Get the best fit circle if radius is specified (constrained) I am trying to get a least-square circles fit. That requires $(x_c, y_c,r)$. I know that for that the system $S = \sum_{i=1}^n{[(x_i-x_c)^2 + (y_i - b)^2 - r^2]^2}$. The way to do this is to equate the partial derivatives of the unknowns $(\partial S / \partial x_c, \partial S / \partial y_c, \partial S / \partial r)$ to $0$ and solve the system of equations.
What to do when $r$ is known? My solution gives a system with two quadratic equations of order $(x_c^3, y_c^3)$ and I was wondering if there is an easier way. Should I move to a numerical solution?
 A: As you wrote, you have $n$ data points $(x_,y_i)$ and you want to minimize
$$S = \sum_{i=1}^n \Big[(X-x_i)^2 + (Y-y_i)^2 - R^2\Big]^2$$ As usual, we need reasonable estimates of $(X,Y)$.
In a first step, consider the $n$ equations
$$F_i=(X-x_i)^2 + (Y-y_i)^2 - R^2=0$$ and buid the $\frac{1}2 n(n-1)$ equations
$$G_{ij}=F_i-F_j=2(x_j-x_i)X+2(y_j-y_i)Y=(x_j^2+y_j^2)-(x_i^2+y_i^2)$$ where $i$ varies from $1$ to $(n-1)$ and $j$ from $(i+1)$ to $n$. This is a very simple problem.
For illustration, using the data taken from this paper (page $2$). this gives
$X_0=3.06030$ and $Y_0=0.743607$ which are almost exactly the results obtained using  Gruntz's procedure.
Now, we can go back to the original problem which means that we need to solve
$$\frac{\partial S}{\partial X}=4\sum_{i=1}^n (X-x_i)\Big[(X-x_i)^2 + (Y-y_i)^2 - R^2\Big]=0$$
$$\frac{\partial S}{\partial Y}=4\sum_{i=1}^n (Y-y_i)\Big[(X-x_i)^2 + (Y-y_i)^2 - R^2\Big]=0$$ which is very simple using Newton-Raphson method starting with $(X_0,Y_0)$ as initial guesses.
Using these values for a few values of $R$
$$\left(
\begin{array}{ccc}
R & X & Y \\
 3.7 &  3.11165 & 0.800459 \\
 3.8 &  3.10106 & 0.788820 \\
 3.9 &  3.08933 & 0.775878 \\
 4.0 &  3.07628 & 0.761420 \\
 4.1 &  3.06172 & 0.745189 \\
 4.2 &  3.04537 & 0.726870 \\
 4.3 &  3.02693 & 0.706070
\end{array}
\right)$$
