A Multi-sided Bezier Patch with a Simple Control Structure

Tamas Varady; Peter Salvi; Gyorgy Kariko

doi:10.1111/cgf.12833

A Multi-sided Bezier Patch with a Simple Control Structure

Tamas Varady, Peter Salvi, Gyorgy Kariko

Research output: Contribution to journal › Article › peer-review

33 Scopus citations

Abstract

A new n-sided surface scheme is presented, that generalizes tensor product Bézier patches. Boundaries and corresponding cross-derivatives are specified as conventional Bézier surfaces of arbitrary degrees. The surface is defined over a convex polygonal domain; local coordinates are computed from generalized barycentric coordinates; control points are multiplied by weighted, biparametric Bernstein functions. A method for interpolating a middle point is also presented. This Generalized Bézier (GB) patch is based on a new displacement scheme that builds up multi-sided patches as a combination of a base patch, n displacement patches and an interior patch; this is considered to be an alternative to the Boolean sum concept. The input ribbons may have different degrees, but the final patch representation has a uniform degree. Interior control points - other than those specified by the user - are placed automatically by a special degree elevation algorithm. GB patches connect to adjacent Bézier surfaces with G1continuity. The control structure is simple and intuitive; the number of control points is proportional to those of quadrilateral control grids. The scheme is introduced through simple examples; suggestions for future work are also discussed.

Original language	English (US)
Pages (from-to)	307-317
Number of pages	11
Journal	Computer Graphics Forum
Volume	35
Issue number	2
DOIs	https://doi.org/10.1111/cgf.12833
State	Published - May 27 2016
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2022-06-01
Acknowledgements: This is a joint work by researchers at the Budapest University of Technology and Economics and a small technology company ShapEx Ltd., Budapest. The pictures were generated by a prototype system called Sketches. The dolphin model has been provided for us by Cindy Grimm (Washington University); the face model was designed by Supriya Chewle (KAUST, Saudi Arabia). The project was partially supported by the Hungarian Scientific Research Fund (OTKA, No. 101845).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

Computer Networks and Communications

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10.1111/cgf.12833

Cite this

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title = "A Multi-sided Bezier Patch with a Simple Control Structure",

abstract = "A new n-sided surface scheme is presented, that generalizes tensor product B{\'e}zier patches. Boundaries and corresponding cross-derivatives are specified as conventional B{\'e}zier surfaces of arbitrary degrees. The surface is defined over a convex polygonal domain; local coordinates are computed from generalized barycentric coordinates; control points are multiplied by weighted, biparametric Bernstein functions. A method for interpolating a middle point is also presented. This Generalized B{\'e}zier (GB) patch is based on a new displacement scheme that builds up multi-sided patches as a combination of a base patch, n displacement patches and an interior patch; this is considered to be an alternative to the Boolean sum concept. The input ribbons may have different degrees, but the final patch representation has a uniform degree. Interior control points - other than those specified by the user - are placed automatically by a special degree elevation algorithm. GB patches connect to adjacent B{\'e}zier surfaces with G1continuity. The control structure is simple and intuitive; the number of control points is proportional to those of quadrilateral control grids. The scheme is introduced through simple examples; suggestions for future work are also discussed.",

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note = "KAUST Repository Item: Exported on 2022-06-01 Acknowledgements: This is a joint work by researchers at the Budapest University of Technology and Economics and a small technology company ShapEx Ltd., Budapest. The pictures were generated by a prototype system called Sketches. The dolphin model has been provided for us by Cindy Grimm (Washington University); the face model was designed by Supriya Chewle (KAUST, Saudi Arabia). The project was partially supported by the Hungarian Scientific Research Fund (OTKA, No. 101845). This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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N2 - A new n-sided surface scheme is presented, that generalizes tensor product Bézier patches. Boundaries and corresponding cross-derivatives are specified as conventional Bézier surfaces of arbitrary degrees. The surface is defined over a convex polygonal domain; local coordinates are computed from generalized barycentric coordinates; control points are multiplied by weighted, biparametric Bernstein functions. A method for interpolating a middle point is also presented. This Generalized Bézier (GB) patch is based on a new displacement scheme that builds up multi-sided patches as a combination of a base patch, n displacement patches and an interior patch; this is considered to be an alternative to the Boolean sum concept. The input ribbons may have different degrees, but the final patch representation has a uniform degree. Interior control points - other than those specified by the user - are placed automatically by a special degree elevation algorithm. GB patches connect to adjacent Bézier surfaces with G1continuity. The control structure is simple and intuitive; the number of control points is proportional to those of quadrilateral control grids. The scheme is introduced through simple examples; suggestions for future work are also discussed.

AB - A new n-sided surface scheme is presented, that generalizes tensor product Bézier patches. Boundaries and corresponding cross-derivatives are specified as conventional Bézier surfaces of arbitrary degrees. The surface is defined over a convex polygonal domain; local coordinates are computed from generalized barycentric coordinates; control points are multiplied by weighted, biparametric Bernstein functions. A method for interpolating a middle point is also presented. This Generalized Bézier (GB) patch is based on a new displacement scheme that builds up multi-sided patches as a combination of a base patch, n displacement patches and an interior patch; this is considered to be an alternative to the Boolean sum concept. The input ribbons may have different degrees, but the final patch representation has a uniform degree. Interior control points - other than those specified by the user - are placed automatically by a special degree elevation algorithm. GB patches connect to adjacent Bézier surfaces with G1continuity. The control structure is simple and intuitive; the number of control points is proportional to those of quadrilateral control grids. The scheme is introduced through simple examples; suggestions for future work are also discussed.

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A Multi-sided Bezier Patch with a Simple Control Structure

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