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Consider two circles with centres on the $x$-axis. What function can describe the line formed on the top of these circles?

enter image description here

By commonsense, I can tell, a variable in that hypothetical function changes from $0-1$ (or vice versa) to produce the exact line on the circles (black), red line, and blue line. But I have no idea where to start.

To provide a physical perspective, imagine we lay a string on the circle. Depending on the string material, it forms the blue line (rigid string), the red line (flexible string), and the black line (ultra soft string).

Or imagine the black circles are some physical fields (electric, magnetic, etc) around particles (centres of the circles). How do we map the field boundaries?

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  • $\begingroup$ Please edit the question to clarify. Do you have equations for the circles? The blue line is a segment of the common tangent. Yes, it can be parameterized by a variable that measures the distance of a point on it from one end. What is the red curve? $\endgroup$
    – Ethan Bolker
    Commented Mar 2, 2022 at 16:28
  • $\begingroup$ @EthanBolker no I do not have any equation. I added some physical examples to clarify my question. $\endgroup$
    – Googlebot
    Commented Mar 2, 2022 at 18:00
  • $\begingroup$ The extra information is both good and bad news. I now know the kind of answer you want. But without more explicit assumptions about the physics I think the question can't be answered. It might help if you edit the question to tell us how you expect to use any answer you get. $\endgroup$
    – Ethan Bolker
    Commented Mar 2, 2022 at 19:13
  • $\begingroup$ @EthanBolker my question is pure math. I look to understand how we merge two circles. The examples from physics only aimed to compensate for my inability to express my question in math language. $\endgroup$
    – Googlebot
    Commented Mar 2, 2022 at 19:40

1 Answer 1

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Let $P$ and $Q$ be the points of tangency at the tops of the circles. Choose $R$ and $S$ on the circles where you want the joining curve to leave the circles. Then you know the value and the slope of your curve at four points. You can use a cubic spline to interpolate. A search for cubic spline will find python and excel implementations.

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  • $\begingroup$ I thought of a cubic spline, as I used one to draw the red line. The point is that I could not find a general rule for the place of control points. I can visually find a rule, but what is the mathematical rule for the changes in the coordinates of the control points changing the black line into the blue line? $\endgroup$
    – Googlebot
    Commented Mar 2, 2022 at 21:05
  • $\begingroup$ Why should there be a rule? You might remove one degree of freedom by requiring the tangent lines at the intermediate control points be reflections of each other across a vertical axis. $\endgroup$
    – Ethan Bolker
    Commented Mar 2, 2022 at 21:57
  • $\begingroup$ Because I look for a general function to describe it. I can think of two ways to do so, (i) with PDF similar to head equation, or (ii) by Fourier series. But there should be a simpler way, as the key is where the $c_1 - c_2 < r_1+r_2$. $\endgroup$
    – Googlebot
    Commented Mar 2, 2022 at 22:38

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