Skeleton curves and surfaces have many applications in computer aided design and analysis. Construction of skeletons is an active area of research. We consider the inverse problem that of recovering boundary surfaces from given skeleton elements. The skeleton of any 3D object will, in general, consist of curves and surfaces. Therefore, any boundary reconstruction algorithm must systematically process the surfaces generated by the skeletal curves and the skeletal surfaces. In this paper (Part I) we present algorithms for reconstructing boundary surfaces corresponding to skeletal curves. Implemented examples are also included. In a companion paper (Part II) we consider skeletal elements that are surfaces.