A simple algorithm for designing developable Bézier surfaces

Aumann, Günter

doi:10.1016/j.cagd.2003.07.001

Cited by 96 publications

(52 citation statements)

References 16 publications

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“…We have therefore characterised developability of a rational ruled surface in terms of blossoms: which is the family of developable surfaces found by Aumann [12], though in that paper the key issue was the use of an affine transformation between adjacent cells of the control net of the surface. This expression states the equality of two (n -l)-atic forms, which is equivalent to the equality of the respective symmetric (n -l)-affine forms, since the correspondence between blossoms and parametrizations is one-to-one,…”

Section: Proposition 3 the Ruled Surface Interpolating Between Two Bmentioning

confidence: 99%

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Non-degenerate Developable Triangular Bézier Patches

Cantón

Fernández–Jambrina

2012

Curves and Surfaces

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Section: Proposition 3 the Ruled Surface Interpolating Between Two Bmentioning

confidence: 99%

“…-Based on the de Casteljau algorithm: Chu and Séquin [11], Aumann [12], [13] and Fernández-J ambrina [14].…”

Section: Introductionmentioning

confidence: 99%

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Non-degenerate Developable Triangular Bézier Patches

Cantón

Fernández–Jambrina

2012

Curves and Surfaces

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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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“…Notably two classes of approaches exist: the first uses the primal representation of surfaces, and the second uses the dual representation of surfaces. The primal representation uses a tensor product surface of degree ð1; nÞ, and usually solves nonlinear characterizing equations to guarantee developability [8,23]. A developable surface can also be viewed as the envelope of a oneparameter family of tangent planes, and thus can be treated as a curve in a dual projective 3-space.…”

Section: D To 2d Pattern Transformationmentioning

confidence: 99%