Engineering Calculus Notes 161

Engineering Calculus Notes 161 - 149 2.2. PARAMETRIZED...

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Unformatted text preview: 149 2.2. PARAMETRIZED CURVES iv i iii ii Figure 2.17: Four-petal 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.287-212 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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