In many computer vision and image understanding problems, it is important to find a smooth surface that fits a set of given unstructured 3D data. Although approaches based on general deformable models give satisfactory results, in particular a local description of the surface, they involve large linear systems to solve when dealing with high resolution 3D images. The advantage of parametric deformable templates like superquadrics is their small number of parameters to describe a shape. However, the set of shapes described by superquadrics is too limited to approximate precisely complex surfaces. This is why hybrid models have been introduced to refine the initial approximation. This article introduces a deformable superquadric model based on a superquadric fit followed by a free-form deformation (FFD) to fit unstructured 3D points. At the expense of a reasonable number of additional parameters, free-form deformations provide a much closer fit and a volumetric deformation field. We first present the mathematical and algorithmic details of the method. Then, since we are mainly concerned with applications for medical images, we present a medical application consisting in the reconstruction of the left ventricle of the heart from a number of various 3D cardiac images. The extension of the method to track anatomical structures in spatio-temporal images (4D data) is presented in a companion article.