# Adjustable Bézier Curves with Simple Geometric Continuity Conditions

## Abstract

**:**

## 1. Introduction

## 2. Blending Functions

#### 2.1. Construction of the Blending Functions

**Definition**

**1.**

#### 2.2. Properties of the Blending Functions

**Proposition**

**1.**

- (a)
- Degeneracy: When $k=s=\alpha =1$, the BL functions are the quartic Bernstein basis functions.
- (b)
- Non-negativity: When $\alpha \in (0,1]$, for any $k\begin{array}{c}(k\ge 1\end{array})$ and $s\begin{array}{c}\begin{array}{c}(\end{array}1\le s\le k\end{array})$, we have ${f}_{i}\ge 0$, $i=0,\dots ,4$.
- (c)
- Normalization: For any $k\begin{array}{c}(k\ge 1\end{array})$, $s\begin{array}{c}\begin{array}{c}(\end{array}1\le s\le k\end{array})$, and $\alpha \in (0,1]$, we have $\sum _{i=0}^{4}{f}_{i}}=1$.
- (d)
- Symmetry: For any $k\begin{array}{c}(k\ge 1\end{array})$, $s\begin{array}{c}\begin{array}{c}(\end{array}1\le s\le k\end{array})$, $\alpha \in (0,1]$, and $\begin{array}{c}t\in [0,1]\end{array}$, we have ${f}_{i}(t)={f}_{4-i}(1-t)$, and $i=0,\dots ,4$.
- (e)
- Endpoint property: For any $k\begin{array}{c}(k\ge 1\end{array})$, $s\begin{array}{c}\begin{array}{c}(\end{array}1\le s\le k\end{array})$, and $\alpha \in (0,1]$ we have$$\{\begin{array}{ccc}{f}_{0}(0)=1,& {f}_{i}(0)=0& (i=1,2,3,4),\\ {f}_{i}(1)=0& (i=0,1,2,3),& {f}_{4}(1)=1,\end{array}$$$$\{\begin{array}{cc}{f}_{0}^{(L)}(0)={(-1)}^{L}L!{C}_{m}^{L}\alpha ,\hspace{1em}{f}_{1}^{(L)}(0)={(-1)}^{L-1}L!{C}_{m}^{L}\alpha ,\hspace{1em}{f}_{i}^{(L)}(0)=0(i=2,3,4),\hfill & \hfill \text{\hspace{1em}\hspace{1em}\hspace{1em}(3a)}\\ {f}_{i}^{(L)}(1)=0(i=0,1,2),\hspace{1em}{f}_{3}^{(L)}(1)=-L!{C}_{m}^{L}\alpha ,\hspace{1em}{f}_{4}^{(L)}(1)=L!{C}_{m}^{L}\alpha .\hfill & \hfill \text{\hspace{1em}\hspace{1em}\hspace{1em}(3b)}\end{array}$$
- (f)
- Linear independence: For any $k\begin{array}{c}(k\ge 1\end{array})$, $s\begin{array}{c}\begin{array}{c}(\end{array}1\le s\le k\end{array})$, and $\alpha \in (0,1]$, the BL functions ${f}_{i}(t;k,s,\alpha )$, $i=0,\dots ,4$ are linearly independent.

**Proof.**

- (e)
- The conclusions in (2) are obvious. We only prove (3a) and (3b). If$$f(t)={a}_{n-1}+{a}_{n}{t}^{n}+{a}_{n+1}{t}^{n+1}+\dots +{a}_{m}{t}^{m},$$$${f}^{(L)}(0)=\{\begin{array}{ll}0,& 1\le L<n,\\ L!{a}_{L},& n\le L\le m.\end{array}$$

- (f)
- Let us consider the linear combination$$\sum _{i=0}^{4}{a}_{i}}{f}_{i}=0,$$$${a}_{0}{B}_{0}^{m}+[(1-\alpha ){a}_{0}+\alpha {a}_{1}]{\displaystyle \sum _{i=1}^{k}{B}_{i}^{m}}+{a}_{2}{\displaystyle \sum _{i=k+1}^{k+s}{B}_{i}^{m}}+[\alpha {a}_{3}+(1-\alpha ){a}_{4}]{\displaystyle \sum _{i=k+s+1}^{2k+s}{B}_{i}^{m}}+{a}_{4}{B}_{m}^{m}=0.$$

## 3. Adjustable Bézier Curves

#### 3.1. Construction of the Adjustable Bézier Curves

**Definition**

**2.**

**Proposition**

**2.**

**Proof.**

**Remark**

**1.**

#### 3.2. Properties of the Adjustable Bézier Curves

- (1)
- Convex hull property: The adjustable Bézier curves lie inside the convex hull of the control points. This is true, since the BL functions are nonnegative on $[0,1]$ and sum to 1.
- (2)
- Geometric invariance: From (13), we know that the adjustable Bézier curves are affine combinations of their control points. Thus, their shape is independent of the choice of the coordinate system.
- (3)
- Symmetry: The points ${\mathit{V}}_{i}$, $i=0,\dots ,4$, and ${\mathit{V}}_{i}$, $i=4,\dots ,0$, define two adjustable Bézier curves with the same shape but different parameterization.
- (4)
- Geometric property at the endpoints: From (2), (3) and (12) we get$$\{\begin{array}{l}\mathit{f}(0)={\mathit{V}}_{0},\\ \mathit{f}(1)={\mathit{V}}_{4}.\end{array}$$$$\{\begin{array}{l}{\mathit{f}}^{(L)}(0)={(-1)}^{L-1}L!{C}_{m}^{L}\alpha ({\mathit{V}}_{1}-{\mathit{V}}_{0}),\\ {\mathit{f}}^{(L)}(1)=L!{C}_{m}^{L}\alpha ({\mathit{V}}_{4}-{\mathit{V}}_{3}),\text{\hspace{1em}}1\le L\le k.\end{array}$$

**Remark**

**2.**

- (5)
- Shape adjustability property: Even if the control points of an adjustable Bézier curve are fixed, its shape can still be adjusted by changing the values of the three parameters $k$, $s$, and $\alpha $.

**Remark**

**3.**

## 4. Composite Adjustable Bézier Curves

^{th}order geometric continuity (${G}^{k}$ continuity) conditions of the adjustable Bézier curves. In [1,16,17], the definition of ${G}^{k}$ continuity is given. Further, the practical Beta-constraints for the geometric continuity of curves are provided in [16,17]. According to the Beta-constraints, we have the following conclusion.

**Lemma**

**1.**

**Proof.**

**Proposition**

**3.**

**Remark**

**4.**

**Remark**

**5.**

## 5. Adjustable Bézier Curves with Tangent Polygon

^{th}segment of the polygon be

^{th}segment

^{th}segment of the composite curve is tangent to the polygon at ${\mathit{T}}_{i}$ and ${\mathit{T}}_{i+1}.$ Moreover, the i

^{th}and (i + 1)

^{th}curve segments are ${G}^{k}$ continuous, where $k=\mathrm{min}\{{k}_{i},{k}_{i+1}\}$.

**Remark**

**6.**

## 6. Conclusions

## Conflicts of Interest

## References

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**Figure 1.**Recursive evaluation process of an adjustable Bézier curve. The red circles, green triangles, blue stars, black stars, yellow triangles, purple rhombuses represent the points received in step 1 to step 6 of the recurrence, respectively. The black square represents the point received in the last step of recurrence.

**Figure 2.**The adjustable Bézier curves with the same parameters $k$ and $s$ but different $\alpha $. (

**A**) $k=2$, $s=1$; (

**B**) $k=s=3$.

**Figure 3.**The adjustable Bézier curves with the same parameters $k$ and $\alpha $ but different $s$. (

**A**) $k=4$, $\alpha ={\scriptscriptstyle \frac{1}{2}}$; (

**B**) $k=5$, $\alpha =1$.

**Figure 4.**The adjustable Bézier curves with the same parameters $s$ and $\alpha $ but different $k$. (

**A**) $s=1$, $\alpha ={\scriptscriptstyle \frac{1}{3}}$; (

**B**) $s=1$, $\alpha ={\scriptscriptstyle \frac{2}{3}}$.

**Figure 6.**Composite adjustable Bézier curves with the same continuity but different shape. (

**A**) $k=s=3$; (

**B**) the parameters are different from each other.

**Figure 7.**Adjustable Bézier curves tangential to the given polygon. (

**A**) $k=s=3$, $\alpha =0$; (

**B**) each segment with different parameters.

© 2016 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Yan, L.
Adjustable Bézier Curves with Simple Geometric Continuity Conditions. *Math. Comput. Appl.* **2016**, *21*, 44.
https://doi.org/10.3390/mca21040044

**AMA Style**

Yan L.
Adjustable Bézier Curves with Simple Geometric Continuity Conditions. *Mathematical and Computational Applications*. 2016; 21(4):44.
https://doi.org/10.3390/mca21040044

**Chicago/Turabian Style**

Yan, Lanlan.
2016. "Adjustable Bézier Curves with Simple Geometric Continuity Conditions" *Mathematical and Computational Applications* 21, no. 4: 44.
https://doi.org/10.3390/mca21040044