22 1. Various Ways of Representing Surfaces and Examples
Figure 1.16. Three curves in
R3.
Exercise 1.11. Find all geodesics on the round cylinder
{ (x, y, z) ∈
R3

x2
+
y2
= 1 }
and the upper half of the round cone
{ (x, y, z) ∈
R3

x2
+
y2
−
z2
= 0, z ≥ 0 }.
c. Parametric representations of curves. We often write a curve
in
R2
as the solution of a particular equation; the unit circle, for ex
ample, is the set of points satisfying
x2
+
y2
= 1. This implicit
representation becomes more diﬃcult in higher dimensions; in gen
eral, each equation we require the coordinates to satisfy will remove
a degree of freedom (assuming independence) and hence a dimension,
so to determine a curve in R3 we require not one, but two equations.
Geometrically, we are obtaining a curve as the intersection of two sur
faces, each specified by one of the equations. For example, the unit
circle lying in the xyplane is the solution set of
x2
+
y2
= 1,
z = 0.
which is the intersection of this plane with a cylinder of unit radius.
This is a simple example for which these equations and the visualisa
tion of the surfaces pose no real diﬃculty; there are many examples
which are more diﬃcult to deal with in this manner, but which can
be easily written down using a parametric representation. That is, we
define the curve in question as the set of all points given by
(x, y, z) = (f1(t), f2(t), f3(t)),