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.
We present a novel and effective method for modeling a developable surface to simulate paper bending in interactive and animation applications. The method exploits the representation of a developable surface as the envelope of rectifying planes of a curve in 3D, which is therefore necessarily a geodesic on the surface. We manipulate the geodesic to provide intuitive shape control for modeling paper bending. Our method ensures a natural continuous isometric deformation from a piece of bent paper to its flat state without any stretching. Test examples show that the new scheme is fast, accurate, and easy to use, thus providing an effective approach to interactive paper bending. We also show how to handle non-convex piecewise smooth developable surfaces. We model developable surfaces by wrapping a planar figure around cones and cylinders. Complicated developables can be constructed by successive mappings using cones and cylinders of different sizes and shapes. We also propose an intuitive control mechanism, which allows a user to select an arbitrary point on the planar figure and move it to a new position. Numerical techniques are then used to find a cone or cylinder that produces the required mapping. Several examples demonstrate the effectiveness of our technique. An effective technique for modeling the bending of paper using cones and cylinders.A methodology for producing complicated developable surfaces from a planar figure.Interactive control of paper bending from a user-specified displacement. We present a new approach to geometric modeling with developable surfaces and the design of curved-creased origami. We represent developables as splines and express the nonlinear conditions relating to developability and curved folds as quadratic equations. This allows us to utilize a constraint solver, which may be described as energy-guided projection onto the constraint manifold, and which is fast enough for interactive modeling. Further, a combined primal-dual surface representation enables us to robustly and quickly solve approximation problems. Motivated by applications in architecture and manufacturing, we discuss the problem of covering a freeform surface by single curved panels. This leads to the new concept of semi-discrete surface representation, which constitutes a link between smooth and discrete surfaces. The basic entity we are working with is the developable strip model. It is the semi-discrete equivalent of a quad mesh with planar faces, or a conjugate parametrization of a smooth surface. We present a B-spline based optimization framework for efficient computing with D-strip models. In particular we study conical and circular models, which semi-discretize the network of principal curvature lines, and which enjoy elegant geometric properties. Together with geodesic models and cylindrical models they offer a rich source of solutions for surface panelization problems. We study the performance of algorithms for freeform surface fitting when different error terms are used as quadratic approximations to the squared orthogonal distances from data points to the fitting surface. We review the TD error term and the SD error term in surface fitting to point clouds, present robust surface fitting algorithms using the TD error term and a new variant of the SD error term. We report experimental results on comparing them with the prevailing PD error term in the setting of fitting B-spline surfaces to point cloud data. We conclude that using the TD error term and the SD error term leads to surface fitting algorithms that converge much faster than using the PD error term.