What is the formula to calculate the distance (arc length) between 2 points $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ on the circumference of a circle of radius $r$ without knowing the angle $\theta$ between them. I found that arc length can be calculated knowing $\theta$. But I know only the $x,y,z$ co-ordinates of 2 points on the circumference of a circle. Please suggest.
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1you can use the formula for length of an arc $l=\int_{a}^{b}\sqrt{<v(t),v(t)>}dt $ – Nebo Alex Jan 01 '16 at 10:37
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Do you mean arclength distance (measured along the circle), or straight-line distance.If the latter, then the circle is irrelevant and the answer is given by @GBeau below. – bubba Jan 01 '16 at 10:37
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I meant arc length – Sangam Jan 01 '16 at 10:41
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@Boris, I am not good at integration. Please can you explain how to use the formula. – Sangam Jan 01 '16 at 10:43
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Assuming you know the radius, find the straight line distance between the points first then you have a triangle with 3 sides known. Use the cosine rule to find $\theta$ – Paul Jan 01 '16 at 10:53
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Let $d$ be the (straight-line) distance between the two points. Then the arclength between them is $$ s = 2r\sin^{-1}\left( \frac{d}{2r} \right) $$ Note that this does not assume that the circle is centered at the origin (as some of the other answers seem to do).
Of course, the problem only makes sense if $d \le 2r$, for otherwise there can be no circle passing through the two given points.
As mentioned in a comment, on any given circle passing through the two points, there is a shorter arc and a longer arc. The formula above gives the arclength $s$ of the shorter arc. The arclength of the longer one is simply $2\pi r - s$.
bubba
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Note that there are actually two arcs on the circle that will go through the two points at a given radius, and the length of the other arc is just the circle circumference minus the arc length calculated above. – IronEagle Apr 17 '22 at 00:26
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