Find the nearest point of intersections of circles

I need to find the nearest point of intersections of 4 circles. Centers of circles don't change, only the radius of circles changes. I know every intersection of each circle but I don't know how to find the average point of intersections 4 circles. Cannot find intersection of 4 circles, because the measurement error is too high, so I need to find some average point.

[Matt J]

Why not use mean?

[Jan]

How is "average point of intersections 4 circles" mathematically defined? The 4 circles have 12 intersection points. If you have their positions, Matt J's auggestion sounds obvious.

Answer 1

[MJFcoNaN]

Hello,

You may need a different method than intersection. For example create a distance function under your own criterion. This is a simple one:

ori_x = [0   , 0   , -6.19, -6.1];
ori_y = [0   , 4.6 ,  4.6 ,  0  ];
r     = [2.55, 4.05,  6.55,  3.65];
[xx, yy] = ndgrid(-20:1e-1:20,-20:1e-1:20);
vv = NaN([size(xx),4]);
for ii=1:4
    vv(:,:,ii) = abs((xx-ori_x(ii)).^2+(yy-ori_y(ii)).^2-r(ii).^2);
end
vs = sum(vv,3);
vs(vs>100) = NaN;
contourf(xx,yy,vs)

[mm,I] = min(vs,[],"all","omitnan");
xx(I)

ans = -1.1000

yy(I)

ans = 0.6000

Answer 2

[Sam Chak]

@Marek Konopka

Some very simple calculations and math concepts (no looping) that you can definitely follow to find the intersections between the circles. Let's try an example with the red circle and the blue circle.

syms x y
% since red & blue circle aligned at the same y-axis, then pick x as variable
% intersections lie at the top of red circle (+) and bottom of blue circle (–)
redC = sqrt(2.55^2 - (x - 0)^2) +  0;
bluC = - sqrt(4.05^2 - (x - 0)^2) + 4.6;
eqn1 = redC == bluC;
solx = solve(eqn1);
solx = double(solx)

solx = 2×1

-2.2371 2.2371

soly = subs(redC, x, solx);
soly = double(soly)

soly = 2×1

1.2239 1.2239

You should be able to find 6 intersections using this concept of coaxial centers.

For non-coaxial centers, let's try finding the intersections between the red circle and the green circle:

% One intersection lie at the top of red circle (+) and bottom of green circle (–)
redC = sqrt(2.55^2 - (x - 0)^2) +  0;
grnC = - sqrt(6.55^2 - (x + 6.19)^2) + 4.6;
eqn2 = redC == grnC;
solx = solve(eqn2);
solx = double(solx)

solx = 0.0309

soly = subs(redC, x, solx);
soly = double(soly)

soly = 2.5498

% One intersection lie at the bottom of red circle (–) and bottom of green circle (–)
redC = - sqrt(2.55^2 - (x - 0)^2) +  0;
grnC = - sqrt(6.55^2 - (x + 6.19)^2) + 4.6;
eqn3 = redC == grnC;
solx = solve(eqn3);
solx = double(solx)

solx = -2.4325

soly = subs(redC, x, solx);
soly = double(soly)

soly = -0.7651

You can find another 6 intersections using this concept of non-coaxial centers.