# A New Quasi Cubic Rational System with Two Parameters

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

## 3. Main Results

#### 3.1. Construction of the QCR-B System

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

#### 3.2. QCR-Bézier Curve

**Definition**

**4.**

#### 3.3. Non-Uniform QCR-B Spline Curves and Their Applications

#### 3.3.1. Structure of the Non-Uniform QCR-B Spline System

**Definition**

**5.**

**Lemma**

**1.**

#### 3.3.2. Properties of the Non-Uniform QCR-B Spline System

- (1)
- Unity: For any $u\in [{u}_{3},{u}_{n+1}]$, ${\sum}_{i=0}^{n}{B}_{i}(u)=1$.

**Proof.**

- (2)
- Nonnegativity: For any ${u}_{i}<u<{u}_{i+4}$, there is ${B}_{i}(u)>0$.

**Proof.**

- (3)
- Symmetry: For any ${\alpha}_{i},{\beta}_{i}$, we have:$${B}_{i,n}(u)={B}_{n-i,n}(1-u)$$

- (4)
- Linear independence: For any ${\alpha}_{i},{\beta}_{i}\in [0,1]$, $\left\{{B}_{0}(u),{B}_{1}(u),\cdots ,{B}_{n}(u)\right\}$ is linear independence on interval $[{u}_{3},{u}_{n+1}]$.

**Proof.**

**Proof.**

- (6)
- Continuity: Given a non-uniform knot vector, for any ${\alpha}_{i}^{},{\beta}_{i}^{}\in [0,1]$, ${B}_{i}^{}(u)$ has ${C}_{}^{2}$ continuity at each knot.

**Proof.**

#### 3.4. Non-Uniform QCR-B Spline Curve

#### 3.4.1. Definition and Properties

**Definition**

**6.**

**Theorem**

**3.**

**Proof.**

#### 3.4.2. Local Adjustment

#### 3.5. QCR-BB System over Triangular Domain

#### 3.5.1. Construction of the QCR-BB System

**Definition**

**7.**

**Lemma**

**2.**

**Proof.**

#### 3.5.2. Properties of the QCR-BB System

- (1)
- Nonnegativity: For any $i,j,k\in N,i+j+k=3$, there are ${T}_{i,j,k}^{3}(\alpha ,\beta ,\gamma ;u,v,w)\ge 0.$
- (2)
- Unity:$${\sum}_{i+j+k=3}{T}_{i,j,k}^{3}}(\alpha ,\beta ,\gamma ;u,v,w)=1.$$
- (3)
- Symmetry:$$\begin{array}{l}{T}_{i,j,k}^{3}(\alpha ,\beta ,\gamma ;u,v,w)={T}_{i,k,j}^{3}(\alpha ,\gamma ,\beta ;u,w,v;)\\ ={T}_{j,k,i}^{3}(\beta ,\gamma ,\alpha ;v,w,u)={T}_{j,i,k}^{3}(\beta ,\alpha ,\gamma ;v,u,w;)\\ ={T}_{k,j,i}^{3}(\gamma ,\beta ,\alpha ;w,v,u)={T}_{k,i,j}^{3}(\gamma ,\alpha ,\beta ;w,u,v;).\end{array}$$
- (4)
- Boundary property:If $w=0$, the ten system ${T}_{i,j,k}^{3}(\alpha ,\beta ,\gamma ;u,v,w)$ described in Equation (14) could degenerate to the QCR-Bernstein system given in Equation (1).
- (5)
- Linear independence: ${T}_{i,j,k}^{3}(\alpha ,\beta ,\gamma ;u,v,w)$ are linear independence.

**Proof.**

#### 3.5.3. The QCR-BB Patch

**Definition**

**8.**

- (1)
- Convex hull and affine invariance: Given that the QCR-BB system has the unity and non-negativity, therefore, the QCR-BB patches have the property of convex hull and affine invariance.
- (2)
- Attribution of corner interpolation: By directly computing, that is:$$\begin{array}{l}R(0,0,1)={P}_{0,0,3},\\ R(0,1,0)={P}_{0,3,0},\\ R(1,0,0)={P}_{3,0,0}.\end{array}$$

- (3)
- Corner point tangent plane: Let $w=1-u-v$, it can:$$\begin{array}{l}{\frac{\delta R(u,v,w)}{\delta u}|}_{(1,0,0)}=(\alpha +3)({P}_{3,0,0}-{P}_{2,0,1}),\\ {\frac{\delta R(u,v,w)}{\delta v}|}_{(1,0,0)}=(\alpha +3)({P}_{2,1,0}-{P}_{2,0,1}),\\ {\frac{\delta R(u,v,w)}{\delta u}|}_{(0,1,0)}=(\beta +3)({P}_{1,2,0}-{P}_{2,0,1}),\\ {\frac{\delta R(u,v,w)}{\delta v}|}_{(0,1,0)}=(\beta +3)({P}_{0,3,0}-{P}_{2,0,1}),\\ {\frac{\delta R(u,v,w)}{\delta u}|}_{(0,0,1)}=(\gamma +3)({P}_{1,0,2}-{P}_{0,0,3}),\\ {\frac{\delta R(u,v,w)}{\delta v}|}_{(0,0,1)}=(\gamma +3)({P}_{0,1,2}-{P}_{0,0,3}).\end{array}$$

- (4)
- Boundary property: When $w=0$, $R(u,v,w)$ degenerated into a QCR-Bézier curve with two parameters $\alpha ,\beta $. When $u=0$, $R(u,v,w)$ degenerated into a QCR-Bézier curve with $\beta ,\gamma $. When $v=0$, $R(u,v,w)$ degenerated into a QCR-Bézier curve with $\alpha ,\gamma $. With the values of $\alpha ,\beta $, and $\gamma $ increasing, the QCR-BB patch will be approached to the control mesh. Hence, the parameters $\alpha ,\beta $, and $\gamma $ have a tension effect.
- (5)
- Shape adjustability: The shape of the $R(u,v,w)$ can be turned-up by modifying the value of the $\alpha ,\beta $, and $\gamma $ when the control mesh is stabled. With the values of $\alpha ,\beta $, and $\gamma $ increasing, the $R(u,v,w)$ would approach to the control mesh. Hence, it is easy for us to get the parameters $\alpha ,\beta $, and $\gamma $ to have a tension effect. According to the boundary property of the $R(u,v,w)$, each boundary curve $R(u,0,w),R(0,v,w)$, and $R(u,v,0)$ only have two related parameters. Thus, changing a parameter can only affect the shape of two boundary curves. Figure 7 shows the effects of different parameter values on the QCR-BB patch when the control mesh is fixed.

#### 3.5.4. De Casteljau-Type Algorithm

#### 3.5.5. Joining of QCR-BB Patches

**Theorem**

**4.**

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Prautzsch, H.; Piper, B. A fast algorithm to raise the degree of spline curves. Comput. Aided Geom. Des.
**1991**, 8, 253–265. [Google Scholar] [CrossRef] - Costantini, P. Curve and surface construction using variable degree polynomial splines. Comput. Aided Geom. Des.
**2000**, 17, 419–466. [Google Scholar] [CrossRef] - Juhász, I.; Hoffmann, M. On the quartic curve of Han. J. Comput. Appl. Math.
**2007**, 223, 123–132. [Google Scholar] [CrossRef] [Green Version] - Salomon, D.; Schneider, F.B. The Computer Graphics Manual; Springer: New York, NY, USA, 2011. [Google Scholar]
- Farin, G. Curves and Surfaces for CAGD: A Practical Guide, 5th ed.; Academic Press: San Diego, CA, USA, 2002; pp. 367–376. [Google Scholar]
- Oruc, H.; Phillips, G.H. q-Bernstein polynomials and Bézier curves. J. Comput. Math.
**2003**, 151, 1–12. [Google Scholar] [CrossRef] [Green Version] - Chen, J.; Wang, G.J. Progressive iterative approximation for triangular Bézier sufaces. Comput. Aided Des.
**2011**, 43, 889–895. [Google Scholar] [CrossRef] - Costantini, P.; Pelosi, F.; Sampoli, M.L. New spline spaces with generalized tension properties. BIT Numer. Math.
**2008**, 48, 665–688. [Google Scholar] [CrossRef] - Xu, G.; Wang, G.Z. Extended cubic uniform B-spline and the cubic trigonometric Bézier curve with two shape parameters. Acta Autom. Sin.
**2008**, 34, 980–984. [Google Scholar] [CrossRef] - Zhang, R.J. Uniform interpolation curves and surfaces based on a family of symmetric splines. Comput. Aided Geom. Des.
**2013**, 30, 844–860. [Google Scholar] [CrossRef] - Delgado-Gonzalo, R.; Thévenaz, P.; Unser, M. Exponential splines and minimal-support bases for curve representation. Comput. Aided Geom. Des.
**2011**, 29, 109–128. [Google Scholar] [CrossRef] [Green Version] - Chen, Q.Y.; Wang, G.Z. A class of Bézier-like curves. Comput. Aided Geom. Des.
**2003**, 20, 29–39. [Google Scholar] [CrossRef] - Tea-wan, K.; Boris, K. A shape-preserving apporximation by weighted cubic splines. J. Comput. Appl. Math.
**2012**, 236, 4383–4397. [Google Scholar] - Yan, L.L.; Han, X.L.; Huang, T. Cubic trigonometric polynomial curves and surfaces with a shape parameter. J. Comput. Aided Des. Comput. Graph.
**2016**, 28, 1047–1058. [Google Scholar] - Wu, X.Q.; Han, X.L. Extension of cubic Bézier curve. J. Eng. Graph.
**2005**, 6, 98–102. [Google Scholar] - Hu, G.; Ji, X.M.; Qin, X.Q. Quickly constuction of quartic λ-Bézier rotation surfaces with shape parameters. Trans. Chin. Soc. Agric. Mach.
**2014**, 45, 304–309. [Google Scholar] - Hu, G.; Ji, X.M.; Guo, L. Quartic generalized Bézier surfaces with multiple shape parameters and its continuity conditions. Trans. Chin. Soc. Agric. Mach.
**2014**, 45, 315–321. [Google Scholar] - Hu, G.; Song, W.J. New method for constructing quasi-Bézier rotation surfaces with multiple shape parameters. J. Xi’an Jiaotong Univ.
**2014**, 28, 74–79. [Google Scholar] - Guo, L.; A, L.S.; Hu, G. Designing car body with blended cubic Q-Bézier surfaces. Mech. Sci. Technol. Aerosp. Eng.
**2017**, 36, 114–118. [Google Scholar] - Juhász, I.; Róth, Á. A class of generalized B-spline curves. Comput. Aided Geom. Des.
**2013**, 30, 85–115. [Google Scholar] [CrossRef] - Mazure, M.L. On dimension elevation in Quasi Extended Chebyshev spaces. Numer. Math.
**2008**, 109, 459–475. [Google Scholar] [CrossRef] - Mazure, M.L. Quasi Extended Chebyshev spaces and weight functions. Numer. Math.
**2011**, 118, 79–108. [Google Scholar] [CrossRef] - Bosner, T.; Rogina, M. Variable degree polynomial splines are Chebyshev splines. Adv. Comput. Math.
**2013**, 38, 383–400. [Google Scholar] [CrossRef] - Carnicer, J.M.; Mainar, E.; Peña, J.M. Critical length for design purpose and extended Chebyshev spaces. Constr. Approx.
**2003**, 20, 55–71. [Google Scholar] [CrossRef] - Mazure, M.L. Which spaces for design? Numer. Math.
**2008**, 110, 357–392. [Google Scholar] [CrossRef] - Mazure, M.L. On a general new class of quasi Chebyshevian splines. Numer. Algorithms
**2011**, 58, 399–438. [Google Scholar] [CrossRef] - Carnicer, J.M.; Peña, J.M. Total positive bases for shape preserving curve design and optimality of B-splines. Comput. Aided Geom. Des.
**1994**, 11, 633–654. [Google Scholar] [CrossRef] - Mazure, M.L. Chebyshev spaces and Bernstein bases. Constr. Approx.
**2005**, 22, 347–363. [Google Scholar] [CrossRef] - 29. Bernstein. S.N. Démonstration du théoème de Weierstrass fondée sur le calcul des probabilités. Comm. Soc. Math. Kharkow
**1912**, 13, 1–2.

**Figure 2.**De Casteljau-type algorithm. (

**a**) The detailed process of the algorithm; (

**b**) QCR-Bézier curve when $\alpha =1,\beta =1,t=0.5$; (

**c**) QCR-Bézier curve when $\alpha =0,\beta =1,t=0.5$; (

**d**) QCR-Bézier curve when $\alpha =0,\beta =0,t=0.5$.

**Figure 4.**Different parameters correspond to the surface of revolution of the vase. (a) The surface of revolution of the vase when $\alpha =0,\beta =0$; (

**b**) The surface of revolution of the vase when $\alpha =1,\beta =1$.

**Figure 5.**Shape adjustable QCR-B spline curve. (

**a**) QCR-B spline curve when ${\alpha}_{i},{\beta}_{i}$ are different; (

**b**) Locally shape adjustable QCR-B spline curve.

© 2020 by the authors. 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/).

## Share and Cite

**MDPI and ACS Style**

Tuo, M.-X.; Zhang, G.-C.; Wang, K.
A New Quasi Cubic Rational System with Two Parameters. *Symmetry* **2020**, *12*, 1070.
https://doi.org/10.3390/sym12071070

**AMA Style**

Tuo M-X, Zhang G-C, Wang K.
A New Quasi Cubic Rational System with Two Parameters. *Symmetry*. 2020; 12(7):1070.
https://doi.org/10.3390/sym12071070

**Chicago/Turabian Style**

Tuo, Ming-Xiu, Gui-Cang Zhang, and Kai Wang.
2020. "A New Quasi Cubic Rational System with Two Parameters" *Symmetry* 12, no. 7: 1070.
https://doi.org/10.3390/sym12071070