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.
Developable surface modeling has been widely studied in various disciplines. In developable object simulation, the triangular meshes are the most frequently used representations. proposed a physically-based method for animating developable surfaces with nonconformal faces @cite_28 . encoded a developable surface as a set of ruling lines and found positions of ruling lines by relaxing mean curvature bending energy @cite_25 . proposed a local operator to modify a triangular mesh into piecewise developales @cite_9 . A planar quadrilateral mesh is another type of discrete developable surface which is based on a solid theoretical foundation. proposed a method for modeling a developable surface with a quadrilateral mesh by optimizing face planarization and performing mesh subdivision in an alternative manner @cite_4 . make use of quadrilateral meshes and define a discrete orthogonal geodesic net to model developable surfaces @cite_19 .
@cite_4 @cite_28 @cite_9 @cite_19 @cite_25
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Developable surfaces are surfaces that can be unfolded into the plane with no distortion. Although ubiquitous in our everyday surroundings, modeling them using existing tools requires significant geometric expertise and time. Our paper simplifies the modeling process by introducing an intuitive sketch-based approach for modeling developables. We develop an algorithm that given an arbitrary, user specified 3D polyline boundary, constructed using a sketching interface, generates a smooth discrete developable surface that interpolates this boundary. Our method utilizes the connection between developable surfaces and the convex hulls of their boundaries. The method explores the space of possible interpolating surfaces searching for a developable surface with desirable shape characteristics such as fairness and predictability. The algorithm is not restricted to any particular subset of developable surfaces. We demonstrate the effectiveness of our method through a series of examples, from architectural design to garments. 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. Introduces a new surface representation for recognizing curved objects. The authors approach begins by representing an object by a discrete mesh of points built from range data or from a geometric model of the object. The mesh is computed from the data by deforming a standard shaped mesh, for example, an ellipsoid, until it fits the surface of the object. The authors define local regularity constraints that the mesh must satisfy. The authors then define a canonical mapping between the mesh describing the object and a standard spherical mesh. A surface curvature index that is pose-invariant is stored at every node of the mesh. The authors use this object representation for recognition by comparing the spherical model of a reference object with the model extracted from a new observed scene. The authors show how the similarity between reference model and observed data can be evaluated and they show how the pose of the reference object in the observed scene can be easily computed using this representation. The authors present results on real range images which show that this approach to modelling and recognizing 3D objects has three main advantages: (1) it is applicable to complex curved surfaces that cannot be handled by conventional techniques; (2) it reduces the recognition problem to the computation of similarity between spherical distributions; in particular, the recognition algorithm does not require any combinatorial search; and (3) even though it is based on a spherical mapping, the approach can handle occlusions and partial views. > 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.