(1)   z = x²/2a + y²/2b, elliptic paraboloid.
(2)   z = x²/2a − y²/2b, hyperbolic paraboloid.
(3)   x²/a² + y²/b² + z²/c² = 1, ellipsoid.
(4)   x²/a² + y²/b² − z²/c² = 1, hyperboloid of one sheet.
(5)   x²/a² + y²/b² − z²/c² = −1, hyperboloid of two sheets.
Fig. 61.

It is at once seen that these are distinct surfaces; and the equations also show very readily the general form and mode of generation of the several surfaces.

In the elliptic paraboloid (fig. 61) the sections by the planes of zx and zy are the parabolas

z = x²,  z = y²,
2a 2b

having the common axes Oz; and the section by any plane z = γ parallel to that of xy is the ellipse

γ = x²+ y²;
2a 2b

so that the surface is generated by a variable ellipse moving parallel to itself along the parabolas as directrices.

Fig. 62.Fig. 63.
Fig. 64.