Does anybody know the equation of this polar curve:
OR can anyone suggest me any cool interesting polar curves. I tried finding it online but didnt get any success.
thanks
Does anybody know the equation of this polar curve:
OR can anyone suggest me any cool interesting polar curves. I tried finding it online but didnt get any success.
thanks
I do not know the equation of this graph, but it reminds me of a graph that I came across last year when I was bored in math class. I was messing around on my calculator, and I found the polar graph setting. So I starting screwing around, just kinda typing in random stuff, and I came across a graph that looks almost exactly like some sort of bug, or misquito! I swear, I came across this graph by myself. I'm guessing that this graph might be online somewhere, but I did actually find it myself as well. Anyway, it doesn't help your question really, so I apologize. But, it is quite interesting. Heres the equation of the graph that looks like a bug:
$\displaystyle r(\theta) = 8[sin(cos(tan(\theta)))]$
Now, thats sorta hard to read. So to make it easier, lets say that:
$\displaystyle sin(x) = f(x)$
$\displaystyle cos(x) = g(x)$
and
$\displaystyle tan(x) = h(x)$
Then it might be easier to read it as:
$\displaystyle r(\theta) = 8[f(g(h(\theta)))]$
This graph shows up almost perfect on a basic Texas Instruments graphing calculator, but on higher quality graphing calculators (like my TI-Nspire) the graph looks alittle less like a bug. And, on graphing calculators that are less detailed then a standard texas instrument, the $\displaystyle theta$-step is too small for the all of the bugs wings to show up. But anyway, its pretty neat when you get it to work right.
Then
The following curve seems to give something close to what you have:
$\displaystyle r = 3\cos(4\theta-\frac{5\pi}{18}) + 3\times 0.07\sin(80\theta) $
The only thing missing is the fact that this curve goes all the way to r = 0 at certain points, whilst the one in your image never does. I'd imagine this could be sorted by adding another sinusoid with suitable parameters.
Basically, it's a high frequency cosine distorted by an even higher frequency sine, and then to get the void in the middle you'll need to find a suitable sinusoid which has the same frequency as the cosine, but out of phase by a suitable amount, and scaled before being added or subtracted to give you the void in the middle.
Heres another kinda cool one, if I rememeber it correctly. It does not look like I rememeber it used to on my standard TI calculator (I'm using a TI-Nspire and it doens't look correct), but maybe I'm remembering it wrong. Anyway, the equation I rememeber, when I typed it into my standard texas instrument graphing calculator, it looked like a bat. Here is the equation:
$\displaystyle r(\theta) = 8[sin(cos(tan(sin(cos(tan(sin(cos(tan(\theta)))))))) )]$
Again, let me simplify this a bit:
Let:
$\displaystyle f(x) = sin(x) \;\;\;\; g(x) = cos(x) \;\;\;\; \mathrm{and} \;\;\;\; h(x) = tan(x)$
Now, define:
$\displaystyle P(x) = f(g(h(x)))$
Then the equation of the bat is:
$\displaystyle r(\theta) = 8[P(P(P(\theta)))]$
Also, I do remember two other graphs that were really cool. Its actually a combination of a six graphs, but I know for sure that these two look correct. You'll need to graph all six on the same set of axes or it doens't look cool (I also came up with these ones myself). The first one is a flower. Type all six of these graphs in:
$\displaystyle r_1(\theta) = 2sin(6\theta)$
$\displaystyle r_2(\theta) = 4sin(6\theta)$
$\displaystyle r_3(\theta) = 6sin(6\theta)$
$\displaystyle r_4(\theta) = 8sin(6\theta)$
$\displaystyle r_5(\theta) = (9/2)[cos(6\theta)]$
$\displaystyle r_6(\theta) = 9cos(6\theta)$
Now the next one is debatable, in terms of what it exactly is. To me, it looks like some sort of Aztec Sun, and thats what I'll call it, either way its cool. Its almost the same as the flower except you change all the $\displaystyle \theta$ coefficients to $\displaystyle 8$. Type all six of these equations into your calculator to see the "Aztec Sun" (I also take claim to finding this graph as well, althouhg its less of a 'find' and more of a 'creation'):
$\displaystyle r_1(\theta) = 2sin(8\theta)$
$\displaystyle r_2(\theta) = 4sin(8\theta)$
$\displaystyle r_3(\theta) = 6sin(8\theta)$
$\displaystyle r_4(\theta) = 8sin(8\theta)$
$\displaystyle r_5(\theta) = (9/2)[cos(8\theta)]$
$\displaystyle r_6(\theta) = 9cos(8\theta)$
Anyway, I hope these graphs look as cool to you as they did to me, plus, if your still in highschool, it'll be fun to show off to your freinds (granted, they might just think your alittle to0 obsessed with your calculator , but these graphs are cool either way).