Calculus: Analytic Geometry and Calculus, with Vectors

6.6 Quadric surfaces 407

examples of surfaces known as saddle surfaces. The origin is a saddle

point, and a huge surprise awaits embryonic mathematical physicists who
suppose that onlycowboys are interested in them.
Finally, we should not forget the cones. When A and B are nonzero
constants not both negative, the graph of the equation

(6.68) Ax2 + Bye = z2

is a nondegenerate quadric cone It is a cone because the point

(Xx,Xy,Xz) lies on the graph whenever X is a number and the point (x,y,z)

lies on the graph. It intersects the plane having the equation z = k in a

point (the vertex of the cone) if k = 0 and in a central conic (circle.

ellipse, or hyperbola) having its center on the z axis if k 0 0. Instead of

exhibiting graphs and photographs of quadric cones, we conclude with a

remark. Since we know that hyperbolas have asymptotes, we need not

be surprised to learn that hyperboloids can have asymptotic cones. The

cone having the equation

(6.69)

x2 y2 x2

a2+b2

• c2

is the common asymptotic cone of the two hyperboloids having the equa-

tions (6.64) and (6.65).

While some of us will meet ellipsoids and other quadric surfaces after

completing this course, we need not invest our time in consideration of

more or less routine problems involving special quadric surfaces. We

have earned the right to think about a little (or big?) problem that can be

of interest to those who slice onions and other things. Let S be a set in

E3 which contains more than one point. Suppose that, for each plane ir,

the intersection of S and rr is a circle or a set consisting of just one point or

the empty set. Do our hypotheses imply that S must be a sphere?

If it is easy to make an incorrect guess and if the problem is difficult, so be

it. If there is just one easy way to solve the problem and only one person

in a million can discover the way, so be it. An anecdote illustrates the

fact that we are sometimes provided opportunities to do our own thinking

and investigating. In a lecture to advanced students at Cambridge,

G. H. Hardy stated that the answer to a particular question is obvious.

Becoming dubious about his assertion, Hardy asked his students to excuse

him while he sat down to think about the matter. Five minutes later,

Hardy arose to report that the answer is obvious and to continue his

lecture.