oy according as γ is positive or negative. The surface is thus generated by a variable hyperbola moving parallel to itself along the parabolas as directrices. The form is best seen from fig. 63, which represents the sections by planes parallel to the plane of xy, or say the contour lines; the continuous lines are the sections above the plane of xy, and the dotted lines the sections below this plane. The form is, in fact, that of a saddle.

In the ellipsoid (fig. 64) the sections by the planes of zx, zy, and xy are each of them an ellipse, and the section by any parallel plane is also an ellipse. The surface may be considered as generated by an ellipse moving parallel to itself along two ellipses as directrices.

In the hyperboloid of one sheet (fig. 65), the sections by the planes of zx, zy are the hyperbolas

x²− z²= 1,  y²− z²= 1,
c² c²b² c²

having a common conjugate axis zOz′; the section by the plane of x, y, and that by any parallel plane, is an ellipse; and the surface may be considered as generated by a variable ellipse moving parallel to itself along the two hyperbolas as directrices. If we imagine two equal and parallel circular disks, their points connected by strings of equal lengths, so that these are the generators of a right circular cylinder, and if we turn one of the disks about its centre through an angle in its plane, the strings in their new positions will be one system of generators of a hyperboloid of one sheet, for which a = b; and if we turn it through the same angle in the opposite direction, we get in like manner the generators of the other system; there will be the same general configuration when a ≠ b. The hyperbolic paraboloid is also covered by two systems of rectilinear generators as a method like that used in § 34 establishes without difficulty. The figures should be studied to see how they can lie.

Fig. 65.Fig. 66.

In the hyperboloid of two sheets (fig. 66) the sections by the planes of zx and zy are the hyperbolas

z²− x²= 1,  z²− y²= 1,
c² a²c² b²

having a common transverse axis along z′Oz; the section by any plane z = ±γ parallel to that of xy is the ellipse