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Research Papers

On Generalized Jacobi–Bernstein Basis Transformation: Application of Multidegree Reduction of Bézier Curves and Surfaces

[+] Author and Article Information
E. H. Doha

Department of Mathematics,
Faculty of Science,
Cairo University,
Giza 12613, Egypt
e-mail: eiddoha@frcu.eun.eg

A. H. Bhrawy

Department of Mathematics,
Faculty of Science,
King Abdulaziz University,
Jeddah 21589, Saudi Arabia
Department of Mathematics,
Faculty of Science,
Beni-Suef University,
Beni-Suef 62511, Egypt
e-mail: alibhrawy@yahoo.co.uk

M. A. Saker

Department of Basic Science,
Institute of Information Technology,
Modern Academy,
Cairo 11931, Egypt
e-mail: m_asaker@yahoo.com

Contributed by the Computers and Information Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received April 21, 2014; final manuscript received September 20, 2014; published online October 8, 2014. Assoc. Editor: Charlie C. L. Wang.

J. Comput. Inf. Sci. Eng 14(4), 041010 (Oct 08, 2014) (15 pages) Paper No: JCISE-14-1153; doi: 10.1115/1.4028633 History: Received April 21, 2014; Revised September 20, 2014

This paper formulates a new explicit expression for the generalized Jacobi polynomials (GJPs) in terms of Bernstein basis. We also establish and prove the basis transformation between the GJPs basis and Bernstein basis and vice versa. This transformation embeds the perfect least-square performance of the GJPs with the geometrical insight of the Bernstein form. Moreover, the GJPs with indexes corresponding to the number of endpoint constraints are the natural basis functions for least-square approximation of Bézier curves and surfaces. Application to multidegree reduction (MDR) of Bézier curves and surfaces in computer aided geometric design (CAGD) is given.

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Topics: Polynomials , Errors
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Figures

Grahic Jump Location
Fig. 1

Degree reduction of Bézier curve of Example 1: (a) using Jn0,0(x), (b) using Jn-1,-1(x), (c) using Jn-2,-2(x), and (d) errors for Jn0,0(x),Jn-1,-1(x), and Jn-2,-2(x)

Grahic Jump Location
Fig. 2

Degree reduction of Bézier curve of Example 2: (a) using Jn0,0(x), (b) using Jn-1,-1(x), (c) using Jn-2,-2(x), and (d) errors for Jn0,0(x),Jn-1,-1(x), and Jn-2,-2(x)

Grahic Jump Location
Fig. 3

Degree reduction of Bézier curve of Example 3: (a) using Jn0,0(x), (b) using Jn-1,-1(x), (c) using Jn-2,-2(x), and (d) errors for Jn0,0(x),Jn-1,-1(x), and Jn-2,-2(x)

Grahic Jump Location
Fig. 4

Degree reduction of Bézier curve of Example 4: (a) using Jn0,0(x), (b) using Jn-1,-1(x), (c) using Jn-2,-2(x), and (d) errors for Jn0,0(x),Jn-1,-1(x), and Jn-2,-2(x)

Grahic Jump Location
Fig. 5

MDR of Bézier Surface of Example 5: (a), (c), and (e) using Jn-1,-1(x), (b), (d), and (f) using Jn0,0(x)

Grahic Jump Location
Fig. 6

MDR of Bézier surface of Example 6: (a), (c), and (e) using Jn-1,-1(x), (b), (d), and (f) using Jn0,0(x)

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