This is a followup to a question I asked earlier.

I have looked for an example on Google and StackExchange, but I have yet to see a clear example of the formula to determine the equation of an Apollonian Circle given two points and a ratio.

For example, if I have points $A = (0, 0)$ and $B = (3, 1)$ and want to maintain a ratio of $11:13$ between them, how could I get the Apollonian Circle?

I am far from being good at interpreting condensed mathematical notation (which is why this answer did not help me), so a step-wise process would be appreciated.

  • $\begingroup$ What have you done? What have you tried? Which step are you stuck at? $\endgroup$ – Calvin Lin Dec 7 '13 at 16:31

Giving a point $(x,y)$, we want to find those that satisfy

$$ 13 \sqrt{x^2+y^2} = 11 \sqrt{(x-3)^2 + (y-1)^2}. $$

Square and expand, and convert it into an equation of the form

$$ (x-h)^2 + (y-k)^2 = r^2. $$

  • $\begingroup$ Awesome! That makes a lot of sense on scaling each side by my ratios. But, I need to extract h, k, and r in code, so expanding and processing that beast in code is daunting. Do you know of a short way, given your original expression, to extract those values? Much appreciated. $\endgroup$ – Eric Olson Dec 7 '13 at 19:39
  • $\begingroup$ @EricOlson Just expand both side and compare coefficients. $\endgroup$ – Calvin Lin Dec 7 '13 at 20:18

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