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Unformatted text preview: 149 2.2. PARAMETRIZED CURVES iv i iii ii Figure 2.17: Fourpetal Rose r = sin 2θ or
x2 + y 2 3/2 = 2xy. While this is an equation in rectangular coordinates, it is not particularly
informative about our curve.
Polar equations of the form
r = sin nθ
deﬁne curves known as “roses”: it turns out that when n is even (as in the
preceding example) there are 2n “petals”, traversed as θ goes over an
interval of length 2π , but when n is odd —as for example n = 1, which was
the previous example—then there are n “petals”, traversed as θ goes over
an interval of length π .
A diﬀerent kind of example is provided by the polar equation
r = aθ
where a > 0 is a constant, which was (in diﬀerent language, of course)
studied by Archimedes of Syracuse (ca.287212 BC) in his work On Spirals
[3] and is sometimes known as the spiral of Archimedes. Here is his own
description (as translated by Heath [26, p. 154]):
If a straight line of which one extremity remains ﬁxed be made
to revolve at a uniform rate in a plane until it returns to the ...
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 Fall '08
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 Calculus, Equations

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