Abstract
Based on a kind of Q-Bézier surfaces with shape parameters, the basic properties of the surfaces and the geometric significance of the shape parameters are analyzed. To resolve the problem of shape control and adjustment of composite surfaces, the continuity conditions for Q-Bézier surfaces of degree (m, n) are investigated. Taking advantage of the terminal properties of generalized Bernstein basis functions, we derive the conditions of \({G}^{1}\) and \({G}^{2}\) continuity between two adjacent Q-Bézier surfaces. In addition, the specific steps of smooth continuity between Q-Bézier surfaces and the shape adjustment function of shape parameters for composite surfaces are discussed. The modeling examples show that the proposed smooth continuity conditions are not only intuitive and easy to implement, but also greatly enhance the shape adjustability, which provide a useful method for constructing complex surfaces in engineering design.
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Acknowledgements
The authors are very grateful to the referees for their helpful suggestions and comments which have improved the paper. This work is supported by the National Natural Science Foundation of China (Nos. 51305344, 11626185). This work is also supported by the Project Supported by Natural Science Basic Research Plan in Shaanxi Province of China (No. 2017JM5048).
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Communicated by Cristina Turner.
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Hu, G., Bo, C. & Qin, X. Continuity conditions for tensor product Q-Bézier surfaces of degree (\(m,\, n\)). Comp. Appl. Math. 37, 4237–4258 (2018). https://doi.org/10.1007/s40314-017-0568-0
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DOI: https://doi.org/10.1007/s40314-017-0568-0