A representation scheme is presented for shapes of three-dimensional curved objects whose surface curvatures are not continuous at all points, i.e., surfaces with C1 continuity only. The representation consists of a subset of lines of curvature and their intersections, called surface structure descriptors, organized in a three-layer hierarchy of increasing stringency. From these descriptors, a hierarchical parameterization of C1 surfaces leading to a global shape representation of the object is obtained in terms of enclosing boundaries of bumps and dents, ridge and valley lines, and peaks and pits (curvature extreme points). To handle the discontinuities of surface curvatures, we introduce a computational scheme in which gradient directions of surface curvatures at the discontinuities can be inferred. Simulation results on the computation of these descriptors from curved surfaces are shown. Potential applications of this representation scheme in computer vision tasks are discussed.