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Computing the area of a circular segment (green part below) from a distance along the radius axis (like $h$, the sagitta distance) is well-known.

circular segment area

What could be the inverse function, that is the $f$ function: $h=f(A)$, or an approximation of it ?

In other words: The circle is filled with a known quantity of a liquid ($A$) - what is the resulting liquid level ($h$) ?

EDIT 1

Considering that:

$h=R(1-\cos(\theta/2))$

and

$A=R^2/2.(\theta - \sin(\theta))$

I would simply need to express $\theta$ as a function of $A$, that is finding the invert function of $(x-\sin(x))$.

EDIT 2

According to this other question pointed out by Saad, it seems an algebraical expression of the inverse function cannot be obtained. An approximation would however be welcome.

EDIT 3

See the motivation behind this question here: https://observablehq.com/@jgaffuri/striped-circle

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  • 1
    $\begingroup$ Related: Why can't the inverse of $F(x)=x+\sin(x)$ have a formula algebraically?. $\endgroup$ Commented Apr 29, 2020 at 12:15
  • $\begingroup$ @Saad: Thanks. It shows that I cannot expect to have an algebraical formula for this inverse. A 'fair' approximation would however be welcome. $\endgroup$
    – julien
    Commented Apr 29, 2020 at 12:21
  • $\begingroup$ An ODE area versus height would be an answer? By CAS an accurate height for given volume ratio can be outputted. $\endgroup$
    – Narasimham
    Commented May 4, 2020 at 21:00

5 Answers 5

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A decent approximate for the sine function over its full variation domain $\theta\in[0,\pi]$ is

$$\sin \theta = \frac{4\theta}\pi\left( 1- \frac \theta\pi\right)$$

(See the plot below.) Then, the area $A$ can be expressed as

$$A =\frac12 R^2\left( \theta - \sin\theta\right) = R^2\left[\left(\frac12-\frac2\pi\right)\theta + \frac2{\pi^2}\theta^2\right] $$

and the corresponding inverse function is

$$\theta(A) = \frac{\pi^2}4\left(\frac12-\frac2\pi\right) \left[\sqrt{1+\frac{32A}{(\pi-2)^2R^2}}-1\right]$$ enter image description here

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  • $\begingroup$ From the plot it is clear $\frac{4\theta}{\pi} \left (1 - \frac{\theta}{\pi} \right ) \geqslant \sin \theta$ for $\theta \in [0,\pi]$, but can you give any hint how one might show this? $\endgroup$
    – omegadot
    Commented Mar 11, 2022 at 18:12
  • $\begingroup$ @omegadot - see the proof in math.stackexchange.com/a/3632012/686284 $\endgroup$
    – Quanto
    Commented Mar 11, 2022 at 18:21
  • $\begingroup$ That will do it, and thanks for the link. $\endgroup$
    – omegadot
    Commented Mar 12, 2022 at 6:17
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Considering $$y=x-\sin(x)$$ you can develop the rhs as a Taylor series to get $$y=\frac{x^3}{6}-\frac{x^5}{120}+\frac{x^7}{5040}-\frac{x^9}{362880}+\frac{x^{11}}{399 16800}-\frac{x^{13}}{6227020800}+O\left(x^{15}\right)\tag 1$$ and use series reversion to obtain $$x=t+\frac{t^3}{60}+\frac{t^5}{1400}+\frac{t^7}{25200}+\frac{43 t^9}{17248000}+\frac{1213 t^{11}}{7207200000}+O\left(t^{13}\right)\tag 2$$ where $t=\sqrt[3]{6y}$.

For sure, we can add as many terms as we wish for $(1)$ and then $(2)$ but we can make $(2)$ more compact transforming the series as a $[2n+1,2n]$ Padé approximant which will be at least as accurate as a Taylor series to $O\left(t^{4n+2}\right)$. For conveniency, they will be written as $$P_n=t \,\frac{1+\sum _{i=1}^n a_i\,t^{2 i} } {1+\sum _{i=1}^n b_i\,t^{2 i}}$$ For example $$P_1=t\,\frac{1-\frac{11}{420} t^2} {1-\frac{3 }{70}t^2 }$$ $$P_2=t\,\frac{1-\frac{1493 }{21120}t^2+\frac{167 }{268800}t^4 } {1-\frac{123 }{1408}t^2+\frac{403 }{295680}t^4 }$$ For testing purposes, assign a value to $x$, compute the corresponding $y$ and $t$ and recompute $x_{(n)}$ given by $P_n$ . Below are the results $$\left( \begin{array}{ccc} x_{\text{given}} & x_{(1)} & x_{(2)} & x_{(3)} \\ 0.1 & 0.100000000 & 0.100000000 & 0.100000000 \\ 0.2 & 0.200000000 & 0.200000000 & 0.200000000 \\ 0.3 & 0.299999998 & 0.300000000 & 0.300000000 \\ 0.4 & 0.399999985 & 0.400000000 & 0.400000000 \\ 0.5 & 0.499999929 & 0.500000000 & 0.500000000 \\ 0.6 & 0.599999745 & 0.600000000 & 0.600000000 \\ 0.7 & 0.699999248 & 0.700000000 & 0.700000000 \\ 0.8 & 0.799998082 & 0.800000000 & 0.800000000 \\ 0.9 & 0.899995616 & 0.899999999 & 0.900000000 \\ 1.0 & 0.999990812 & 0.999999995 & 1.000000000 \\ 1.1 & 1.099982049 & 1.099999986 & 1.100000000 \\ 1.2 & 1.199966902 & 1.199999964 & 1.200000000 \\ 1.3 & 1.299941871 & 1.299999913 & 1.300000000 \\ 1.4 & 1.399902050 & 1.399999801 & 1.400000000 \\ 1.5 & 1.499840739 & 1.499999570 & 1.499999999 \\ 1.6 & 1.599748976 & 1.599999116 & 1.599999997 \\ 1.7 & 1.699615013 & 1.699998260 & 1.699999992 \\ 1.8 & 1.799423695 & 1.799996701 & 1.799999981 \\ 1.9 & 1.899155774 & 1.899993951 & 1.899999957 \\ 2.0 & 1.998787130 & 1.999989239 & 1.999999906 \\ 2.1 & 2.098287918 & 2.099981371 & 2.099999800 \\ 2.2 & 2.197621631 & 2.199968535 & 2.199999590 \\ 2.3 & 2.296744092 & 2.299948039 & 2.299999183 \\ 2.4 & 2.395602386 & 2.399915947 & 2.399998417 \\ 2.5 & 2.494133744 & 2.499866600 & 2.499997012 \\ 2.6 & 2.592264408 & 2.599791977 & 2.599994490 \\ 2.7 & 2.689908485 & 2.699680877 & 2.699990060 \\ 2.8 & 2.786966855 & 2.799517867 & 2.799982431 \\ 2.9 & 2.883326151 & 2.899281948 & 2.899969535 \\ 3.0 & 2.978857863 & 2.998944916 & 2.999948113 \end{array} \right)$$

Edit

As a Padé like approximant, $$x=t \, \frac{1-\frac{1267 }{9428}t^2+\frac{31}{6507}t^4-\frac{1}{32473} t^6} {1-\frac{2251}{14902}t^2+\frac{63 }{9593}t^4-\frac{1}{13890}t^6}\qquad (t=\sqrt[3]{6y})$$ leads to errors lower than $10^{-7}$ over all the range.

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I prefer to add another answer since it is based on a totally different approach.

"Thanks" to the containment, I wondered what we could get using the superb approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad \qquad(0\leq x\leq\pi)$$ proposed, more than $1,400$ years ago, by Mahabhaskariya of Bhaskara I, a seventh-century Indian mathematician.

This leads to a cubic equation in $x$ $$4 x^3-4 (y+\pi -4)x^2+\pi\left(4 y+5 \pi -16 \right)x-5 \pi ^2 y=0$$ Let $t=y+\pi-4$ $$4 x^3-4 t x^2+\pi\left(4 t+\pi \right) x-5 \pi ^2 (t+4-\pi)=0$$ which has only one real root as soon as $y > 0.016$; we can compute the solution using the hyperbolic method.

The nasty but amazing result is

$$x=\frac t 3 -2 P(t) \sinh \left(\frac{1}{3} \sinh ^{-1}\left(\frac{Q(t)}{\big[P(t)\big]^3}\right)\right)$$ where $$P(t)=\sqrt{\frac{\pi ^2}{12}+\frac{\pi }{3}t-\frac{1}{9}t^2}$$ $$Q(t)=\frac{5}{8} (\pi -4) \pi ^2-\frac{7 \pi ^2 }{12}t+\frac{\pi }{6}t^2-\frac{1}{27}t^3$$ which gives quite good results as soon as $y>1-\sin(1)\sim 0.16$.

Repeating the same procedure as in my other answer $$\left( \begin{array}{cc} x_{\text{given}} & x_{\text{calc}} \\ 1.0 & 0.99726 \\ 1.1 & 1.09810 \\ 1.2 & 1.19884 \\ 1.3 & 1.29941 \\ 1.4 & 1.39978 \\ 1.5 & 1.49996 \\ 1.6 & 1.59999 \\ 1.7 & 1.69990 \\ 1.8 & 1.79974 \\ 1.9 & 1.89954 \\ 2.0 & 1.99936 \\ 2.1 & 2.09922 \\ 2.2 & 2.19917 \\ 2.3 & 2.29921 \\ 2.4 & 2.39936 \\ 2.5 & 2.49961 \\ 2.6 & 2.59994 \\ 2.7 & 2.70029 \\ 2.8 & 2.80061 \\ 2.9 & 2.90081 \\ 3.0 & 3.00076 \\ 3.1 & 3.10033 \end{array} \right)$$

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Exact inversion is not possible because the equation is transcendental, so we have to resort to numerical methods.

We approximate the exact curve, in blue,

$$y=x-\sin x$$

in the range $[0,\pi]$. Other ranges follow by symmetry/periodicity.

Using the first two terms of the Taylor development, we get the green curve

$$y\approx \frac{x^3}6,$$ which matches the function well for small $x$.

For larger $x$, we can fix the deviation (up to $\dfrac{\pi^3}6$ instead of $\pi$) by adding a damping factor such that the function value is exact at $0$ and $\pi$. We also require that for small $x$, the formula remains asymptotically exact. For convenience, we chose the form $1-ax^3$ and obtained the magenta curve

$$y\approx\frac{x^3}6\left(1-\left(1-\dfrac6{\pi^2}\right)\frac{x^3}{\pi^3}\right).$$

enter image description here

As you can notice, both

$$y=ax^3$$ and $$y=ax^3(1-bx^3)$$ are easily inverted (for the second, solve the quadratic equation in $x^3$).

From these inital approximations, you can start Newton's iterations,

$$x_{n+1}=x_n-\frac{x_n-\sin x_n-y}{1-\cos x_n}.$$

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  • $\begingroup$ @ClaudeLeibovici: hi Claude. I should have mentioned (shame on me), that I want the approximation to be unbiased for small small $x$, i.e. the cubic coefficient should be $a=1/6$. $\endgroup$
    – user65203
    Commented May 3, 2020 at 8:54
  • $\begingroup$ @ClaudeLeibovici: I am not surprised to see a Padé approximant ;-) As usual, this is a matter of a tradeoff between accuracy and run-time cost, and can only be decided by timing tests on a real platform. $\endgroup$
    – user65203
    Commented May 3, 2020 at 8:59
  • $\begingroup$ @ClaudeLeibovici: I see. Yep, the cubic equation can be solved with hyperbolic functions (just as is it solved with trigonometry in the casus irreductibilis). Amazing how early mathematicians could obtain such advanced results "with bare hands" ! $\endgroup$
    – user65203
    Commented May 3, 2020 at 9:39
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Due to their transcendental nature an exact inverse relation cannot be defined. A numerical iteration supplies exact heights.

In the following a differential equation approach is chosen and set up in Mathematica. Started with parametrization on $t$ but ended up as normal ODE to output height $h$ as dependent variable. It is integrated to read off y-axis water heights at desired x-axis volume fraction of cylindrical tank ( three outputs shown) :

$$ x= \dfrac{A}{\pi R^2}=\dfrac{t-\sin t}{2 \pi};\, y= \dfrac{h}{R}=1-\cos t/2\, $$ $$ x'=\dfrac{1 - \cos t}{2 \pi}; \,y'= \dfrac12 \,\sin t/2 $$

$$ \dfrac{dy}{dx}=\dfrac{y'}{x'}=\pi \dfrac {\sqrt{1-(1-y)^2}}{2 ( 1- \cos^2 t/2})= \dfrac{\pi/2}{\sqrt{y(1-y)}}$$

 niveau = {Y'[x] == (Pi/2)/Sqrt[Y[x] (2 - Y[x])], Y[0] == 10^-8};
    NDSolve[niveau, Y, {x, 0, xm}];
    y[u_] = Y[u] /. First[%];
    Plot[y[x], {x, 0., 1}, GridLines -> Automatic, AspectRatio -> 1, 
     PlotStyle -> {Blue, Thick}, Axes -> True, 
     AxesLabel -> {AreaFraction, Height}]
    {y[0.2].y[0.6], y[0.85]}
    ParametricPlot[{(t - Sin[t])/(2 Pi), (1 - Cos[t/2])}, {t, 0, 2 Pi}, 
     AspectRatio -> 1, PlotStyle -> {Blue, Thick}, GridLines -> Automatic,
      Axes -> True, AxesLabel -> {AreaFraction, Height}]

enter image description here

Integrated and direct plots are seen to be identical.

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