Smooth quasi-developable surfaces bounded by smooth curves

Computing a quasi-developable strip surface bounded by design curves finds wide industrial applications. Existing methods compute discrete surfaces composed of developable lines connecting sampling points on input curves which are not adequate for generating smooth quasi-developable surfaces. We propose the first method which is capable of exploring the full solution space of continuous input curves to compute a smooth quasi-developable ruled surface with as large developability as possible. The resulting surface is exactly bounded by the input smooth curves and is guaranteed to have no self-intersections. The main contribution is a variational approach to compute a continuous mapping of parameters of input curves by minimizing a function evaluating surface developability. Moreover, we also present an algorithm to represent a resulting surface as a B-spline surface when input curves are B-spline curves.Comment: 18 page

Cartan Ribbonization of Surfaces and a Topological Inspection

We develop the concept of Cartan ribbons and a method by which they can be used to ribbonize any given surface in space by intrinsically flat ribbons. The geodesic curvature along the center curve on the surface agrees with the geodesic curvature of the corresponding Cartan development curve, and this makes a rolling strategy successful. Essentially, it follows from the orientational alignment of the two co-moving Darboux frames during the rolling. Using closed center curves we obtain closed approximating Cartan ribbons that contribute zero to the total curvature integral of the ribbonization. This paves the way for a particular simple topological inspection -- it is reduced to the question of how the ribbons organize their edges relative to each other. The Gauss-Bonnet theorem leads to this topological inspection of the vertices. Finally, we display two examples of ribbonizations of surfaces, namely of a torus using two ribbons, and of an ellipsoid using its closed curvature lines as center curves for the ribbons. The topological inspection of the torus ribbonization is particularly simple as it has no vertex points, giving directly the Euler characteristic $0$. The ellipsoid has $4$ vertices -- corresponding to the $4$ umbilical points -- each of degree one and each therefore contributing one-half to the Euler characteristic

Caravan Awnings: a Geometrical Problem

Two questions regardingthe design of caravan awnings were posed by a company.The company wishes to produce awnings with a pretty appearance. When an awning is attached to a caravan, some wrinkles could appear. We developed some methods to avoid the wrinkles. The problem is restricted to awnings which are made from one piece of cloth