A framework for the visual representation of three-dimensional free-form curved surfaces based on a special class of surface curves, herein called the surface structure curves, is in progress. By analyzing their properties, we attempt to construct a basis for describing the topographical structures of curved surfaces which give a global description of the surface geometry. Surface structure curves are a set of surface curves defined by using viewpoint-invariant features-surface curvatures (and their gradients and asymptotes) from differential geometry. From these surface structure curves, surface sketches by means of the topographical structure of ridge lines, valley lines and the enclosing boundaries of bumps and dents can be inferred. This paper proposes a view-point invariant representation scheme which provides a smooth surface sketch",--a natural parameterization of free-form curved surfaces. We define three types of surface structure points and five types of surface structure curves in terms of zero-crossings, asymptotes and gradients of the Gaussian and mean curvatures. We discuss their properties and usefulness in edge based segmentation and description of free-form curved surfaces. Some examples of surface sketches using the surface structure curves are shown.