# Make 3 circles intersect in only one point by changing their radius as little as possible

I have 3 circles in a 2D space with known center coordinates ((xa,ya),(xb,yb),(xc,yc)).

The circles don't intersect at a unique point but they should be close to it. I would like to find the point (x,y) at which they would intersect if we were to change their radiuses the slightest possible to make them intersect in only one point.

eg: circle A has a radius of Ra, B of Rb, C of Rc. I would like to find the intersection of 3 circles with radiuses (1+Da)Ra, (1+Db)Rb, (1+Dc)Rc with the minimum values for {Da, Db, Dc} (according to norm2?).

I can't get my head around it, I have

(1+Da)^2+(1+Db)^2+(1+Dc)^2 = ((y-ya)^2+(x-xa)^2)/Ra^2+((y-yb)^2+(x-xb)^2)/Rb^2+((y-yc)^2+(x-xc)^2)/Rc^2

But if I minimize the right part of the equation I am actually finding the point which would minimize the circles' radiuses, not the deformation of the radiuses, and I don't get a satisfying result.

Is it possible to solve this problem?

• You need to be more precise about what you want to minimize. You can obviously minimize $D_a,D_b$ by taking $D_c$ so that the third circle passes through a point of intersection of the first two. – almagest Jun 2 '16 at 10:59
• You have now been a member for a year. Don't you think it is time you learned something about Mathjax meta.math.stackexchange.com/questions/5020/… – almagest Jun 2 '16 at 11:00
• @almagest I have been a member for 5 min. – Ben Jun 2 '16 at 11:12
• Ah, I guess the 1 year refers to other SE sites. You still need to learn some basic math formatting quickly. People here get impatient with badly formatted questions. – almagest Jun 2 '16 at 11:20

Let $(x_i,y_i)$ be the centers of the three given circles, $r_i$ their radii, $s_i$ the envisaged correction of $r_i$, and $(x,y)$ the prospective point of intersection of the corrected circles. Then you want to minimize $$f(s_1,s_2,s_3):=\sum_i s_i^2$$ under the three constraints $$(x-x_i)^2+(y-y_i)^2-(r_i+s_i)^2=0\qquad(1\leq i\leq3)\ .\tag{1}$$You therefore have to set up the Lagrangian $$\Phi(x,y,s_1,s_2,s_3):=f(s_1,s_2,s_3)-\sum_i\lambda_i\bigl((x-x_i)^2+(y-y_i)^2-(r_i+s_i)^2\bigr)$$ and solve the system consisting of $(1)$ and the five equations $${\partial\Phi\over\partial x}=0,\quad {\partial\Phi\over\partial y}=0,\qquad {\partial\Phi\over\partial s_i}=0 \quad(1\leq i\leq3)\ .$$ Good luck!