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

Infinitesimal isometries on developable surfaces and asymptotic theories for thin developable shells

We perform a detailed analysis of first order Sobolev-regular infinitesimal isometries on developable surfaces without affine regions. We prove that given enough regularity of the surface, any first order infinitesimal isometry can be matched to an infinitesimal isometry of an arbitrarily high order. We discuss the implications of this result for the elasticity of thin developable shells

Spontaneous curvature cancellation in forced thin sheets

In this paper we report numerically observed spontaneous vanishing of mean curvature on a developable cone made by pushing a thin elastic sheet into a circular container. We show that this feature is independent of thickness of the sheet, the supporting radius and the amount of deflection. Several variants of developable cone are studied to examine the necessary conditions that lead to the vanishing of mean curvature. It is found that the presence of appropriate amount of radial stress is necessary. The developable cone geometry somehow produces the right amount of radial stress to induce just enough radial curvature to cancel the conical azimuthal curvature. In addition, the circular symmetry of supporting container edge plays an important role. With an elliptical supporting edge, the radial curvature overcompensates the azimuthal curvature near the minor axis and undercompensates near the major axis. Our numerical finding is verified by a crude experiment using a reflective plastic sheet. We expect this finding to have broad importance in describing the general geometrical properties of forced crumpling of thin sheets.Comment: 13 pages, 12 figures, revtex