Graphing polar curves: limacon and 2 oddballs

[rocomath]

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? $$r=a\pm\cos\theta$$ and $$r=a\pm\sin\theta$$

$$r=1+2\cos\theta$$ inner loop

$$r=1.5+\cos\theta$$ no inner loop

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

$$r=1+2\cos\theta$$ inner loop, testing if b=even

$$r=a+3\cos\theta$$ 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 $$[1,\infty)U(-\infty,-1]$$, 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 $$-\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}$$, and Cosine is symmetric with the x-axis, so I use $$0\leq\theta\leq\pi$$.

Now my main problem:

$$r=\csc\theta+2$$ (conchoid of Nicomedes)

Cosecant is also symmetric with the y-axis, so I choose my theta interval to be $$-\frac{\pi}{2}<\theta<\frac{\pi}{2}$$.

 

[dynamicsolo]

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

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 $$\theta$$ over a full cycle

$$r=1+2\cos\theta$$ inner loop

$$r=1.5+\cos\theta$$ 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 $$r =a + b\cos\theta$$ or $$r =a + b\sin\theta$$ , there will be a loop if a < b ; the curve will have a "dimple" if a = b ; for a > b , there is no loop.

Now my main problem:

$$r=\csc\theta+2$$ (conchoid of Nicomedes)

Cosecant is also symmetric with the y-axis, so I choose my theta interval to be $$-\frac{\pi}{2}<\theta<\frac{\pi}{2}$$.

I'm a little unclear on what you're asking here, but the radius function will be negative when $$\sin\theta$$ < -(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.