Calculus: Analytic Geometry and Calculus, with Vectors

6.6 Quadric surfaces 403

are notrequired to read it. As is easy to see by considering such special
examples as

(6.61) x2+y2+z2+1=0, x2+y2+z2=0, x2+y2=0,

the graph in E3 of an equation of the form

(6.62) 4x2 + Bye + Cz2 + Dxy + Eyz + Fzx + Gx + Hy

+Iz+J=0

can be the empty set or a point or a line. In case 11, B, C, D, E, F are not
all zero and the graph is a surface, the surface is called a quadric surface.
Examples of the form

(6.621) Axe + Bye + Cxy + D = 0,

where 14, B, C are not all 0, show that some quadric surfaces are cylinders
which may be planes or pairs of planes. Our interest in this section lies
in quadric surfaces that are not cylinders.
As we study quadric surfaces, it will be helpful to have some results of
Section 2.6 in mind. A line which does not lie completely on a quadric
surface can intersect the quadric surface in at most two points. Each
plane section of a quadric surface must be the empty set or a single point
or a single line or two lines or a circle or a parabola or an ellipse or a

hyperbola. From this we can conclude that each nonempty bounded

plane section of a quadric surface must be either a point or a circle or an

ellipse.

When the equation (6.62) of a quadric surface is given, it is possible to

introduce a new coordinate system in such a way that the equation in the

new coordinates has one or another of several standard forms. There are

places, even in applied mathematics, where it is necessary to know pro-

cedures for putting given equations into standard forms, but the pro-

cedures are much too complicated for coverage in elementary courses.

In this course we can be content with a little information about the

standard forms and their graphs. In what follows, it is always assumed

that a, b, c are given positive constants. The standard forms are

selected in such a way that, insofar as is possible, the coordinate planes

are planes of symmetry and the curves in which the surface intersects

planes parallel to the coordinate planes are easily described and sketched

The graph of the equation

(6.63) x2 y2

z2

a2+b+CZ=1l

is an ellipsoid except when a = b = c and the graph is a sphere. Putting

z = k shows that the ellipsoid intersects the plane having the equation