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Unformatted text preview: 180 CHAPTER 2. CURVES Regular Curves in the Plane
The graph of a function deﬁned on an interval I is a very special kind of
planar curve; in particular it must pass the vertical line test:16 for every
x0 ∈ I , the vertical line x = x0 must meet the curve in exactly one point.
Certainly there are many curves in the plane which fail this test. A large
class of such curves are polar curves, given by an equation of the form
r = f (θ ) .
It is easy to check that as long as f is a C 1 function and f (θ ) = 0, the
− (θ ) = (f (θ ) cos θ, f (θ ) sin θ )
is a regular parametrization of this curve (Exercise 3).
One example is the spiral of Archimedes, given by the polar equation
r = θ , and hence parametrized by
− (θ ) = (θ cos θ, θ sin θ ).
In this case, even though the vertical and horizontal line tests fail, the
parametrization (see Exercise 4) is one-to-one: each point gets “hit” once.
It will be useful for future discussion to formalize this notion; for technical
reasons that will become clear later, we impose an additional condition in
the deﬁnition below:
Deﬁnition 2.4.6. An arc is a curve that can be parametrized by a
one-to-one vector-valued function17 on a closed interval [a, b].18
The careful reader will note that if the spiral of Archimedes is deﬁned on
the inﬁnite interval θ > 0 then the second condition fails; however, it holds
for any (closed) ﬁnite piece. We explore some aspects of this concept in
A diﬀerent example is the (full) circle, given by r = 1, for which the
related parametrization is the standard one
− (θ ) = (cos θ, sin θ ).
16 This is the test for a curve to be the graph of y as a function of x; the analogous
horzontal line test checks whether the curve is the graph of x as a function of y .
We have not speciﬁed that the function must be regular, although in practice we deal
only with regular examples.
For the importance of this assumption, see Exercise 10b. ...
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- Fall '08