Least squares approximation for two circles and two lines

Can anyone please tell me how to do a least squares approximation for two circles and two(or one) line to find their approximate intersection point? Thank you.

Answer 1

[Torsten]

Use fmincon to solve the optimization problem

min: sqrt((x*-x1*)^2+(y*-y1*)^2)+sqrt((x*-x2*)^2+(y*-y2*)^2)+sqrt((x*-x3*)^2+(y*-y3*)^2)+sqrt((x*-x4*)^2+(y*-y4*)^2)

under the constraints

(x1*-x1)^2+(y1*-y1)^2=R1^2
(x2*-x2)^2+(y2*-y2)^2=R2^2
x3*=x3+lambda*l_x3
y3*=y3+lambda*l_y3
x4*=x4+mu*l_x4
y4*=y4+mu*l_y4

Here,

x1,y1,R1,x2,y2,R2,x3,l_x3,y3,l_y3,x4,l_x4,y4,l_y4

are known parameters and

x1*,y1*,x2*,y2*,x3*,y3*,x4*,y4*,lambda, mu, x*,y*

are the unknowns to be solved for.

(x*,y*) will be the approximate intersection point.

Best wishes

Torsten.

[arisish roy]

Thank you for the solution but can you please tell me what lambda and mu are? (x1*-x1)^2+(y1*-y1)^2=R1^2 (x2*-x2)^2+(y2*-y2)^2=R2^2 are circle equations but I can't understand what lambda and mu are? Are they derived from line equations? My lines are of the form y=mx+b. Can you please tell me how to derive the parameters from that?

[Torsten]

Then replace constraints (3-6) by

y3*=m1*x3*+b1
y4*=m2*x4*+b2

and remove lambda and mu from the list of unknowns.

Best wishes

Torsten.