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Graphing polar curves: limacon and 2 oddballs

  1. May 16, 2008 #1
    I'm trying to find patterns for polar curves. I just reviewed and feel comfortable with taking advantage of symmetry, but I still have trouble with some type of curves.

    Limacons: Two types

    1) inner loop
    2) no inner loop

    Is there a general formula that tells me whether there will be an inner loop? [tex]r=a\pm\cos\theta[/tex] and [tex]r=a\pm\sin\theta[/tex]

    [tex]r=1+2\cos\theta[/tex] inner loop

    [tex]r=1.5+\cos\theta[/tex] no inner loop

    I tried to find a pattern myself, but I didn't find one.

    [tex]r=1+2\cos\theta[/tex] inner loop, testing if b=even

    [tex]r=a+3\cos\theta[/tex] inner loop, testing if a=k, b=odd

    Now the reciprocal curves threw me off. I had forgotten about the range of cosecant, which is [tex][1,\infty)U(-\infty,-1][/tex], and I looped my curve back inwards, which is incorrect.

    In general, when I'm graphing polar. Sine is symmetric with the y-axis, so the values of theta that I choose are from [tex]-\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}[/tex], and Cosine is symmetric with the x-axis, so I use [tex]0\leq\theta\leq\pi[/tex].

    Now my main problem:

    [tex]r=\csc\theta+2[/tex] (conchoid of Nicomedes)

    Cosecant is also symmetric with the y-axis, so I choose my theta interval to be [tex]-\frac{\pi}{2}<\theta<\frac{\pi}{2}[/tex].
    Last edited: May 16, 2008
  2. jcsd
  3. May 29, 2008 #2


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    Homework Helper

    I don't think of it as a general formula so much as a method for sorting out what the curve will do.

    Plot each function on a graph of r versus [tex]\theta[/tex] over a full cycle

    [tex]r=1+2\cos\theta[/tex] inner loop

    [tex]r=1.5+\cos\theta[/tex] no inner loop

    The first of these has a radius function which ranges from 3 to -1. The fact that the radius is negative in part of the second and third quadrants tells you there will be an inner loop. Solving for r = 0 tells us that the loop runs from (2/3)pi to (4/3)pi.

    The second curve has a radius function ranging from 2.5 to 0.5. Since the radius is never negative (or even zero), there will be no loop.

    For a cardioid [tex]r =a + b\cos\theta[/tex] or [tex]r =a + b\sin\theta[/tex] , there will be a loop if a < b ; the curve will have a "dimple" if a = b ; for a > b , there is no loop.

    I'm a little unclear on what you're asking here, but the radius function will be negative when [tex]\sin\theta[/tex] < -(1/2) , so there ought to be a loop for the interval from
    (7/6)pi to (11/6)pi . There's a nice little animation for conchoids at http://mathworld.wolfram.com/ConchoidofNicomedes.html

    Since you are working with cosecant, your conchoid will run "parallel" to the x-axis, but otherwise will behave like the one in the illustration I refer to.
    Last edited: May 29, 2008
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