It sounds to me that you are using the following equations for a point on a circle at time t:
x(t) = x0 + r sin(omega*t)
y(t) = y0 + r cos(omega*t)
where omega is the frequency that you are zipping around the circle. If you know the (fixed) radius, r, and the frequency omega, then giving any pair of x, y points at a known value of t, it is trivial to unambiguously figure out x0 and y0 (it is basically a subtraction problem). No need to worry about what quadrant you are in, just do the arithmetic.