In the famous "Problem of Apollonius" (https://en.wikipedia.org/wiki/Problem_of_Apollonius), the goal is to draw three circles that are tangential to another circle:
Algebraically, we can write this problem as a system of equations:
$$(x_s-x_1)^2+(y_s-y_1)^2=(r_s-s_1 r_1)^2$$ $$(x_s-x_2)^2+(y_s-y_2)^2=(r_s-s_2 r_2)^2$$ $$(x_s-x_3)^2+(y_s-y_3)^2=(r_s-s_3 r_3)^2$$
My Question: Can we consider this above problem as a "constraint optimization problem"?
For instance, from the above equations:
- we want to find out values of "x1, y1, s1, r1, x2, y2, s2, r2, x3, y3, s3, r3"
- such that all "black circles" intersect with the "purple circle"
Thus: we could make a objective function
that takes inputs "x1, y1, s1, r1, x2, y2, s2, r2, x3, y3, s3, r3"
and that outputs a number between "0 and 1" corresponding to the proportion of "black circles" are tangential to the "purple circle" (e.g. 0, 0.33, 0.66, 0.99)
Thus, we could use some Optimization Algorithm such as the Genetic Algorithm for trying to generate "feasible combinations" of "x1, y1, s1, r1, x2, y2, s2, r2, x3, y3, s3, r3" such that "the proportion of black circles tangential to the purple circle" are optimized (hoping that we eventually get a value of 0.99, signifying that the Problem of Apollonius is solved).
Is this correct?
Note Does anyone know why the "Problem of Apollonius" was considered to be so difficult? Was the original objective of this problem to prove that circles can be drawn this way? Or was the original objective of this problem to prove that this can be done for more than 3 circles?