Wikipedia:Reference desk/Archives/Mathematics/2014 February 15

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February 15

Area of a window with 2 curved sides of given radius and a horizontal base of given length

I need a formula for the area of a window, The dimensions vary in particular cases, but one example has a horizontal base 10 feet wide, then two curved sides which meet at the top. Each of the curved sides is of a 10 foot radius. A general formula would be very helpful for base =b and radius = r. Thanks. Edison (talk) 03:00, 15 February 2014 (UTC)

Are the two curved sides each semi-circles ? In that case, you have a full circle plus a triangle with at least 2 sides the same. Try splitting that large triangle into two smaller right triangles to get the area. The hypotenuses should be the diameter of the circle, and the horizontal legs are each half the width of the base.

If the two curved sides are not semi-circles, then it's quite a bit more complex to solve.

It also occurs to me that you might mean a rectangle with a semi-circle on each side. In that case the area is 2rb + πr². StuRat (talk) 03:35, 15 February 2014 (UTC)

If the centers of the circles are on the baseline, in other words if each arc meets the baseline at a right angle, it's feasible but (at this late hour) I'm too lazy to do it without assurance that that is the case. —Tamfang (talk) 08:59, 15 February 2014 (UTC)

There is not quite enough information, but if we add the assumption that the arcs do meet the base at right angles (so that the arc centres-of-curvature are level with the baseline, and the height of the window is $\scriptstyle {h={\sqrt {b\left(r-{\frac {b}{4}}\right)}}}$ ), then (I reckon) the area is

$A=r^{2}\arccos \left(1-{\frac {b}{2r}}\right)-\left(r-{\frac {b}{2}}\right){\sqrt {b\left(r-{\frac {b}{4}}\right)}}$

In the special case of $\scriptstyle {b=2r}$, which makes the whole window a semicircle, this reduces to $\scriptstyle {A={\frac {1}{2}}\pi r^{2}}$ as expected. In another special case of $\scriptstyle {b=4r}$, which make the window in to two side-by-side semicircles, so that the height at the centre of the window is zero, the formula reduces to $\scriptstyle {A=\pi r^{2}}$ - again as expected, so it could be right --catslash (talk) 14:59, 15 February 2014 (UTC)

And $\scriptstyle {r=b=10\mathrm {ft} }$ gives $\scriptstyle {A=25\left(4\arccos \left({\frac {1}{2}}\right)-{\sqrt {3}}\right)\mathrm {ft} ^{2}=25\left(4{\frac {\pi }{3}}-{\sqrt {3}}\right)\mathrm {ft} ^{2}=61.4185\mathrm {ft} ^{2}}$ --catslash (talk) 15:10, 15 February 2014 (UTC)

I ought to have answered by reference to WP articles: so we understand your window to be two halves of a circular segment, cut along its line of symmetry (the base of the window), and joined along the chord (the vertical centre-line of the window). --catslash (talk) 17:04, 15 February 2014 (UTC)

Thanks for the prompt and detailed responses. Some clarification: there is a horizontal bottom of length b (in the sample case, 10 feet). Each of the two curved segments starts at one side of the base, with the radius or origin of the curve at the other baseline, starting at a right angle to the base. The radiusof each arc is equal to the baseline (10 feet) in the sample case, but in the general case it could be greater, and the two arcs intersect at the top. They are arched rather than semicircles. An approximation would be to draw it on graph paper, then count the squares and approximate fractions of squares it covers. Another approximation would be to draw it on paper, cut it out and weigh it with an accurate balance, and compare it to the weight of a rectangle of known area of the paper. But given an appropriate equation it will be easy to run a spreadsheet to calculate each of the several windows I am studying. Thanks. Edison (talk) 21:33, 15 February 2014 (UTC)

Still not 100% understanding the shape. Is it "Think of an equilateral triangle. Draw a circular arc from bottom right corner to top corner radius one side. Draw a circuar arc from bottom left corner to top corner radius 1 side. Then erase the two straight lines that meet at the top corner"? Also I assume that the window is flat and the curcvature is in the xy plane. We aren't talking 3d ciurvature (like a car windscreen). -- SGBailey (talk) 00:10, 16 February 2014 (UTC)

Yea, a picture is worth 1000 words here. Is there any way you can upload a pic or link to one online ? StuRat (talk) 00:29, 16 February 2014 (UTC)

Perhaps like the middle windows here?

Perhaps like the middle windows here? --catslash (talk) 02:28, 16 February 2014 (UTC)

• I don't understand why people are having such difficulty -- this seems straightforward to me. Suppose you replace one of the curved edges with a straight line -- then you get a sector of a circle swept out by an angle of 60 degrees. Therefore its area is 1/6 of the area of the circle. Now do the same thing replacing the other curved edge with a straight line, and add the two results. This is the area of your region, except that the equilateral triangle has been counted twice. So subtract the area of the equilateral triangle, and you get the answer. (This would probably be easier to understand with a picture, but I'm hoping that the description is clear enough.) Looie496 (talk) 15:43, 16 February 2014 (UTC)

That gives

$A=2{\frac {\pi r^{2}}{6}}-\left({\frac {1}{2}}r{\frac {\sqrt {3}}{2}}r\right)=\left({\frac {\pi }{3}}-{\frac {\sqrt {3}}{4}}\right)r^{2}$

-the same as is obtained by putting $\scriptstyle {b=r}$ into the formula given earlier. --catslash (talk) 17:26, 16 February 2014 (UTC)

Probability question

n shapes each of area a are placed randomly on area A (equally likely to be placed at any position; all orientations equally likely; shapes overlapping as necessary). a is negligibly small compared to A, while n is large enough so that the total area covered is non-negligible. Ignoring negligible effects, are probability results concerning the total area covered given n independent of the actual shape of a (e.g. whether circular, rectangular, etc.)? (I assume they are independent of the shape of A.) 86.160.221.91 (talk) 23:46, 15 February 2014 (UTC)

I believe so, yes. Although very long, thin shapes are more likely to overlap each other, the area of each overlap is likely to be less. I think it would therefore all even out. Also, if the "a" shapes can only fit in region "A" in one direction, that might affect the probabilities, such as if parts of A can not possibly be covered by the "a" shapes. StuRat (talk) 00:33, 16 February 2014 (UTC)

Unless you only want lower bounds on the probability, it seems to me you need some condition on the the shapes, e.g. connected or convex. Say A is the unit square and a is the unit square with a square of side 1-e cut out of the center, The area of a is 2e-e² which can be made as small as you want, but a can only fit in A in one way so the probability of an intersection is 1 when n>1. Or let a be the unit square and A be the same union a rectangle of height 1/2 and length M attached to the unit square by its short side. The area of A is 1+M/2 which can be made as large as you want but again, a only fits in A one way. --RDBury (talk) 06:03, 16 February 2014 (UTC)

I meant to exclude all "how/where does it fit?" considerations like this, but the statement that a is negligibly small compared to A seems not to be sufficient. Please add another condition, something like the greatest distance between any two points in a is negligible compared to the smallest relevant dimension of A. 86.160.221.91 (talk) 12:49, 16 February 2014 (UTC)

However, looking at this again, I think that both your scenarios might be excluded by the original condition that the total area covered must be non-negligible. 86.160.221.91 (talk) 22:01, 16 February 2014 (UTC)

No. For certain conditions on a, A, n, you might me able to model the probabilities as independent of shape - that is, convince a reviewer that any errors coming from the independence assumption are small. But RDBury gives some good examples where shape matters, and it only takes one exception to break this conjecture. Things get even weirder if you have collections of different shapes, or if you allow "shapes" such as a fat cantor set (a.k.a Smith–Volterra–Cantor_set). Finally, even "a is negligibly small compared to A" isn't very meaningful, unless you can express the relation rigorously. I don't know what further conditions will give you true independence of shapes (if any), but connectedness, convexity, compactness, and restricting the diameter in terms of A would be a place to start. SemanticMantis (talk) 17:39, 16 February 2014 (UTC)

I think that your "diameter" condition is essentially what I was referring to in my last post (above yours). I want to include concave shapes in the everyday meaning of concave. I'm not sure about compactness -- I don't understand the concept clearly. I want to include the kind of shapes you could cut out of a piece of paper. If there are any other conditions that could be applied to exclude weird and exotic constructions but include all "sensible" shapes them please apply them. 86.160.221.91 (talk) 20:35, 16 February 2014 (UTC)

Yes, I see that you were getting at generalized diameter, so that's good. You probably don't have to worry about compactness is you do stipulated that the pieces are connected. If you have more specific questions, it might be best to open up a new question below, that will get more attention. SemanticMantis (talk) 15:40, 18 February 2014 (UTC)

So, just to clarify, if I cut lots of "ordinary" small shapes out of paper, and drop these randomly on a large area, then as far as coverage probability results are concerned, the actual shapes do not matter (or matter only negligibly), only the areas matter, is that right? 86.160.221.91 (talk) 03:50, 19 February 2014 (UTC)