Interpolation with developable Bézier patches

Aumann, Günter

doi:10.1016/0167-8396(91)90014-3

Cited by 102 publications

(39 citation statements)

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“…Research related to Computer Aided Geometric Design, in particular those concerning the design and approximation of developable surfaces, can be found in [18][19][20][21][22][23][24][25][26][27]. Most of them are in terms of NURBS or its special case -B-spline or Bézier surfaces [18][19][20][21][22][23][24]. Aumann [18] proposed the condition under which a developable Bézier surface can be constructed with two boundary curves.…”

Section: Related Workmentioning

confidence: 99%

“…Most of them are in terms of NURBS or its special case -B-spline or Bézier surfaces [18][19][20][21][22][23][24]. Aumann [18] proposed the condition under which a developable Bézier surface can be constructed with two boundary curves. The boundary curves in his approach are restricted to lie in parallel planes; the projection of the boundary curves on the x-y plane must be a rectangle.…”

Section: Related Workmentioning

confidence: 99%

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Achieving developability of a polygonal surface by minimum deformation: a study of global and local optimization approaches

Wang

Tang

2004

Vis Comput

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Surface developability is required in a variety of applications in product design, such as clothing, ship hulls, automobile parts, etc. However, most current geometric modeling systems using polygonal surfaces ignore this important intrinsic geometric property. This paper investigates the problem of how to minimally deform a polygonal surface to attain developability, or the so called developability-by-deformation problem. In our study, this problem is first formulated as a global constrained optimization problem, and a penalty function based numerical solution is proposed for solving this global optimization problem. Next, as an alternative to the global optimization approach which usually requires lengthy computing time, we present an iterative solution based on a local optimization criterion which achieves near real-time computing speed. Both approaches preserve the topology and continuity of the original polygonal surface in the case when more than one individual polygonal patches comprise the surface. Experimental examples are provided to demonstrate the functionality of the proposed two approaches as well as their comparison in terms of computing cost, effectiveness of attaining developability, dimensional difference between the surfaces before and after the optimization, and other important aspects.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Achieving developability of a polygonal surface by minimum deformation: a study of global and local optimization approaches

Wang

Tang

2004

Vis Comput

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“…Basically, an exact surface-parametric or algebraic-is defined to interpolate the given points or curves. Bezier or B-spline surfaces are the most used ones, and the developability is enforced by nonlinear constraints [10], [11], [12], [13], [14], [15]. To facilitate the modeling task, some novel algebraic tools were proposed.…”

Section: Background and Previous Workmentioning

confidence: 99%

Quasi-Developable Mesh Surface Interpolation via Mesh Deformation

Tang

Chen

2009

IEEE Trans. Visual. Comput. Graphics

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Abstract-We present a new algorithm for finding a most "developable" smooth mesh surface to interpolate a given set of arbitrary points or space curves. Inspired by the recent progress in mesh editing that employs the concepts of preserving the Laplacian coordinates and handle-based shape editing, we formulate the interpolation problem as a mesh deformation process that transforms an initial developable mesh surface, such as a planar figure, to a final mesh surface that interpolates the given points and/or curves. During the deformation, the developability of the intermediate mesh is maintained by means of preserving the zero-valued Gaussian curvature on the mesh. To treat the high nonlinearity of the geometric constrains owing to the preservation of Gaussian curvature, we linearize those nonlinear constraints using Taylor expansion and eventually construct a sparse and overdetermined linear system that is subsequently solved by a robust least squares solution. By iteratively performing this procedure, the initial mesh is gradually and smoothly "dragged" to the given points and/or curves. The initial experimental tests have shown some promising aspects of the proposed algorithm as a general quasi-developable surface interpolation tool.

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“…One approach is to prove characterizing equations for the free form surfaces to be developable. The readers can refer to papers (Nolan, 1971, Aumann, 1991, Chalfant and Maekawa, 1998. However, the complex system of coupled equations is very difficult for the design of developable Bézier surfaces in a CAD system.…”

Section: Introductionmentioning

confidence: 99%