12/12/2003

Other references

J. F. Crawford

A NON-ITERATIVE METHOD FOR FITTING CIRCULAR ARCS TO MEASURED POINTS

NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH  211: (1) 223-225 1983

Equation 4:

X0^2 + Y0^2 – R^2 = (1/N)*(2*X0*Sx + 2*Y0*Sy – Sxx – Syy)

Simultaneous equations to solve for center:

X0*(Sxx – Sx^2/N) + Y0*(Sxy – Sx*Sy/N) = .5 * (Sxxx + Sxyy – Sx*(Sxx + Syy)/N)

X0*(Sxy – Sx*Sy/N) + Y0*(Syy – Sy^2/N) = .5 * (Sxxy + Syyy – Sy*(Sxx + Syy)/N)

Let:

F1 = Sxy - Sx*Sy/N

F2 = (Sxx + Syy)/N

F3 = (1/N)*(2*X0*Sx + 2*Y0*Sy – Sxx – Syy)

Then

X0^2 + Y0^2 – R^2 = F3

X0 * (Sxx – Sx^2/N) + Y0*F1 = .5 * (Sxxx + Sxyy – Sx*F2)

X0*F1 + Y0 * (Syy – Sy^2/N) = .5 * (Sxxy + Syyy – Sy*F2)

Let:

F4 = Sxx – Sx^2/N

F5 = Syy – Sy^2/N

F6 = Sxxx + Sxyy – Sx*F2

F7 = Sxxy + Syyy – Sy*F2

Then:

X0^2 + Y0^2 – R^2 = F3

X0 * F4 + Y0 * F1 = .5 * F6

X0 * F1 + Y0 * F5 = .5 * F7

Let:

Det(a, b, c, d) = a * d – b * c

F8 = Det (F4, F1, F1, F5)

Then:

X0 = Det(.5 * F6, F1, .5 * F7, F5) / F8

Y0 = Det(F4, .5 * F6, F1, .5 * F7) / F8

R = sqrt (X0^2 + Y0^2 – F3)