I'm trying to measure the compactness (roundness) of some 800 polygons of different shape and size. I created the centroids and what I want to do now is to buffer those centroids with the exact same area of their polygons.  Sorry if this is too simple but I just can't find the way to do it.
I am trying to find ways to quantitatively describe the shapes of various polygons. For my project, these polygons represent lakes, rivers, lagoons, and parks. So they can be almost any shape. One easy metric is to calculate perimeter vs. area, which is at best an only slightly useful metric. But I would also very much like to be able to say something about the 'roundness' of a polygon. Or how 'compact' the shape is on a map.  The only way I can think of doing this easily is to calculate the area of each polygon in relation to a bounding box for that polygon (which I already have). But this seems like a poor solution.  So now I am thinking of something more like this - take the centroid of the polygon, add on a series of buffers of increasing areas (say 50%, 100%, 150%), then compare how much overlap there is between each buffer and the original polygon. A perfect circle will have perfect overlap at 100%, and I can use the 50% and 150% buffers to judge how much and in what way each polygon differs.  But even that feels cumbersome, and like a poor workaround for what somebody else has probably already figured out far better.  For reference, at a minimum I will need to be able to look at the resulting indices for the shape of various polygons, and be able to make an educated guess as to their source (River? Reservoir with dendritic shape? Lake/Lagoon? Park?)
I use a lot of lists and arrays but I have yet to come across a scenario in which the array list couldn't be used just as easily as, if not easier than, the linked list.  I was hoping someone could give me some examples of when the linked list is notably better. How can I get an underbrace and an overbrace to partially overlap in an equation? Sometimes I want to describe portions of an equation using \underbrace and \overbrace in such a way that the portions are not completely distinct, for example:          /-------\ x = A + B + C + D + E     \-------/   \---/      What is the best way to do this? Prove that for all integers $n$, $n \geq 1$,   $$1 + 3 + 5 + \cdots + (2n - 1) = n\cdot n$$   How would I go about proving this? How to vote on friends / colleagues ' questions (or answers) Almost every time a colleague of mine asks a question, we have discussed this problem beforehand. Also, we sometimes meta-discuss the question for stackoverflow (why did I get the &quot;this might be subjective&quot; warning, is this understandable enough, etc). This is less so with answers, but I could believe for some groups this works the same. For now I'll stick to questions, but you could s/question/answer if you like. The resulting question is almost always  Interesting for me, and a problem I don't know the answer to. I probably would favorite it According to me a valid and good question (as we talked about it beforehand) I would probably want to upvote it.  Now I believe to read that &quot;ring voting&quot; is a thing, and I tried to find some questions about this, but didn't find many. I did find and read this blogpost from 2008:  . It reads to me that upvoting 'fraud' is more commonplace then downvoting and it might seem anomalous that I would upvote almost all questions from one user. If all colleagues would do this to their team, you'd get some mediocre questions getting a lot of votes, and some users getting a lot of attention, points etc from that. If you would -as seems to be done -, sort the amount of upvotes for my colleague, you'd get me being 'anomalous' in front of the crowd. On the other hand, as explained above, I do want to upvote almost all stuff some of my colleagues put on here for (I believe) the right reasons: they are questions I consider good and interesting. It's just that I know of all their questions (as they tell me about them), which obviously isn't the case for a random user. So what is the deal?  When is upvoting content from one user frowned upon? Would my usecase be in danger of being a (what I consider 'false') positive for the fraud-detection? (and would you agree with that or not) How to prevent this false positive (if you agree with me it is one, otherwise this is a moot question) -&gt; As noted in the comments: I'm not looking for tricks, so an answer could be &quot;don't vote for your colleagues&quot;. I agree that there should be no to 'tricks' around that could be used for real fraud of course Prove by contradiction (not using a calculator) that $\sqrt6 + \sqrt2 < \sqrt{15}$? What category does the 'the recent perfect' belong to? A scholar has said that there are 4 usages of the present past tense:   The Universal sense of the Perfect, used to indicate that a state of affairs prevailed throughout some interval stretching from the past into the present       I've known Max since 1960.  The Existential sense of the Perfect, used to indicate the existence of past events     I have read Principial Mathematica five times.  The Stative/Resultative sense of the Perfect, used to indicate that the direct effect of a past event still continues     I can't come to your party tonight - I've caught the flu.  The Hot News sense of the Perfect, used to report hot news     Malcolm X has just been assassinated.    To what category does the recent past belong? Does there exist a polynomial $p(x) \in \mathbb C[x]$ such that $p(x) \notin \mathbb R[x]$ and $p(x)p(-x)=p(x^2)$ ?  I have noticed that if $a_n$ is the leading coefficient of $p(x)$ then $a_n=(-1)^n $, so the leading coefficient cannot be imaginary; also $p(0)^2=p(0)$ so the constant term also is real. Please help. Thanks in advance. How do I print bold text in Python? Should the user be forced to sign in for a wish list on an ecommerce site? I believe that signing into a site can be a big barrier to users so prefere e-commerce sites that offer guest access.  The question is should wishlists be a guest feature or do the business advantages enforcing the user to sign up outweigh the potential increases in profits.   Does anyone have any research or views on this? How to parse JSON in iOS App How does this wiring work? I have a Trane XV90 and XL 14i. The wiring looks a little strange as in the picture bellow.                    _____________                   |           |   BK W2  W1   R  G  Y  B/C  YLO O     |___||   |__|  ||   |          |   |     ||   |_____>|----------|         |   |     ||_________>|other wire|         |   |     |           ------------         V   V     V     ---------------     |thermostat   |     ---------------  The R and G are jumpered together and the G wire is connected on the board  but is actually not connecting to the thermostat on the other end. So people told me that means the furnace fan is always on! (In my diagram above, I switched the location of Y and B/C to make diagram easier to draw. You can refer to the photo added below to see the details) My questions are the following  is the jumper between O and Y doing anything? I don't think I have heat pump I need to install Honeywell wifi thermostat and I am thinking of doing either one of the following:   remove jumper between R and G, and use G to connect B/C and then to thermostat C and put jumper between G and Y  leave jumper between R and G, and use G to connect B/C and then to thermostat C and put jumper between G and Y  remove jumper between R and G, connect G on control board to thermostat G, and finally have the spare wire to connect B/C and then thermostat C  leave jumper between R and G, does not connect G to thermostat,  and just use a spare wire to connect B/C on board to thermostat C   Can anyone comment on those plans? Thanks a lot! What Is the Real Name of the #? I have added a custom/secondary query to a template file/custom page template; how can I make WordPress use my custom query for pagination, instead of using the main query loop's pagination?  Addendum  I have modified the main loop query via . Why isn't pagination working, and how do I fix it? MVC DateTime binding with incorrect date format Need a comma before an adverb as last word in sentence? Say I have a code to draw a flow chart, I want now a arrow from node 3 to node 1, how can I do that?  1&lt;--| |   | 2   | |   | 3----   MWE  \documentclass[class=article,border=0pt]{standalone}  \usepackage{tikz} \usetikzlibrary{shapes,arrows}  % Define block styles \tikzset{     block/.style={rectangle, draw, line width=0.5mm, black, text width=5em, text centered,                   rounded corners, minimum height=2em},     line/.style={draw, -latex} }% &lt;- if you insist in using this in the document add this % here. \begin{document} \begin{tikzpicture}[node distance = 1cm, auto]     % Place nodes     \node [block] (BLOCK1) {a};     \node [block, below of=BLOCK1] (BLOCK2) {b};     \node [block, below of=BLOCK2, node distance=1cm] (BLOCK3) {c};     % Draw edges     \path [line] (BLOCK1) -- (BLOCK2);     \path [line] (BLOCK2) -- (BLOCK3); \end{tikzpicture} \end{document} $n^2 + 3n +5$ is not divisible by $121$ Question:  Show that $n^2 + 3n + 5$ is not divisible by $121$, where $n$ is an integer. How to merge two arrays in JavaScript and de-duplicate items What command should I use to rotate wood to S,N,W,E,D and U?  I read the entire world edit guide and tried some things but all the results are not satisfactory. What is the limit of cabin baggage on Wizz Air flights? What's the weight limit on Wizz Air, for cabin baggage?    I can't find anything about the weight on their website. I know that for Ryanair is 10kg, and EasyJet say they don't have a limit for weight.