2

I followed https://en.wikipedia.org/wiki/Problem_of_Apollonius#Algebraic_solutions.
I tried to solve it using https://mathworld.wolfram.com/ApolloniusProblem.html.

I got to the point, where a, b, c, d, a', b', c' and d' were used for substitution and I computed x and y correctly.

Those x and y can then be plugged back into the quadratic equation (x-x1)^2 + (y-y1)^2 = (r(+or-)r1)^2 and solved using the quadratic formula.

I think I get the point that it should give me a formula to r, but when solving the quadratic equation, I tried more approaches and ended up with 2 full lists of paper and nowhere near to the end.

On wikipedia, there is a hint:
x = M + N*r
y = P + Q*r
where M, N, P and Q are known functions of the given circles and the choice of signs. Substitution of these formulae into one of the initial three equations gives a quadratic equation for rs, which can be solved by the quadratic formula.

But I totally don't get, what is the M, N, P and Q aka "known functions of the given circles".

I want to use it in an C++ algorithm and there is a Java solution: https://rasmusfonseca.github.io/implementations/apollonius.html, where you can see the individual computation steps and I can just use it as it is, but I want to get an understanding of the solution.

So
1) How to get those M, N, P and Q?
2) Knowing the a, b, c, d, a', b', c', d', x and y, how I get the r from quadratic formula (x-x1)^2 + (y-y1)^2 = (r(+or-)r1)^2, step by step?

I tried wolphramalpha (because I remember it giving steps of computation in its solution) with no success, but I don't use it very often, so maybe my input has not been suitable.

@Jean Marie My scenario Let's say I have those 3 circles with radii 1, 2 and 3.
What are those zk (z1..z4) centers in your question? Can you give me an example? Or what would be their value in my example?
How in this scenario can I use that (2) formula from your question to help me?

Salda
  • 121
  • 1
    See the very efficient solution using complex numbers as described in the first part of this recent question of mine. – Jean Marie May 04 '20 at 12:25
  • 1
    I just read your profile, and I find some convergence with my way of thinking ans acting ! – Jean Marie May 04 '20 at 14:21
  • 1
    @JeanMarie I added some content related to your question. – Salda May 04 '20 at 19:00
  • A key point in these issues is that, for each contact point $P$ between circles centered in $C_1$ and $C_2$, (be it an external or internal contact), points $P, C_1, C_2$ are aligned. – Jean Marie May 04 '20 at 23:11
  • Thanks, I finally understand it, but have another problem now :D – Salda May 05 '20 at 17:16
  • @Salda what exactly do you need? Do you want to construct the fourth circle? Do you rather need to compute something? If yes, what exactly? The radius? The center? Both? – Viera Čerňanová May 10 '20 at 18:31
  • @user376343 Radius and center. I think it's best solved with https://en.wikipedia.org/wiki/Descartes%27_theorem, but I postponed that problem and do something else now. Will do it in a week probably. – Salda May 10 '20 at 20:49

0 Answers0