Nonlinear Weighted Average and Blossoming
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Nonlinear Weighted Average and Blossoming Rongin Uwitije1
· Xuhui Wang2 · Ammar Qarariyah1 · Jiansong Deng1
Received: 26 September 2019 / Revised: 9 December 2019 / Accepted: 8 January 2020 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we introduce a new averaging rule, the nonlinear weighted averaging rule. As an application, this averaging rule is used to replace the midpoint averaging in the de Casteljau evaluation algorithm and with this scheme we can also generate transcendental functions which cannot be generated by the classical de Casteljau algorithm. We also investigate the properties of the curves of the functions generated by blossoming, where the results show that these curves and the classical Bézier curves have some similar properties, including variation diminishing property and endpoint interpolation. However, the curves obtained by blossoming using nonlinear weighted averaging rules induced by certain functions violate some properties like convex hull property. Keywords Nonlinear weighted averaging rule · Midpoint averaging · De Casteljau algorithm · Bézier curves · Blossoming Mathematics Subject Classification 65D10
1 Introduction A Bézier curve is a parametric curve P(t) which is obtained from an approximation of a set of control points and this curve is always a polynomial function of a parameter t. This curve does not necessarily pass through all of its control points, it rather passes through its first and last control points. Further properties of a Bézier curve are well
B
Jiansong Deng [email protected] Rongin Uwitije [email protected]
1
School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui, People’s Republic of China
2
School of Mathematics, Hefei University of Technology, Hefei 230009, Anhui, People’s Republic of China
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R. Uwitije et al.
discussed in [3]. A Bézier curve generated by n + 1 control points is a degree-n polynomial function in t defined by: P(t) =
n
Pi Bi,n (t) ,
(1.1)
i=0
n i where the Bernstein polynomial function Bi,n (t) is given by Bi,n (t) = t i (1 − t)n−i and Pi are the control points. de Casteljau [8] developed an approach for generating the Bézier curve known as the de Casteljau algorithm which is based on repeated linear interpolation (see Fig. 1). The de Casteljau algorithm is one of the known linear recursive algorithms for generating smooth freeform curves and surfaces from piecewise linear functions by recursively averaging adjacent coefficient using arithmetic mean: A(x, y) =
x+y , 2
(1.2)
and in the limit this algorithm generates polynomial and piecewise polynomial functions. Referring to the algorithm of Paul de Casteljau and Ramshaw [19] developed another labeling scheme for generating Bézier curve. In his scheme, a degree-n polynomial in one variable t is transformed into a multivariate function f (u 1 , u 2 , . . . , u n ) in n variables with degree 1 in
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