Discontinuities in the wavefront phase or amplitude affect curvature sensor (CS) response. These discontinuities may be the result of an improper alignment of the elements of a segmented mirror in either piston or tip-tilt. We describe the CS response with an analytical approach based on the Fresnel approximation of the diffracted field and show that a CS is capable of detecting misalignment among segments. The response model that we present leads to a fast algorithm for the measurement of segment positions. As an application, we focus on piston co-phasing. The paper concludes with the characterization of the integrated curvature signal, which simplifies the analysis of the CS response to misalignments. The low processing time and principal memory requirements of the algorithm make it suitable for extremely large telescopes (ELTs).