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Unformatted text preview: 146 CHAPTER 2. CURVES or, in terms of coordinates,
x = 1 + a 2 3 cos θ − 2 sin θ
y = 2 − a cos θ + b 2 3 sin θ
2 The general relation between a plane curve, given as the locus of an
equation, and its possible parametrizations will be clariﬁed by means of
the Implicit Function Theorem in Chapter 3. Analyzing a Curve from a Parametrization
The examples in the preceding section all went from a static expression of a
curve as the locus of an equation to a dynamic description as the image of
a vector-valued function. The converse process can be diﬃcult, but given a
function − : R → R2 , we can try to “trace out” the path as the point moves.
As an example, consider the function − : R → R2 deﬁned by
x(t) = t3
y (t) = t2
with domain (−∞, ∞). We note that y (t) ≥ 0, with equality only for t = 0,
so the curve lies in the upper half-plane. Note also that x(t) takes each
real value once, and that since x(t) is an odd function and y (t) is an even
function, the curve is symmetric across the y -axis. Finally, we might note
that the two functions are related by
(y (t))3 = (x(t))2
y (t) = (x(t))2/3
so the curve is the graph of the function x2/3 —that is, it is the locus of the
y = x2/3 .
This is shown in Figure 2.15: as t goes from −∞ to ∞, the point moves to
the right, “bouncing” oﬀ the origin at t = 0.
A large class of curves can be given as the graph of an equation in polar
coordinates. Usually, this takes the form
r = f (θ ) . ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
- Fall '08