*spirals*. The basic concept of what a spiral should be, in polar coordinates, is any plot $r$ varying with $\theta$, $r=f(\theta)$, where $f$ is always increasing, and $\theta$ is allowed to go beyond the usual range of $0\leq \theta < 2\pi$ (or $-\pi < \theta \leq \pi$). The most fundamental one, of course, is $r = \theta$, which says the distance from the origin is proportional to the amount you've wound around the origin:

(here the angle is in radians and the equation is $r = \theta$). Of course, some people thought this was a bit boring, so Fermat spiced it up and thought what if we make the spirals wind tighter as you go around? Basically, make it a concave-down, increasing function, $r = \sqrt{\theta}$. Actually, it's more fun to also consider $r = -\sqrt{\theta}$:

($r=\sqrt{\theta}$ is in blue, and $r = -\sqrt{\theta} is in yellow$)

This of course could more succinctly be written $\theta = r^2$. But perhaps the second-most famous, due to its ubiquity in nature (because stuff tends to grow exponentially) is the famous logarithmic spiral $r = e^{c \theta}$ or $\theta = \frac{1}{c} \log r$, which magnifies every time you spin it.

($r=e^{\theta/5}$)

And finally, where would we be in polar graphs if we didn't have the analogue of stuff with asymptotes? Consider $r^{-2} = \theta$ (instead of $r^+2$):

This is called the

*lituus of Cotes.*There's all sorts of curves from back in the day with very funny names; I used to look for a lot of them on the MacTutor History of Math Page (more specifically, its Curves Index).
There's one other spiral that's my favorite, but (1) it isn't polar, and (2) there's a lot of interesting differential geometry going on in it, so I'm saving it for another post.

Someone essentially lend a hand to make severely articles I might state. This is the first time I frequented your website page and so far? I surprised with the research you made to create this actual post amazing. Magnificent job! gmail login

ReplyDeleteThanks for another magnificent post. The place else may anyone get that kind of information in such a perfect method of writing? I have a presentation subsequent week, and I am on the search for such information. paypal credit login

ReplyDelete