2 1. Various Ways of Representing Surfaces and Examples

Figure 1.1. Three familiar surfaces.

distance is equal to R, the equation for the cylinder is

(1.2)

x2

+

y2

=

R2.

Another surface familiar from elementary geometry (and also

from ice-cream parlours) is the cone, which is obtained by rotating

a straight line around another line which intersects it. If the axis of

rotation is again the z-axis and the initial line lies in the xz-plane,

with the equation x = az, then the cone is given by the equation

(1.3)

x2

+

y2

=

a2z2.

Exercise 1.1. If we construct a surface of revolution using parallel

lines instead of intersecting lines (as we did with the cone), we obtain

a cylinder. There is a third possibility; the lines may be skew, that

is, neither intersecting nor parallel. Describe the surface obtained in

this case, and derive its equation.

We feel immediately that the three objects expressed by equations

(1.1), (1.2), and (1.3), which are shown in Figure 1.1, are very different

in a variety of robust ways. For example, the sphere is bounded—

in fact, compact—while the cylinder and cone are not (contrary to

what the picture might suggest). The sphere and cylinder are smooth

everywhere, while the cone has a special point, the intersection of the

two lines in the construction, which is the origin in (1.3).

These differences are qualitative, and would not be changed if

we deformed each surface by a small amount—this reflects the fact

that the three surfaces in question have different topologies. Such a

deformation would, however, change the quantitative properties of a

surface, which constitute its geometry. For example, stretching or