Wikipedia:Reference desk/Archives/Mathematics/2021 March 18

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March 18

Circle measure problem.

C is at the center of a circle (call it one unit in radius). A and B are on the circle edge. The points in the circle that are less than one unit from either A or B (or both) make up exactly half of the Circle, what is the measure of the angle ACB?

(The amount of the Circle that is less than one unit from A is slightly more than 1/3, so if A & B are at the same point, the amount would be slightly more than 1/3 and if A & B are opposite, it would be slightly more than 2/3. As B moves around the circle from being equal to A to being opposite from A the amount increases monotonically)Naraht (talk) 21:14, 18 March 2021 (UTC)

Another quilting problem?

I must be missing something, but as long as ∠ACB ≤ 2π/3, isn't your area easily divided into a sector and two segments, all with easy to calculate areas, with that of the segments fixed and that of the sector a function of your angle? -- ToE 01:59, 19 March 2021 (UTC)

Elementary, but not simple. Given a point $P$ in the plane, let $\Theta _{P}$ stand for the unit-radius disk with centre $P$, and $\mathrm {O} _{P}$ for the corresponding unit circle, the boundary of the disk. Furthermore, for a measurable planar shape $S$, let $|S|$ denote its measure (area). We have three points $A,B$ and $C$ for which $A,B\in \mathrm {O} _{C}$ (which is equivalent with $C\in \mathrm {O} _{A}\cap \mathrm {O} _{B}$). The task is, given $|(\Theta _{A}\cup \Theta _{B})\cap \Theta _{C}|=u$, to determine the angle $\alpha =\angle ACB$. More specifically, we are given that $u$ has to equal the quantity ${\tfrac {1}{2}}|\Theta _{C}|={\tfrac {1}{2}}\pi$.

In general, $|(S\cup T)\cap U|=|S\cap U|+|T\cap U|-|S\cap T\cap U|$, so

$u=|\Theta _{A}\cap \Theta _{C}|+|\Theta _{B}\cap \Theta _{C}|-|\Theta _{A}\cap \Theta _{B}\cap \Theta _{C}|$.

Moreover, for reasons of symmetry,

$|\Theta _{A}\cap \Theta _{C}|=|\Theta _{B}\cap \Theta _{C}|$,

for which the formula for the area of a circular segment (really, a disk segment) gives us

$v=|\Theta _{A}\cap \Theta _{C}|={\frac {2\pi }{3}}-{\frac {1}{2}}{\sqrt {3}}.$

Note that this quantity is independent of the placements of $A$ and $B$. If we have an expression for $w=|\Theta _{A}\cap \Theta _{B}\cap \Theta _{C}|$ as a function of $\alpha$, so that $u=2v-w$, we can solve the equation $u={\tfrac {1}{2}}\pi$ for $\alpha$. For the special case that $\alpha =0$, so $A$ and $B$ coincide, we find $u=w=v\approx 1.22837<{\tfrac {1}{2}}\pi$, so these points are distinct. For the special case that $\alpha =\pi$, so $A$ and $B$ are diametrically opposite, $w=0$, so $u=2v\approx 2.45674>{\tfrac {1}{2}}\pi$, so we may assume that $\alpha <\pi$. Without loss of generally, we put $C$ at the origin of the Cartesian plane, and $A$ and $B$ symmetrically with respect to the x-axis, so $A=(c,s),B=(c,{-}s)$ for values $c$ and $s$ such that $c^{2}+s^{2}=1$. Assume, still without loss of generality, that $c>0$ and $s>0$, so $A$ is above $B$, in the first quadrant. Putting $\varphi ={\tfrac {1}{2}}\alpha$, we have, of course, that $c=\cos \varphi ,s=\sin \varphi$. $\mathrm {O} _{C}$ intersects the x-axis in $M=(1,0)$, and the line ${\overline {CM}}$ is perpendicular to the line ${\overline {AB}}$. $\mathrm {O} _{C}$ intersects $\mathrm {O} _{A}$ in two points; the lower one is $U=({\tfrac {1}{2}}c+{\tfrac {1}{2}}{\sqrt {3}}\,s,-{\tfrac {1}{2}}{\sqrt {3}}\,c+{\tfrac {1}{2}}s)$. The upper point of intersection of $\mathrm {O} _{C}$ and $\mathrm {O} _{B}$ is the mirror point $V=({\tfrac {1}{2}}c+{\tfrac {1}{2}}{\sqrt {3}}\,s,{\tfrac {1}{2}}{\sqrt {3}}\,c-{\tfrac {1}{2}}s)$. $U$ and $V$ coincide when $\varphi ={\tfrac {1}{3}}\pi ,c={\tfrac {1}{2}},s={\tfrac {1}{2}}{\sqrt {3}}$, and then $U=V=M$. For this case, using again the disk segment area formula, we find $w={\tfrac {1}{3}}-{\tfrac {1}{2}}{\sqrt {3}}$, so $u\approx 2.27557>{\tfrac {1}{2}}\pi$, from which we may conclude that $\varphi <{\tfrac {1}{3}}\pi$. $\mathrm {O} _{A}$ and $\mathrm {O} _{B}$ intersect in $C$; their second point of intersection is at $(2c,0)$, which is outside $\Theta _{C}$ if $0<\varphi <{\tfrac {1}{3}}\pi$. The x-axis divides $\Theta _{A}\cap \Theta _{B}\cap \Theta _{C}$ into two equal parts, so it suffices to find the area of the upper part, which is bounded by the line segment ${\overline {CM}}$ and the circular arcs from $M$ to $V$ along $\mathrm {O} _{C}$ and from $V$ back to $C$ along $\mathrm {O} _{B}$. The triangle area $|\triangle CMV|={\tfrac {1}{2}}({\tfrac {1}{2}}{\sqrt {3}}\,c-{\tfrac {1}{2}}s)$. To this we need to add the areas of the disk segments for the arcs from $M$ to $V$ and from $V$ to $C$. Angle $\angle MCV=\angle BCV-\angle BCM={\tfrac {1}{3}}\pi {-}\varphi$, so for the first of these two disk segments we find area ${\tfrac {1}{2}}({\tfrac {1}{3}}\pi {-}\varphi -\sin({\tfrac {1}{3}}\pi {-}\varphi ))$. Triangle $\triangle VBC$ is equilateral, so $\angle VBC={\tfrac {1}{3}}\pi$, and the area of the remaining disk segment is ${\tfrac {1}{2}}({\tfrac {1}{3}}\pi -{\tfrac {1}{2}}{\sqrt {3}})$. Combining all, we have

$u=2v-\left(({\tfrac {1}{2}}{\sqrt {3}}\,c-{\tfrac {1}{2}}s)+({\tfrac {1}{3}}\pi {-}\varphi -\sin({\tfrac {1}{3}}\pi {-}\varphi ))+({\tfrac {1}{3}}\pi -{\tfrac {1}{2}}{\sqrt {3}})\right)$,

where $c=\cos \varphi ,s=\sin \varphi ,\varphi ={\tfrac {1}{2}}\alpha$. The transcendental equation $u={\tfrac {1}{2}}\pi$ can only be solved numerically, so there is no point in attempting to massage this into a more pleasing form. The numeric solution is $\alpha \approx 0.68485$, putting $A$ at approximately $(0.94194,0.33577)$.  --Lambiam 11:40, 19 March 2021 (UTC)

Wow! That's a lot of work to determine α = √3 - π/3 ≈ 0.68485 or 39.239°. It makes me wonder if I've oversimplified things, but since our answers match, here goes. I'd already typed this with θ = ∠ACB, so I'll keep my notation.

If θ ≤ 2π/3, then the your area can be easily divided into a sector (of circle C) and two segments (of Circles A & B).

The sector is of a unit circle with central angle 2π/3 + θ, and thus has area of (1²/2)(2π/3 + θ) = π/3 + θ/2.

The segments are of a unit circle with central angle π/3, and thus each have an area of (1²/2)(π/3 - sin(π/3)) = (π/3 - √3/2)/2. (Each segment is ~ 0.029 · π or 2.9% the area of your circle.)

Summing the three areas yields A(θ) = 2π/3 + θ/2 - √3/2, for 0 ≤ θ ≤ 2π/3.

Sanity checks:

A(0) = 2π/3 - √3/2 ≈ 0.39 · π, or "slightly more than 1/3" the area of your circle.

Also: A(2π/3) = π - √3/2 ≈ 0.72 · π, or more than 1/2 your circle, which is good as the area becomes more complicate for larger angles where we have to deal with the intersection of segments.

Note that the area becomes simple again at θ = π, with A(π) = 2A(0) = 4π/3 - √3 ≈ 0.78 · π.

Also: A(π) - A(2π/3) = π/3 - √3/2 should be the area of two segments, which it is.

Solving for A = π/2 yields θ = (√3 - π/3) radians or ≈ 39.2°

-- ToE 12:45, 19 March 2021 (UTC)

Yeah, that is a lot simpler. I guess I did not fully understand your approach. It requires θ not to exceed 2π/3; otherwise the decomposition into a sector and two segments is not possible.  --Lambiam 13:45, 19 March 2021 (UTC)

FWIW I concur. If θ is between 2π/3 and π, then the formula for the area covered is different, but I didn't think it was worth working out because it's clear that it increasing as a function of θ (since the overlap decreases) and the value is already greater than π/2 when θ is 2π/3. According to my calculations the area covered ranges from .391π at θ=0 and .782π at θ=π. If the question was to find θ so that the area covered is 3/4 π then you'd need (I assume) the other formula. --RDBury (talk) 14:30, 19 March 2021 (UTC)