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$C_1$ and $C_2$ are concentric circles, with radii $R_1$ and $R_2$, $R_1$ being shorter. Point "$P_1$" on $C_1$, and Point "$P_2$" on $C_2$. The radii to the points form an unknown angle, but $\text{Arc}_1$ (the arc length on $C_1$) is known. How to calculate the $P_1$ to $P_2$ distance? Thanksdistance in red?

I see related posts, but not clear how to convert to my problem. (solving for spiral distance, or given the skew angle)

Mike Rowave
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  • When you say the arclength on $c_1$ is known, are you talking about the arc between $R-1$ and $R_2$ in the figure? – Andrei Nov 02 '22 at 18:25
  • Correct, thanks. This is actually a geography application, to determine the direct distance between points with different x,y,z coordinates. I'm getting Arc length (at sea level) from Vincenty's formulae, and calculating Earth radii from here: ([link] (https://rechneronline.de/earth-radius/)) – Mike Rowave Nov 02 '22 at 18:31
  • The arc length in radians is the same as the angle between $R_1$ and $R_2$. Now apply the law of cosines. – Ross Millikan Nov 02 '22 at 18:36
  • Thanks @RossMillikan. If not too much bother, what would that formula look like in practice? – Mike Rowave Nov 02 '22 at 18:43
  • https://en.wikipedia.org/wiki/Law_of_cosines – Ross Millikan Nov 02 '22 at 18:45
  • ok, i see it, duh. Since I have $R_1$ and $R_2$, and the angle (in radians), then i just need to convert radians to degrees and then solve the triangle. Thanks. – Mike Rowave Nov 02 '22 at 18:47
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    Not quite right, but I've found the missing piece: "Arc Length" = "Radius" x "Angle in Radians". Now I can solve with law of cosines. :-) – Mike Rowave Nov 02 '22 at 19:08

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