New rational cubic trigonometric B-spline curves with two shape parameters
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New rational cubic trigonometric B-spline curves with two shape parameters Abdul Majeed1 · Faiza Qayyum1 Received: 30 December 2019 / Revised: 26 March 2020 / Accepted: 14 May 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract Trigonometric B-spline curves have gained a remarkable attention in computer-aided geometric design (CAGD). This paper presents the cubic and rational cubic trigonometric B-spline curves using new trigonometric functions and shape parameter η ∈ (1/2, 2]. The proposed curves inherit the basic properties of classical B-spline and have been proved. For uniform knots, both curves are C 2 continuous. On non-uniform knots, cubic trigonometric curves are C 3 and C 5 continuous, whereas rational trigonometric curves are C 3 continuous and have been derived. The applicability of proposed curves has been checked by constructing open and closed curves. Different models like glass, kettle, human hand, and vase have been designed by both schemes and compared. Keywords Trigonometric B-spline and rational trigonometric B-spline basis · Curves and its properties · Continuity of trigonometric B-spline and rational trigonometric B-spline · Application of trigonometric and rational B-spline Mathematics Subject Classification 65D17 · 68U07 · 65D10 · 65D18
1 Introduction With the advent of modern technologies in various fields and technology oriented business atmosphere, the usage of computers has become the essential need. To implement the algorithms and 3D designing of several products, computer-aided geometric design (CAGD) has played a vital role. CAGD is also called geometric modeling, which refers to the research stream for the development and representation of different forms of free form curves and surfaces. Different researchers have introduced variety of curves for modeling in different areas like Bosner and Rogina (2007) has introduced cycloidal splines to get the corner cutting algorithm.
Communicated by Antonio José Silva Neto.
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Abdul Majeed [email protected] Division of Science and Technology, Department of Mathematics, University of Education, Lahore, Pakistan 0123456789().: V,-vol
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A. Majeed, F. Qayyum
Rational Ball curves have been used in Majeed and Piah (2014) for image reconstruction. The nth-order uniform trigonometric B-spline with shape parameters is presented by Wang and Wang (2005). The basis functions used in this paper can be used to design circular and elliptic objects. In Yan (2016), class of non-uniform B-spline basis functions with local shape parameter is demonstrated. C 2 continuity attained for single knot, and for unique shape parameter, C 3 and C 5 continuity can be attained using proposed bases. Cubic trigonometric B-spline curve with two exponential shape parameters has been proposed by Zhu and Han (2015). The uniform cubic B-spline with shape parameter is introduced by Mishra (2016). Trigonometric spline with variable shape parameter is used Choubey and Ojha (2008). Using these bases, one can control
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