Lecture 2 13

Figure 1.10. The sphere as a union of graphs.

What good is all this? What benefit do we gain from representing

the torus, or any other surface, by an equation? Of course, it allows

us to plug the equation into a computer and look at pretty pictures of

our surface, but what we are really after is coordinates on our surface.

After all, the surface is a two-dimensional affair, and so we should be

able to describe its points using just two variables, but the equations

we obtain are written in three variables.

To address this, we first backtrack a bit and discuss graphs of

functions. Recall that given a function f : R2 → R, the graph of f is

graph f = { (x, y, z) ∈

R3

| z = f(x, y) }.

If f is ‘nice’, its graph is a ‘nice’ surface sitting in

R3.

Of course,

most surfaces cannot be represented globally as the graph of such a

function; the sphere, for instance, has two points on the z-axis, and

hence we require at least two functions to describe it in this manner.

In fact, more than two functions are required if we adopt this

approach. The unit sphere is given as the solution set of

x2 +y2 +z2

=

1, so we can write it as the union of the graphs of f1 and f2, where

f1(x, y) = 1 − x2 − y2,

f2(x, y) = − 1 − x2 − y2.