Degree elevation and developable Bézier surfaces

“…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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“…However, each control point pair is limited to be coplanar in the algorithm, reducing its modeling capability. The first and third groups of methods are likely to suffer from fewer number of free design parameters compared to the second one, but this drawback can be improved using the technique of degree elevation [15]. Chu and Chen [17] proposed that developable composite surfaces are constructed by joining adjacent patches along their end rulings while maintaining certain continuity and developability across the boundary, but the DOF were not characterized for all the continuity conditions.…”

Section: Introductionmentioning

confidence: 99%