The objective of this paper is to present a significant improvement to the approach of Duncan et al. to analyze the deformations of curves in sequences of 2-D images. This approach is based on the paradigm that high curvature points usually possess an anatomical meaning and are therefore good landmarks to guide the matching process, especially in the absence of a reliable physical or deformable geometric model of the observed structures. As Duncan's team, we therefore propose a method based on the minimization of an energy which tends to preserve the matching of high curvature points, while ensuring a smooth field of displacement vectors everywhere. The innovation of our work stems from the explicit description of the mapping between the curves to be matched, which ensures that the resulting displacement vectors actually map points belonging to the two curves, which was not the case in Duncan's approach. Moreover, the energy minimization is obtained through the mathematical framework of finite element analysis, which provides a rigorous and efficient numerical solution. Moreover we show that the generalization of our approach can be easily generalized in 3-D to analyze the deformations of surfaces. We have actually implemented the method in 2-D and we present the results of the tracking of a heart structure in a sequence of ultrtasound images.