# Generalized Developable Cubic Trigonometric Bézier Surfaces

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## Abstract

**:**

## 1. Introduction

## 2. Definition and Properties of Cubic Trigonometric Bézier Curves

#### 2.1. Cubic Trigonometric Bézier Basis Functions

**Definition**

**1.**

**Theorem**

**1.**

- (a)
- Non-negativity: ${w}_{i,3}\left(u\right)\ge 0,i=0,\dots ,3$.
- (b)
- Partition of unity: ${\sum}_{i=0}^{3}{w}_{i,3}\left(u\right)=1$.
- (c)
- Monotonicity: with the given parameter u, ${w}_{0,3}\left(u\right)$ and ${w}_{3,3}\left(u\right)$ are monotonically decreasing while ${w}_{1,3}\left(u\right)$ and ${w}_{2,3}\left(u\right)$ are monotonically increasing for the shape parameters ${\gamma}_{1}$ and ${\gamma}_{2}$, respectively.
- (d)
- Symmetry: ${w}_{i,3}(u,{\gamma}_{1},{\gamma}_{2})={w}_{3-i,3}(1-u,{\gamma}_{1},{\gamma}_{2})$ for $i=0,\dots ,3$.

#### 2.2. Construction of CT-Bézier Curve

**Definition**

**2.**

- (a)
- Boundary properties:$B(0;{\gamma}_{1},{\gamma}_{2})={b}_{0}$,$B(1;{\gamma}_{1},{\gamma}_{2})={b}_{3}$,${B}^{\prime}(0;{\gamma}_{1},{\gamma}_{2})=\frac{\pi}{2}(2+{\gamma}_{1})({b}_{1}-{b}_{0})$,${B}^{\prime}(1;{\gamma}_{1},{\gamma}_{2})=\frac{\pi}{2}(2+{\gamma}_{2})({b}_{3}-{b}_{2})$,${B}^{\u2033}(0;{\gamma}_{1},{\gamma}_{2})=\frac{{\pi}^{2}}{2}\left((1+2{\gamma}_{1}){b}_{0}-(2+2{\gamma}_{1}){b}_{1}+{b}_{2}\right)$,${B}^{\u2033}(1;{\gamma}_{1},{\gamma}_{2})=\frac{{\pi}^{2}}{2}\left((1+2{\gamma}_{2}){b}_{3}-(2+2{\gamma}_{2}){b}_{2}+{b}_{1}\right).$
- (b)
- Symmetry: ${b}_{0},{b}_{1},{b}_{2},{b}_{3}$ and ${b}_{3},{b}_{2},{b}_{1},{b}_{0}$ define the same CT-Bézier curve in different parameterizations, i.e.,$B(u;{\gamma}_{1},{\gamma}_{2};{b}_{0},{b}_{1},{b}_{2},{b}_{3})$ = $B(1-u;{\gamma}_{1},{\gamma}_{2};{b}_{3},{b}_{2},{b}_{1},{b}_{0})$, $u\in [0,1]$, ${\gamma}_{1},{\gamma}_{2}\in [-2,1]$.
- (c)
- Geometric invariance: The CT-Bézier curve has a shape that is independent of the selection of the coordination, i.e., Equation (2) satisfies the following two equations:$B(u;{\gamma}_{1},{\gamma}_{2};{b}_{0}+q,{b}_{1}+q,{b}_{2}+q,{b}_{3}+q)$ = $B(u;{\gamma}_{1},{\gamma}_{2};{b}_{0},{b}_{1},{b}_{2},{b}_{3})+q$,$B(u;{\gamma}_{1},{\gamma}_{2};{b}_{0}\ast T,{b}_{1}\ast T,{b}_{2}\ast T,{b}_{3}\ast T)$ = $B(u;{\gamma}_{1},{\gamma}_{2};{b}_{0},{b}_{1},{b}_{2},{b}_{3})\ast T$.where q is an arbitrary vector in ${\mathbb{R}}^{2}$ or ${\mathbb{R}}^{3}$ and T is an arbitrary $d\times d$ matrix, $d=2$ or 3.
- (d)
- Convex hull property: The entire segment of the CT-Bézier curve must lie inside the control polygon.

## 3. Construction of GDCT-Bézier Surfaces

#### 3.1. Dual Generation of Single-Parameter Family of Planes

#### 3.2. Generalized Enveloping Developable CT-Bézier Surface

#### 3.3. Generalized Spine Curve Developable CT-Bézier surface

#### 3.4. Developable Surface Interpolating Geodesic CT-Bézier Curve with Parameters

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

#### 3.5. Analysis Properties of the GDCT-Bézier Surface

## 4. Continuity Conditions between GDCT-Bézier Surfaces

#### 4.1. The ${G}^{1}$ Continuity Conditions of GDCT-Bézier Surfaces

**Theorem**

**4.**

**Proof.**

#### 4.2. Farin–Boehm ${G}^{2}$ Continuity Conditions of GDCT-Bézier Surfaces

**Theorem**

**5.**

**Proof.**

#### 4.3. ${G}^{2}$ Beta Continuity Conditions of GDCT-Bézier Surfaces

**Theorem**

**6.**

**Proof.**

## 5. Design Examples of GDCT-Bézier Surface

#### 5.1. Examples of Enveloping GDCT-Bézier Surfaces

- When modifying the value of ${\gamma}_{1}$ and keeping ${\gamma}_{2}$ unchanged, the $T(1;{\gamma}_{1},{\gamma}_{2})$ generator retains the same length and location. However, the length of the generator $T(0;{\gamma}_{1},{\gamma}_{2})$ becomes longer when we increase the value of ${\gamma}_{1}$, but its position remains unchanged.
- If the value of shape parameter ${\gamma}_{1}$ is constant and ${\gamma}_{2}$ is adjusted, the generator $T(0;{\gamma}_{1},{\gamma}_{2})$ retains the same length and position. However, the position of generator $T(1;{\gamma}_{1},{\gamma}_{2})$ also remains unchanged, but its length increases when we increase the value of ${\gamma}_{2}$.

#### 5.2. Examples of Spine Curve GDCT-Bézier Surfaces

- When modifying the value of ${\gamma}_{1}$, and keeping ${\gamma}_{2}$ unchanged, the $T(0;{\gamma}_{1},{\gamma}_{2})$ generator retains the same length and location. However, the length of the generator $T(1;{\gamma}_{1},{\gamma}_{2})$ becomes shorter when we increase the value of ${\gamma}_{1}$, but its position remains unchanged.
- If the value of shape parameter ${\gamma}_{1}$ is constant and ${\gamma}_{2}$ is adjusted, the generator $T(1;{\gamma}_{1},{\gamma}_{2})$ retains the same length. However, the length of the generator $T(0;{\gamma}_{1},{\gamma}_{2})$ becomes shorter when we increase the value of ${\gamma}_{2}$, but its position remains unchanged.

#### 5.3. Example of Developable Surface Interpolating Geodesic Cubic Trigonometric Bézier Curve with Parameters

#### 5.4. Example of Smooth Continuity Between Two Adjacent GDCT-Bézier Surfaces

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Cubic trigonometric (CT)-Bézier basis functions for different combinations of shape parameters. (

**a**) ${\gamma}_{1}=1$, ${\gamma}_{2}=1$, (

**b**) ${\gamma}_{1}=1$, ${\gamma}_{2}=-1$, (

**c**) ${\gamma}_{1}=-1$, ${\gamma}_{2}=1$, (

**d**) ${\gamma}_{1}=-2$, ${\gamma}_{2}=-2$.

**Figure 2.**The effect of parameter ${\gamma}_{1}$ on the $1\times 3$ enveloping developable surface (

**a**) ${\gamma}_{1}=1$, ${\gamma}_{2}=1$, (

**b**) ${\gamma}_{1}=0$, ${\gamma}_{2}=1$, (

**c**) ${\gamma}_{1}=-1$, ${\gamma}_{2}=1$, (

**d**) ${\gamma}_{1}=-2$, ${\gamma}_{2}=1$.

**Figure 3.**The effect of parameter ${\gamma}_{2}$ on the $1\times 3$ enveloping developable surface (

**a**) ${\gamma}_{1}=1$, ${\gamma}_{2}=1$, (

**b**) ${\gamma}_{1}=1$, ${\gamma}_{2}=0$, (

**c**) ${\gamma}_{1}=1$, ${\gamma}_{2}=-1$ (

**d**) ${\gamma}_{1}=1$, ${\gamma}_{2}=-2$.

**Figure 4.**The effects of parameter ${\gamma}_{1}$ on the $1\times 3$ tangent of the spine curve developable surface (

**a**) ${\gamma}_{1}=0$, ${\gamma}_{2}=0$, (

**b**) ${\gamma}_{1}=-0.7$, ${\gamma}_{2}=0$, (

**c**) ${\gamma}_{1}=-1.4$, ${\gamma}_{2}=0$, (

**d**) ${\gamma}_{1}=-2$, ${\gamma}_{2}=0$.

**Figure 5.**The effects of parameter ${\gamma}_{2}$ on the $1\times 3$ tangent of the spine curve developable surface (

**a**) ${\gamma}_{1}=0$, ${\gamma}_{2}=0$, (

**b**) ${\gamma}_{1}=0$, ${\gamma}_{2}=-0.7$, (

**c**) ${\gamma}_{1}=0$, ${\gamma}_{2}=-1.4$, (

**d**) ${\gamma}_{1}=0$, ${\gamma}_{2}=-2$.

**Figure 6.**The effects of parameter ${\gamma}_{1}$ on the developable surface through the geodesic (

**a**) ${\gamma}_{1}=1$, ${\gamma}_{2}=1$, (

**b**) ${\gamma}_{1}=0$, ${\gamma}_{2}=1$, (

**c**) ${\gamma}_{1}=-1$, ${\gamma}_{2}=1$, (

**d**) ${\gamma}_{1}=-2$, ${\gamma}_{2}=1$.

**Figure 7.**The effects of parameter ${\gamma}_{2}$ on the developable surface through the geodesic (

**a**) ${\gamma}_{1}=1$, ${\gamma}_{2}=1$, (

**b**) ${\gamma}_{1}=1$, ${\gamma}_{2}=0$, (

**c**) ${\gamma}_{1}=1$, ${\gamma}_{2}=-1$, (

**d**) ${\gamma}_{1}=1$, ${\gamma}_{2}=-2$.

**Figure 8.**${G}^{1}$ smooth continuity between two enveloping generalized developable cubic trigonometric (GDCT)-Bézier surfaces. (

**a**) ${\gamma}_{1,1}=0$, ${\gamma}_{2,1}=0$, ${\gamma}_{1,2}=0$, ${\gamma}_{2,2}=1$, (

**b**) ${\gamma}_{1,1}=0$, ${\gamma}_{2,1}=0$, ${\gamma}_{1,2}=0$, ${\gamma}_{2,2}=-2$.

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**MDPI and ACS Style**

Ammad, M.; Misro, M.Y.; Abbas, M.; Majeed, A.
Generalized Developable Cubic Trigonometric Bézier Surfaces. *Mathematics* **2021**, *9*, 283.
https://doi.org/10.3390/math9030283

**AMA Style**

Ammad M, Misro MY, Abbas M, Majeed A.
Generalized Developable Cubic Trigonometric Bézier Surfaces. *Mathematics*. 2021; 9(3):283.
https://doi.org/10.3390/math9030283

**Chicago/Turabian Style**

Ammad, Muhammad, Md Yushalify Misro, Muhammad Abbas, and Abdul Majeed.
2021. "Generalized Developable Cubic Trigonometric Bézier Surfaces" *Mathematics* 9, no. 3: 283.
https://doi.org/10.3390/math9030283