Chapter 1

Various Ways of

Representing Surfaces

and Basic Examples

Lecture 1

a. First examples. For many people, one of the most basic images

of a surface is the surface of the Earth. Although it looks flat to

the naked eye (at least in the absence of any striking geographic

features), we learn early in our lives that it is in fact round, and that

its shape is very well approximated by a sphere. Geometrically, the

sphere is defined as the locus of points at a fixed distance, called the

radius, from a given point, the centre. Using Cartesian coordinates

and putting the origin at the centre, we derive the familiar equation

(1.1)

x2

+

y2

+

z2

=

R2,

where R is the radius; the sphere is the set of all points in R3 whose

coordinates (x, y, z) satisfy this equation.

Many other familiar shapes can also be defined geometrically and

represented as the set of solutions of a single equation, as in (1.1). For

example, the (round) cylinder is the locus of points at a fixed distance

from a given straight line. If the line is taken to be the z-axis and the

1

http://dx.doi.org/10.1090/stml/046/01