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 investigate how to define a triangulated ruled surface interpolating two polygonal directrices that will meet a variety of optimization objectives which originate from many CAD CAM and geometric modeling applications. This optimal triangulation problem is formulated as a combinatorial search problem whose search space however has the size tightly factorial to the numbers of points on the two directrices. To tackle this bound, we introduce a novel computational tool called multi-layer directed graph and establish an equivalence between the optimal triangulation and the single-source shortest path problem on the graph. Well known graph search algorithms such as the Dijkstra’s are then employed to solve the single-source shortest path problem, which effectively solves the optimal triangulation problem in O(mn) time, where n and m are the numbers of vertices on the two directrices respectively. Numerous experimental examples are provided to demonstrate the usefulness of the proposed optimal triangulation problem in a variety of engineering applications. AbstractSurface development is used in many manufacturing planning operations, e.g. for garments, ships and automobiles. However, most freeform surfaces used in design are not developable, and therefore the developed patterns are not isometric to the designed surface. In some domains, the CAD model is created by skinning operations that interpolate smooth strips between a specified set of skeleton curves. In this paper, we propose a method to approximate a strip with a developable surface between the two space curves bounding it. We allow one of the bounding curves to be perturbed within a controllable tolerance and meet some other special engineering requirements. We formulate the problem as a combination of a discrete combinatorial optimization problem and a constrained nonlinear optimization problem, and propose an efficient iterative approach to solve the problem. We present the first sketch-based modeling method for developable surfaces with pre-designed folds, such as garments or leather products. The main challenge we address for building folded surfaces from sketches is that silhouette strokes on the sketch correspond to discontinuous sets of non-planar curves on the 3D model. We introduce a new zippering algorithm for progressively identifying silhouette edges on the model and tying them to silhouette strokes. Our solution ensures that the strokes are fully covered and optimally sampled by the model. This new method, interleaved with developability optimization steps, is implemented in a multiview sketching system where the user can sketch the contours of internal folds in addition to the usual silhouettes, borders, and seam lines. All strokes are interpreted as hard constraints, while developability is only optimized. The developability error map we provide then enables users to add local seams or darts where needed and progressively improve their design. This makes our method robust, even to coarse input for which no fully developable solution exists. Abstract The use of developable surfaces in ship design is of engineering importance because they can be easily manufactured without stretching or tearing, or without the use of heat treatment. In some cases, a ship hull can be entirely designed with the use of developable surfaces. In this paper, a method to create a quasi-developable B -spline surface between two limit curves is presented. The centreline, chines and sheer lines of a vessel are modelled as B -spline curves. Between each pair of these boundary curves or directrix lines, the generator lines or rulings are created and a quasi-developable B -spline surface containing the rulings is defined. A procedure based on multiconic development is used to modify the directrix lines in case the rulings intersect inside the boundary curves, avoiding non-developable portions of the surface. B -spline curves and surfaces are widely used today in practically all the design and naval architecture computer programs. Some examples of ship hulls entirely created with developable surfaces are presented. A common operation in clothing and shoe design is to design a folding pattern over a narrow strip and then superimpose it with a smooth surface; the shape of the folding pattern is controlled by the boundary curve of the strip. Previous research results studying folds focused mostly on cloth modeling or in animations, which are driven more by visual realism, but allow large elastic deformations and usually completely ignore or avoid the surface developability issue. In reality, most materials used in garment and shoe industry are inextensible and uncompressible and hence any feasible folded surface must be developable, since it eventually needs to be flattened to its 2D pattern for manufacturing. Borrowing the classical boundary triangulation concept from descriptive geometry, this paper describes a computer-based method that automatically generates a specialized boundary triangulation approximation of a developable surface that interpolates a given strip. The development is achieved by geometrically simulating the folding process of the sheet as it would occur when rolled from one end of the strip to the other. Ample test examples are presented to validate the feasibility of the proposed method.