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
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
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