We introduce the linearized perturbation method to find the center of a set of concentric circular fringes. The mathematical foundations of the method presented here are instrumental in the design of a two-dimensional interferometric position sensor, which uses the geometry of the fringe pattern rather than the wavelength. A wavelength-independent physical characteristic of the fringe pattern is derived, which relates the radii of successive fringes. The linearized perturbation method is formulated for multiple fringes as a constrained quadratic optimization problem. Experimental calibrations have shown that using physical constraint information significantly improves the precision on the center position.