The geometry of plane-based calibration methods is well understood, but some user interaction is often needed in practice for feature detection. This paper presents a fully automatic calibration system that uses patterns of pairs of concentric circles. The key observation is to introduce a geometric method that constructs a sequence of points strictly convergent to the image of the circle center from an arbitrary point. The method automatically detects the points of the pattern features by the construction method, and identify them by invariants. It then takes advantage of homological constraints to consistently and optimally estimate the features in the image. The experiments demonstrate the robustness and the accuracy of the new method.