# Construction of Local Shape Adjustable Surfaces Using Quintic Trigonometric Bézier Curve

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Quintic Trigonometric Bézier Basis Function

**Theorem**

**1.**

- (a)
- Non-negativity: ${f}_{i}\left(t\right)\ge 0,i=0,1,2,3,4,5$
- (b)
- Partition of unity: ${\sum}_{i=0}^{5}{f}_{i}\left(t\right)=1$
- (c)
- Symmetry: ${f}_{i}(t,\alpha ,\beta )={f}_{5-i}(1-t,\alpha ,\beta )$ for $i=[0,5]$

**Proof.**

- (a)
- For $t\in [0,1]$ and $\alpha ,\beta $$\in [-4,1]$, then $1-sin\frac{\pi t}{2}\ge 0$, $1-cos\frac{\pi t}{2}\ge 0$,$sin\frac{\pi t}{2}\ge 0$, $cos\frac{\pi t}{2}\ge 0$, $1-\alpha sin\frac{\pi t}{2}\ge 0,\phantom{\rule{1.em}{0ex}}$$1-\beta cos\frac{\pi t}{2}\ge 0$,$4+\alpha -\alpha sin\frac{\pi t}{2}\ge 0$, $4+\beta -\beta sin\frac{\pi t}{2}\ge 0$.
- (b)
- ${\sum}_{i=0}^{5}{f}_{i}\left(t\right)={f}_{0}\left(t\right)+{f}_{1}\left(t\right)+{f}_{2}\left(t\right)+{f}_{3}\left(t\right)+{f}_{4}\left(t\right)+{f}_{5}\left(t\right)=1$.
- (c)
- $\begin{array}{cc}\hfill {f}_{0}(t,\alpha ,\beta )& ={(1-sin\frac{\pi t}{2})}^{4}(1-\alpha sin\frac{\pi t}{2})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={(1-cos\frac{\pi (1-t)}{2})}^{4}(1-\beta cos\frac{\pi (1-t)}{2})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={f}_{5}(1-t,\beta ,\alpha )\hfill \end{array}$

## 3. Construction of Biquintic Trigonometric Bézier Surface with Shape Parameter

**Remark**

**1.**

**Remark**

**2.**

## 4. G^{2} Continuity Conditions for Biquintic Trigonometric Bézier Surfaces

#### 4.1. Continuity in the u Direction

#### 4.2. Continuity in u and v Direction

#### 4.3. Continuity in the v Direction

## 5. Examples of G^{2} Smooth Continuity between Two Biquintic Trigonometric Bézier Surfaces

## 6. Constructing Swept Surfaces with Shape Parameters

## 7. Constructing Swung Surfaces with Shape Parameters

**Remark**

**3.**

## 8. Effect of Shape Parameters on Surface Using Mean Curvature Nephogram

#### 8.1. Coons Patch

## 9. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**G

^{2}continuity condition in the u-direction between two adjacent biquintic trigonometric Bézier surfaces at different scale factor and shape parameter values.

**Figure 9.**Effect of shape parameters ${\alpha}_{u}$, ${\beta}_{u}$ and ${\alpha}_{v}$, ${\beta}_{v}$ in surface designing shown with mean curvature without fixing boundary curves.

**Figure 10.**(

**a**) The surface ${R}_{1}(u,v)$ defined in Equation (44) with traditional Bézier basis function in the u-direction (blue color) and quintic trigonometric Bézier basis function in the v-direction (red color). (

**b**) The surface ${R}_{2}(u,v)$ defined in Equation (45) with traditional Bézier basis function in the v-direction and quintic trigonometric Bézier basis function in the u-direction. (

**c**) The Coon surface $T(u,v)$ defined in Equation (46), constructed by four quintic trigonometric Bézier curves. (

**d**) The final biquintic-Coons surface $S(u,v)$.

**Figure 11.**Effect of shape parameters ${\alpha}_{u}$, ${\beta}_{u}$ in surface design shown with mean curvature with fixed boundary curves.

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**MDPI and ACS Style**

Ammad, M.; Misro, M.Y.
Construction of Local Shape Adjustable Surfaces Using Quintic Trigonometric Bézier Curve. *Symmetry* **2020**, *12*, 1205.
https://doi.org/10.3390/sym12081205

**AMA Style**

Ammad M, Misro MY.
Construction of Local Shape Adjustable Surfaces Using Quintic Trigonometric Bézier Curve. *Symmetry*. 2020; 12(8):1205.
https://doi.org/10.3390/sym12081205

**Chicago/Turabian Style**

Ammad, Muhammad, and Md Yushalify Misro.
2020. "Construction of Local Shape Adjustable Surfaces Using Quintic Trigonometric Bézier Curve" *Symmetry* 12, no. 8: 1205.
https://doi.org/10.3390/sym12081205