A novel technique on flexibility and adjustability of generalized fractional Bézier surface patch

Syed Ahmad Aidil Adha Said Mad Zain; Md Yushalify Misro; Syed Ahmad Aidil Adha Said Mad Zain; Md Yushalify Misro

doi:10.3934/math.2023026

AIMS Mathematics

2023, Volume 8, Issue 1: 550-589. doi: 10.3934/math.2023026

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Research article

A novel technique on flexibility and adjustability of generalized fractional Bézier surface patch

School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Gelugor, Pulau Pinang, Malaysia

Received: 06 July 2022 Revised: 10 September 2022 Accepted: 27 September 2022 Published: 09 October 2022
MSC : 65D17, 68U07

Designing complex surfaces is one of the major problems in industries such as the automotive, shipbuilding and aerospace industries. To solve this problem, continuity conditions between surfaces are applied to construct the complex surfaces. The geometric and parametric continuities are the two metrics that usually have been used in connecting surfaces. However, the conventional geometric and parametric continuities have significant limitations. The existing continuity conditions only allow the two surfaces to be joined at the end of the boundary point. Therefore, if the designers want to connect at any arbitrary line of the first surface, the designers must use the subdivision method to splice the surfaces. Nevertheless, this method is tedious and involves a high computational cost, especially when dealing with a higher degree order of surfaces. Thus, this paper presents fractional continuity of degree two (or $ F^2 $) for generalized fractional Bézier surfaces. The fractional parameter embedded in the generalized fractional Bézier basis functions will solve the mentioned limitation by introducing fractional continuity. The generalized fractional Bézier surface also has excellent shape parameters that can alter the shape of the surface without changing the control points. Thus, the shape parameters enable the control of the shape flexibility of the surfaces, while fractional parameters enable the control of the adjustability of the surfaces' size. The $ F^2 $ continuity for generalized fractional Bézier surfaces can become an easier and faster alternative to the subdivision method. Therefore, the fractional continuity for generalized fractional Bézier surfaces will be a good tool to generate complex surfaces due to its flexibility and adjustability of shape and fractional parameters.
- fractional Bézier basis functions,
- Bézier surfaces,
- shape and fractional parameters,
- fractional continuity,
- smooth continuity conditions
Citation: Syed Ahmad Aidil Adha Said Mad Zain, Md Yushalify Misro. A novel technique on flexibility and adjustability of generalized fractional Bézier surface patch[J]. AIMS Mathematics, 2023, 8(1): 550-589. doi: 10.3934/math.2023026

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Abstract

Designing complex surfaces is one of the major problems in industries such as the automotive, shipbuilding and aerospace industries. To solve this problem, continuity conditions between surfaces are applied to construct the complex surfaces. The geometric and parametric continuities are the two metrics that usually have been used in connecting surfaces. However, the conventional geometric and parametric continuities have significant limitations. The existing continuity conditions only allow the two surfaces to be joined at the end of the boundary point. Therefore, if the designers want to connect at any arbitrary line of the first surface, the designers must use the subdivision method to splice the surfaces. Nevertheless, this method is tedious and involves a high computational cost, especially when dealing with a higher degree order of surfaces. Thus, this paper presents fractional continuity of degree two (or $ F^2 $) for generalized fractional Bézier surfaces. The fractional parameter embedded in the generalized fractional Bézier basis functions will solve the mentioned limitation by introducing fractional continuity. The generalized fractional Bézier surface also has excellent shape parameters that can alter the shape of the surface without changing the control points. Thus, the shape parameters enable the control of the shape flexibility of the surfaces, while fractional parameters enable the control of the adjustability of the surfaces' size. The $ F^2 $ continuity for generalized fractional Bézier surfaces can become an easier and faster alternative to the subdivision method. Therefore, the fractional continuity for generalized fractional Bézier surfaces will be a good tool to generate complex surfaces due to its flexibility and adjustability of shape and fractional parameters.

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