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Editorial Team
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Asked: 2026-05-18T19:47:46+00:00

I want to visually join two circles that are overlapping so that becomes I

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I want to visually join two circles that are overlapping so that

AltText

becomes

alt text

I already have methods for partial circles, but now I need to know how large the overlapping angle for earch circle is, and I don’t know how to do that.

Anyone got an Idea?

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1 Answer

  1. Editorial Team

    Phi= ArcTan[ Sqrt[4 * R^2 - d^2] /d ]

    HTH!

    Edit

    For two different radii:

    Simplifying a little: 

    Phi= ArcTan[Sqrt[-d^4 -(R1^2 - R2^2)^2 + 2*d^2*(R1^2 + R2^2)]/(d^2 +R1^2 -R2^2)]

    Edit

    If you want the angle viewed from the other circle center, just exchange R1 by R2 in the last equation.

    Here is a sample implementation in Mathematica: 

    f[center1_, d_, R1_, R2_] := Module[{Phi, Theta},

   Phi=  ArcTan[Sqrt[-d^4-(R1^2-R2^2)^2 + 2*d^2*(R1^2 + R2^2)]/(d^2 +R1^2 -R2^2)]

   Theta=ArcTan[Sqrt[-d^4-(R1^2-R2^2)^2 + 2*d^2*(R1^2 + R2^2)]/(d^2 -R1^2 +R2^2)]

   {Circle[{center1, 0}, R1, {2 Pi - Phi,   Phi}], 
    Circle[{d,       0}, R2, {Pi - Theta,  -Pi + Theta}]}

   ];
Graphics[f[0, 1.5, 1, 1]]

    alt text

    Graphics[f[0, 1.5, 1, 3/4]]

    alt text

    And…

    ImageMultiply[
 Binarize@FillingTransform[#], 
 ImageResize[Import@
 "http://i305.photobucket.com/albums/nn235/greeneyedgirlox/blondebabybunny.jpg", 
   ImageDimensions@#]] &@
 Rasterize@Graphics[f[0, 1.5, 1, 1], Background -> Black]

    alt text

    🙂

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