Wikipedia:Reference desk/Archives/Mathematics/2021 September 23

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

September 23

"Curviness" of lines

Looking at (for instance) a UK Ordnance Survey map, we see a number of different linear features. Even disregarding the colour and thickness of the lines, different types of features can to a large extent be distinguished by examining their "curviness", for instance a modern A-road, a Roman road, a motorway, a railway, a watercourse, a coastline.

Is there a mathematical way of describing this curviness?

rossb (talk) 14:48, 23 September 2021 (UTC)

I think you need curvature, which is a formal mathematical way (or many ways) to approach this problem.--2A00:23C8:4583:9F01:2996:B9C7:BE9D:7CEB (talk) 14:54, 23 September 2021 (UTC)

This is an interesting question (and I have no idea of the answer). A naïve measure of the curviness is a comparison of the length of a line against the "straight line" distance between two points, but this might well give a larger answer for some large sweeping curves than for shorter tight curves, making a road that you would not consider bendy to drive down being classed as bendier than one that you would! Also it does not easily allow for the "bendiness" of lines which have to avoid certain geographical obstacles. The curvature articles gives a number of possible measures, but deciding which apply seems to me to be subjective -- Q Chris (talk) 15:01, 23 September 2021 (UTC)

Sinuosity seems to be the (or an) answer. It should be noted that curvature is a local measure, it describes a curve at a given point. The question seems to be looking for a global measure that describes a curve as a whole, or at least a finite segment of a curve. --Wrongfilter (talk) 15:16, 23 September 2021 (UTC)

For issues in defining the sinuosity of coastlines, see Coastline paradox.  --Lambiam 20:29, 23 September 2021 (UTC)

It's not mentioned in the article, but it seems obvious from looking at the references, that "Sinuosity" is mainly used in reference to rivers. River meanders tend to follow the same family of curves, approximated by "meander curves" (See the mathcurve.com entry.) I believe, however, that the "ideal" meander, is the "elastica". (See this mathcurve.com entry.) For both of these families, the sinuosity can be used as a parameter to define the curve up to similarity. (Side note, before computers, designers would use splines, thin pieces of springy metal or rubber, bent according to need, to create curvy shapes. These also follow elastica curves, but physical splines are computationally difficult to model, and since most drafting is done on a computer now, Bezier splines, actually piecewise cubic polynomial curves, are used instead.) I have an irresistible urge to mention artist Andy Goldsworthy here, see for example File:"Taking a Wall for a Walk" - Grizedale Forest - geograph.org.uk - 36895.jpg.

I believe people who who design highways, especially ramps, have to think about his kind of thing as well. In that case you want the path of the road to be the opposite of sinuous. A large factor is (or should be) the comfort and expectations of the driver, and you have to take the acceleration into account as well as the curvature. At the opposite end of the spectrum, racetracks are often described as "technical" or "nontechnical" depending on the driving skill they require. This seems to be a somewhat subjective term though, and I don't think there's an agreed upon definition.

My point here is that mathematical definitions usually depend on the application you're working on or the problem you're trying to solve. Definitions succeed if they help formulate a mathematical model of a particular situation. Definitions that seek only to measure some vague adjective are less interesting and have dubious value. --RDBury (talk) 21:52, 23 September 2021 (UTC)

For a discussion of design approaches to the shape of ramps and such, see Track transition curve, which mentions both the elastic curve (which if I'm not mistaken minimizes $\textstyle {\int \kappa ^{2}ds}$, where $\kappa$ denotes the curvature) and the clothoid (which minimizes $\sup |{\dot {\kappa }}|$, fluxion with respect to curve length).  --Lambiam 08:52, 24 September 2021 (UTC)

Area in center "rectangle" - 2n+1

I took a look at how clean the 2n solution was, and eventually my thought was how can we make it messier. Given a 2n+1 polygon, draw a rectangle with a base on points n, n+1 with the other two points on the lines between points 1 & 2 and between points 1 and 2n+1. So for a Triangle, this is 0, and for a Pentagon, this is from the flat bottom up to the two lines that meet at the top... That *has* to be messier...Naraht (talk) 19:16, 23 September 2021 (UTC)

I think you probably mean points n+1 and n+2. If you start numbering (with 1) the vertices of a pentagon (n=2) at the top, then the bottom two vertices are 3 and 4. --RDBury (talk) 22:03, 23 September 2021 (UTC)

It seems that the case of a polygon with an odd number of sides requires recourse to trig functions. The ratio of the rectangle area to that of the polygon is (I reckon)

${\frac {R}{P}}={\frac {2\left(2-{\frac {r}{h}}\right)\left(1+{\frac {r}{h}}\right)}{2n+1}}={\frac {2\left(2-\sec \left({\frac {\pi }{2n+1}}\right)\right)\left(1+\sec \left({\frac {\pi }{2n+1}}\right)\right)}{2n+1}}$

where $r$ and $h$ are the circumradius and apothem.

For a triangle $n=1$,

${\frac {R}{P}}=0$

For a pentagon $n=2$,

${\frac {R}{P}}={\frac {2\left(2-{\frac {4}{{\sqrt {5}}+1}}\right)\left(1+{\frac {4}{{\sqrt {5}}+1}}\right)}{5}}=2\left({\frac {3}{\sqrt {5}}}-1\right)=0.683282$

For large $n$, $h=r$

${\frac {R}{P}}={\frac {4}{2n+1}}$

catslash (talk) 00:27, 30 September 2021 (UTC)