Enhancement of a multi-sided Bézier surface representation. (English) Zbl 1375.65032
Summary: A new multi-sided surfacing scheme – the Generalized Bézier (GB) patch – has been introduced recently by the first author et al. [“A multi-sided Bézier patch with a simple control structure”, Comput. Graph. Forum 35, No. 2, 307–317 (2016; doi:10.1111/cgf.12833)]. The patch is created over a polygonal domain, its parameterization is defined by means of generalized barycentric coordinates. It has a simple control structure; the control points are associated with a combination of bi-parametric Bernstein functions, multiplied by rational terms. GB patches are compatible with adjacent quadrilateral Bézier patches and inherit most of their properties.
In this paper we present an enhanced version of the former scheme. The control structure has been slightly modified, yielding a perfect generalization of quadrilateral patches. The parameterization has also been altered, matching a concept of how quadrilateral domains transform to \(n\)-sided polygons. We propose improved blending functions and investigate how the weight deficiency of the basis functions can be distributed amongst control points. The former rational weighting functions have been modified to support not only \(G^1\), but higher degree continuity between adjacent patches. After briefly discussing how degree reduction and elevation proceed, we present algorithms to automatically create and optimize the internal control points of GB patches. A few simple examples and suggestions for future work conclude the paper.
In this paper we present an enhanced version of the former scheme. The control structure has been slightly modified, yielding a perfect generalization of quadrilateral patches. The parameterization has also been altered, matching a concept of how quadrilateral domains transform to \(n\)-sided polygons. We propose improved blending functions and investigate how the weight deficiency of the basis functions can be distributed amongst control points. The former rational weighting functions have been modified to support not only \(G^1\), but higher degree continuity between adjacent patches. After briefly discussing how degree reduction and elevation proceed, we present algorithms to automatically create and optimize the internal control points of GB patches. A few simple examples and suggestions for future work conclude the paper.
MSC:
| 65D17 | Computer-aided design (modeling of curves and surfaces) |
| 65D10 | Numerical smoothing, curve fitting |
Keywords:
general topology surfaces; multi-sided Bézier patches; barycentric parameterization; surface fitting; numerical example; algorithmReferences:
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