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In a circular segment, can we depend on the chord length, arch length, and add to it the angle between the tangent at the arc endpoints and the chord (if at all needed), to define the height of the segment.

We want the height to equal zero and the formula is still defined for chord length equal to arc length (and the angle between the tangent and chord is zero).

In other words, we need to deliberately not use radius, arc angle, or divide by the height.

The reason being that I'm using the problem solution in a computer software, and I can not deal with infinitely large radii, or even too big radii in float arithmetics. And intuitively it seems totally possible to depend on tangent angle with very small or zero value, to define a height of zero whenever encountered.

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  • $\begingroup$ Height is always zero if arc and chord have same length (with infinite radius of arc),no need to define it in any other way $\endgroup$
    – Narasimham
    Jan 3, 2020 at 10:35
  • $\begingroup$ You need to read carefully before commenting. We are trying to define a formula that is still correct when the arc length equals the chord. Nobody is talking about "defining" the height, neither about figuring out whether it is zero in such conditions. Your comment simply misses the very point of the question $\endgroup$
    – Physician
    Jan 3, 2020 at 11:18
  • $\begingroup$ @Physician Then you shouldn't write "to define the height of the segment", but to "calculate the height of the segment". $\endgroup$
    – Paul Frost
    Jan 3, 2020 at 11:32
  • $\begingroup$ Does this answer your question? Calculating the height of a circular segment at all points provided only chord and arc lengths $\endgroup$
    – Paul Frost
    Jan 3, 2020 at 11:37
  • $\begingroup$ @Paul Frost, the formula is not "defined" if we'll divide by zero at infinite radius. I can still calculate the height using a limit calculation, but I can't define it using the typical formula that involves the radius and angle of the arc because dividing by zero is not defined, that's what I meant by defining it. Regarding the similar question you suggested, first off, thanks for your effort, second, unfortunately it doesn't help as it is dependent on radius, which will be infinitely large in my mentioned problem. $\endgroup$
    – Physician
    Jan 3, 2020 at 11:45

1 Answer 1

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The answer to Calculating the height of a circular segment at all points provided only chord and arc lengths shows that you have to solve the equation (where $\phi = \theta /2\in [0,\pi/2]$) $$\phi= \frac{s}{a} \sin \phi$$ which in general can only be done numerically. If you have determined $\phi$, then $$h = \frac{a}{2 \sin \phi} ( 1 - \cos \phi) = \frac{a \sin\phi}{2(1+ \cos \phi)} $$ since $\frac{1 - \cos \phi}{ \sin \phi} = \frac{\sin \phi}{1 + \cos \phi}$. This applies also for $\phi = 0$ which is the "limit angle" for $s = a$. It gives $h = 0$ as desired.

Note that $\phi= \frac{s}{a} \sin \phi$ is equivalent to $$\frac{a}{s} = \frac{\sin \phi}{\phi} = \sum_{n=1}^\infty (-1)^n\frac{\phi^{2n}} {(2n+1)!} = 1 - \frac{\phi^{2}}{3!} + \frac{\phi^{4}}{5!} - \ldots$$ which allows to get suitable numerical approximations.

By elementary geometry you see that the angle between the tangent at the arc endpoints and the chord is nothing else than $\phi$ (draw a picture). Thus, if you know this angle, you are done.

Remark: If you know $\phi$, then you do not need to calculate $\sin \phi$ and $\cos \phi$ by using the sine and cosine functions. In fact, you have $$\sin \phi = \frac{a}{s} \phi \\ \cos \phi = \sqrt{1 - \frac{a^2}{s^2} \phi^2} \\ h = \frac{a^2 \phi}{2(s + \sqrt{s^2 - a^2\phi^2})}$$

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  • $\begingroup$ Just a perfect answer. I can't thank you enough. $\endgroup$
    – Physician
    Jan 3, 2020 at 14:15

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