Calculate coordinates of circle that best approximates intersection between 2 circles

Question

I have two circles that could intersect, and if is it the case, I would calculate the coordinates of the largest circle that approximates that intersection area.

I drew a sketch in svg to represent the problem:

<svg width="400" height="400">
    <line x1="85" y1="50" x2="140" y2="70" stroke="rgb(50,50,50)" stroke-width="1" />
    <circle cx="85" cy="50" r="50" stroke="rgba(50,50,50,0.8)" stroke-width="1" fill="rgba(0,0,150,0.3)" />
    <circle cx="140" cy="70" r="40" stroke="rgba(50,50,50,0.8)" stroke-width="1" fill="rgba(150,0,0,0.3)" />

    <circle cx="117" cy="62" r="16" stroke="rgba(50,50,50,0.8)" stroke-width="1" fill="rgba(00,150,0,0.3)" />
</svg>

you can try it online here: http://www.w3schools.com/svg/tryit.asp?filename=trysvg_line .

What I would, having the first two circles, is the center and the radius of third one (that in the example I drew by hand).

What do you mean by "approximating" the intersection area? - Your online SVG does not work in my browser, perhaps you can add an image to your question. — Martin R
The code in tryit does not match the code above, if you replace the code with that above you do get required diagram — jing3142
I'm sorry for the misunderstanding, but If I remove that parameter, the page doesn't work, it requires a 'filename' — marka.thore

Martin R Martin R · Accepted Answer · 2014-03-29T10:45:27

Let

A, r1 = center, radius of first circle
B, r2 = center, radius of second circle

be the given input data and

C, r3 = center, radius of third circle

be the largest circle that fits into the intersection of the first two circles.

Denote by

D = intersection point of first with third circle
E = intersection point of second with third circle

D and E are points on the line connecting the centers A and B. D has distance r1 from A and E has distance r2 from B. Therefore

D = A + r1 * (B - A)/dist(A, B)
E = B - r2 * (B - A)/dist(A, B)

from which follows

C  = (D + E)/2    = (A + B + (r1 - r2)*(B - A)/dist(A, B)) / 2
r3 = dist(D, E)/2 = (r1 + r2 - dist(A, B)) / 2

If r3 < 0 then the circles do not intersect at all.

(The above calculation assumes that none of the circles lies completely within the other circle.)

Calculate coordinates of circle that best approximates intersection between 2 circles

1 Answers