Circle Center's Cartesian Coordinates

Date: 03/24/99 at 09:40:08
From: Andreu Cuartiella
Subject: Finding the center of a circle with two points and radius

The problem is to find the cartesian coordinates of a circle's centers 
knowing two points on its perimeter. I guess it will have two 
solutions. 

I've tried applying the general circle equation and/or right 
triangles, but the resulting formula is  really huge. I'm looking for 
the easiest solution.

Thank you very much for your help.

Date: 03/24/99 at 11:42:03
From: Doctor Rob
Subject: Re: Finding the center of a circle with two points and radius

Thanks for writing to Ask Dr. Math!

Sorry, but the general solution is, in fact, rather complicated.
If the equation is

   (x-h)^2 + (y-k)^2 = r^2

and the two points are (x1,y1) and (x2,y2), then you have two
simultaneous equations in the two unknowns h and k:

   (x1-h)^2 + (y1-k)^2 = r^2,
   (x2-h)^2 + (y2-k)^2 = r^2.

Now I would substitute H = h - x2, K = k - y2, X = x1 - x2,
Y = y1 - y2, and let D^2 = X^2 + Y^2, so D is the distance between the 
two points. Then

   (X-H)^2 + (Y-K)^2 = r^2
           H^2 + K^2 = r^2

Subtract the second from the first, and the quadratic terms in
H and K will be removed:

   X^2 + Y^2 - 2*X*H - 2*Y*K = 0
   D^2 = 2*X*H + 2*Y*K
   H = (D^2-2*Y*K)/(2*X)

Now you substitute that into either of the quadratic equations above,
and you will have one quadratic equation in the single unknown K:

   [(D^2-2*Y*K)/(2*X)]^2 + K^2 = r^2
     (D^2-2*Y*K)^2 + 4*X^2*K^2 = 4*X^2*r^2
                 D^2*(2*K-Y)^2 = X^2*(4*r^2-D^2)
                       2*K - Y = +-X*sqrt(4*r^2/D^2-1)
                             K = [Y+-X*sqrt(4*r^2/D^2-1)]/2

You have the two values of K for the two possible centers.  Putting
that back into the formula for H, you get the corresponding values
of H.

   H = (X^2-Y*[2*K-Y])/(2*X)
     = [X-+Y*sqrt(4*r^2/D^2-1)]/2

Now you can substitute their equals for H, K, X, Y, and D, and you
will have the formulae in terms of the original data for h and k.

If 2*r < D, and the diameter of the circle is less than the distance 
between the points, then the values of H and K are not real, for 
obvious geometric reasons. If 2*r = D, the center is (h,k) = 
([x1+x2]/2,[y1+y2]/2), the midpoint between the two given points. If 
2*r > D, there are two real solutions.

Sorry that the results are complicated! Of course, when numerical 
values are used, this method is not so messy.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/