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.
A particular problem in industrial design is modeling developable strips bounded by design curves. This problem is frequently encountered in fabrication with inextensible materials @cite_5 . One example is the ship-hull design with a few dominating feature curves as input and developable surface patches are computed to interpolate the feature curves @cite_20 . Other industrial applications include shoe design and garment design. When the input curves are polylines, a developable mesh surface is created. Tang and Wang @cite_12 simulated the unfolding process of a bendable sheet to generate a bridge triangulation of vertices of input polylines to form a developable surface. In order to find a global optimum in the solution space of all reasonable triangulations, a developable triangulation problem is formulated as a graph problem and the Dijkstra's algorithm is utilized to solve it @cite_18 . proposed a local-global method which allows a perturbation of mesh vertices to optimize surface developability @cite_0 .
@cite_18 @cite_0 @cite_5 @cite_20 @cite_12
1987836319 2157953584 2023544486 2222430368
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. Design using free-form developable surfaces plays an important role in the manufacturing industry. Currently most commercial systems can only support converting free-form surfaces into approximate developable surfaces. Direct design using developable surfaces by interpolating descriptive curves is much desired in industry. In this paper, by enforcing a propagation scheme and observing its nesting and recursive nature, a dynamic programming method is proposed for the design task of interpolating 3D boundary curves with a discrete developable surface. By using dynamic programming, the interpolatory discrete developable surface is obtained by globally optimizing an objective that minimizes tangent plane variations over a boundary triangulation. The proposed method is simple and effective when used in industry. Experimental results are presented that demonstrate its practicality and efficiency in industrial design. 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. In CAGD the design of a surface that interpolates an arbitrary quadrilateral mesh is definitely a challenging task. The basic requirement is to satisfy both criteria concerning the regularity of the surface and aesthetic concepts.With regard to aesthetic quality, it is well known that interpolatory methods often produce shape artifacts when the data points are unevenly spaced. In the univariate setting, this problem can be overcome, or at least mitigated, by exploiting a proper non-uniform parametrization, that accounts for the geometry of the data. Recently the same principle has been generalized and proven to be effective in the context of bivariate interpolatory subdivision schemes.In this paper, we propose a construction for parametric surfaces of good aesthetic quality and high smoothness interpolating quadrilateral meshes of arbitrary topology. In the classical tensor product setting the same parameter interval must be shared by an entire row or column of mesh edges. Conversely, in this paper, we assign a different parameter interval to each mesh edge. This particular structure, which we call an augmented parametrization, allows us to interpolate each section polyline at parameters values that prevent wiggling of the resulting curve or other interpolation artifacts and yields high quality interpolatory surfaces.The proposed method is generalization of the local univariate spline interpolants introduced in (2013a) and (2014), that have arbitrary continuity and arbitrary order of polynomial reproduction. The generated surfaces retain the same smoothness of the underlying class of univariate splines in the regular regions of the mesh (where, locally, all vertices have valence 4). Mesh regions containing vertices of valence other than 4 are covered with suitably defined surface patches joining the neighboring regular ones with G 1 - or G 2 -continuity. Interpolatory methods often produce shape artifacts for unevenly-spaced data points.A proper non-uniform parametrization can help overcome interpolation artifacts.Parametric surfaces of good aesthetic quality and high smoothness are constructed.The method relies on non-uniform parametrization and locally supported splines.The surfaces interpolate quadrilateral meshes with arbitrary topology.