Multidegree reduction of disk Bézier curves with \(G^{0}\) and \(G^{1}\)continuity
 Abedallah Rababah^{1}Email author and
 Yusuf Fatihu Hamza^{2}
https://doi.org/10.1186/s136600150833y
© Rababah and Hamza 2015
Received: 17 February 2015
Accepted: 21 September 2015
Published: 30 September 2015
Abstract
In this paper, we propose methods to find a \(G^{k}\)multidegree reduction of disk Bézier curves for \(k=0,1\). The methods are based on degree reducing the center and radius curves using \(G^{k}\)continuity and minimizing the corresponding errors. Some examples and comparisons are given to illustrate the efficiency and simplicity of the proposed methods. The examples show that by using our proposed methods, we get \(G^{0}\), and \(G^{1}\)degree reductions, while having less errors than existing methods, which are without any continuity conditions.
Keywords
disk Bézier curves degree reduction \(G^{0}\)continuity \(G^{1}\)continuity1 Introduction and preliminaries
Lack of robustness is a fundamental issue in computer aided design and solid modeling. Taking disks in the plane as control points of Bézier curves is an appropriate approach toward solving this issue because it gives a Bézier curve with tolerance; see [1]. The degree reduction of curves is an important issue; in \(G^{k}\)degree reduction, we approximate a disk Bézier curve of degree n by a disk Bézier curve of degree m, \(m< n\), under the satisfaction of boundary conditions and minimum error requirement. The issue of degree reduction of Bézier curves has been tackled by many researchers; see [2–9]. Unlikely, degree reduction of disk Bézier curves has not been tackled by many researchers.
We end this section by addressing related preliminaries like defining the disk Bézier curves, the Gram matrix, and the delta operator.
A disk Bézier curve is defined as follows.
Definition 1
(Disk Bézier curves)
For \(n=m\), the matrix \(G_{m,m}\) is real, symmetric, and positive definite [13].
In the sections on \(G^{0}\), and \(G^{1}\)continuity, the submatrices of the Gram matrix \(G_{m,n}\) are defined. Each section uses the same notation for these submatrices, but with different dimensions. It should be clear that the form used within a section is the one defined within that section.
The delta operator Δ on the disks \((p_{i})\) is defined as follows: \(\Delta^{0} (p_{i}) = (p_{i})\), \(\Delta^{k} (p_{i}) = \Delta^{k1}(p_{i+1})  \Delta^{k1}(p_{i})\), \(k \geq1\), \(i=0,1 , \ldots,nk\).
For more on the disk and interval Bézier curves, see [14–20].
2 Geometric continuity of disk Bézier curves
\(G^{k}\)continuity furnishes the shape of the approximating curve with additional design parameters that are used in \(G^{k}\)degree reduction as additional parameters to reduce the error.
3 Degree reduction of disk Bézier curves
 (1)
\((P_{n})(t)\) and \((Q_{m})(t)\) are \(G^{k}\)continuous at the end disks, \(t=0,1\), for \(k=0,1\),
 (2)
the \(L_{2}\)error between \((P_{n})(t)\) and \((Q_{m})(t)\) is minimum, and
 (3)
\((P_{n})(t) \subseteq(Q_{m} )(t)\), \(0 \leq t \leq1\).
In the following sections, we investigate, in particular, the cases of \(G^{0}\), and \(G^{1}\)continuity with degree reduction of disk Bézier curves.
4 \(G^{0}\)Degree reduction
5 \(G^{1}\)Degree reduction
6 Examples and comparisons

longdashed: WBdegree reduction without any boundary condition,

shortdashed: \(G^{0}\)degree reduction,

dotted: \(G^{1}\)degree reduction.
Example 1
(see [15])
The methods in [15] of linear programming (LP1, LPM) and constrained linear programming (CLP1, CLPM) degree reductions give errors of 0.14, 0.25, 0.15, 0.18, respectively, while the proposed methods of WB, \(G^{0}\), and \(G^{1}\)degree reductions have errors of 0.04, 0.025, 0.029, respectively. This example shows that the methods proposed in this paper give better results than existing methods besides satisfying additional boundary conditions.
Example 2
(see [15])
The methods in [15] are based on linear programming (LP1, LPM) and constrained linear programming (CLP1, CLPM) degree reduction methods and give errors of 6.2, 6.2, 12.6, 11.6, respectively. They also approached the problem by making each control point of the degree reducing disk Bézier curve bound the original one and got an error of 70. The proposed methods in this paper with WB, \(G^{0}\), and \(G^{1}\)degree reductions have errors of 7, 5.1, 4.9, respectively.
Example 3
(see [17])
Example 4
(see [16])
Examples 14 show that the proposed WB, \(G^{0}\), \(G^{1}\)degree reduction methods in this paper give errors that are less than existing methods with and without continuity conditions; moreover, our methods are the first methods of this kind that consider geometric continuity with degree reductions.
Imposing boundary conditions consumes free parameters that can be used to minimize the error. That is, using the same method of degree reduction without boundary conditions gives less error than with boundary conditions.
7 Conclusions

Continuity conditions are considered, while most existing methods do not consider any boundary conditions.

Geometric conditions are considered with the method of degree reduction for the first time, which makes the methods novel and new.

The degree reduction is done for the center Bézier curve and the radius curve simultaneously, which minimizes the computational cost of degree reducing disk Bézier curves.

The numerical results show that our proposed methods have error less than existing methods besides the advantages mentioned above.

Existing methods are impractical because disk Bézier curves do not exist alone; they are pieces of splines and degree reducing them without boundary conditions gives a spline that is not continuous.
Declarations
Acknowledgements
The authors would like to thank the referees for valuable comments that lead to improve this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Lin, Q, Rokne, J: Disk Bézier curves. Comput. Aided Geom. Des. 15, 721737 (1998) MATHMathSciNetView ArticleGoogle Scholar
 Bogacki, P, Weinstein, SE, Xu, Y: Degree reduction of Bézier curves by uniform approximation with endpoint interpolation. Comput. Aided Des. 27(9), 651661 (1995) MATHView ArticleGoogle Scholar
 Brunnett, G, Schreiber, T, Braun, J: The geometry of optimal degree reduction of Bézier curves. Comput. Aided Geom. Des. 13, 773788 (1996) MATHMathSciNetView ArticleGoogle Scholar
 Eck, M: Least squares degree reduction of Bézier curves. Comput. Aided Des. 27(11), 845853 (1995) View ArticleGoogle Scholar
 Lachance, MA: Chebyshev economization for parametric surfaces. Comput. Aided Geom. Des. 5(3), 195205 (1988) MATHMathSciNetView ArticleGoogle Scholar
 Rababah, A, Lee, BG, Yoo, J: A simple matrix form for degree reduction of Bézier curves using ChebyshevBernstein basis transformations. Appl. Math. Comput. 181, 310318 (2006) MATHMathSciNetView ArticleGoogle Scholar
 Rababah, A, Lee, BG, Yoo, J: Multiple degree reduction and elevation of Bézier curves using JacobiBernstein basis transformations. Numer. Funct. Anal. Optim. 28(910), 11791196 (2007) MATHMathSciNetView ArticleGoogle Scholar
 Rababah, A, Mann, S: Iterative process for \(G^{2}\)multidegree reduction of Bézier curves. Appl. Math. Comput. 217(20), 81268133 (2011) MATHMathSciNetView ArticleGoogle Scholar
 Rababah, A, Mann, S: Linear methods for \(G^{1}\), \(G^{2}\), and \(G^{3}\)multidegree reduction of Bézier curves. Comput. Aided Des. 45, 405414 (2013) MathSciNetView ArticleGoogle Scholar
 Farin, G: Curves and Surfaces for Computer Aided Geometric Design. Academic Press, Boston (1993) Google Scholar
 Farouki, RT: The Bernstein polynomial basis: a centennial retrospective. Comput. Aided Geom. Des. 29, 379419 (2012) MATHMathSciNetView ArticleGoogle Scholar
 Farouki, RT, Goodman, TNT: On the optimal stability of the Bernstein basis. Math. Comput. 65, 15531566 (1996) MATHMathSciNetView ArticleGoogle Scholar
 Rababah, A: Distance for degree raising and reduction of triangular Bézier surfaces. J. Comput. Appl. Math. 158, 233241 (2003) MATHMathSciNetView ArticleGoogle Scholar
 Chen, F, Lou, W: Degree reduction of interval Bézier curves. Comput. Aided Des. 32(6), 571582 (2000) MATHView ArticleGoogle Scholar
 Chen, F, Yang, W: Degree reduction of disk Bézier curves. Comput. Aided Geom. Des. 21, 263280 (2004) MATHView ArticleGoogle Scholar
 Hu, Q, Wang, G: Multidegree reduction of disk Bézier curves in \(L_{2}\) norm. J. Inf. Comput. Sci. 7(5), 10451057 (2010) Google Scholar
 Jiang, P, Tan, J: Degree reduction of disk SaidBall curves. J. Comput. Inf. Syst. 1(3), 389398 (2005) Google Scholar
 Mudur, SP, Koparkar, PA: Interval methods for processing geometric objects. IEEE Comput. Graph. Appl. 4(2), 717 (1984) View ArticleGoogle Scholar
 Patrikalakis, NM: Robustness issues in geometric and solid modeling. Comput. Aided Des. 32, 629689 (2000) View ArticleGoogle Scholar
 Sederberg, TW, Farouki, RT: Approximation by interval Bézier curves. IEEE Comput. Graph. Appl. 15(2), 8795 (1992) View ArticleGoogle Scholar
 Rababah, A: Taylor theorem for planer curves. Proc. Am. Math. Soc. 119(3), 803810 (1993) MATHMathSciNetView ArticleGoogle Scholar
 Rababah, A: High order approximation method for curves. Comput. Aided Geom. Des. 12, 89102 (1995) MATHMathSciNetView ArticleGoogle Scholar
 Rababah, A: High accuracy Hermite approximation for space curves in \(\mathbb{R}^{d}\). J. Math. Anal. Appl. 325, 920931 (2007) MATHMathSciNetView ArticleGoogle Scholar