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.

 

[Architeuthis Dux]

However, since the cosecant function has asymptotes at \theta=0 and \theta=\pi, the curve will not be continuous and will have breaks at those points. This is why you may have experienced difficulty in finding a pattern for this type of curve.

To overcome this, you can graph the curve in separate sections, using different theta intervals for each section. For example, you can graph the curve for -\frac{\pi}{2}<\theta<0 and then for 0<\theta<\frac{\pi}{2}, and then connect the two sections to get the full graph. You can also use a graphing calculator or software to help you visualize the curve.

In terms of finding patterns for polar curves, it is important to understand the behavior of the trigonometric functions and their ranges. For example, the sine function has a range of [-1,1], so when it is added to a constant, the resulting curve will have a maximum and minimum value. On the other hand, the cosecant function has a range of [1,\infty)U(-\infty,-1], so when it is added to a constant, the resulting curve will have breaks at the asymptotes.

Overall, graphing polar curves can be challenging and may require breaking down the curve into sections or using technology. It is also important to have a strong understanding of the trigonometric functions and their ranges to accurately graph these types of curves. Keep practicing and experimenting, and you will become more comfortable with graphing polar curves.