A New Unified Model of Univariate and Bivariate Bases for Curves, Rectangular surfaces and Triangular surfaces Jaratpong Jangchai, Natasha Dejdumrong Department of Computer Engineering, King Mongkut’s University of Technology Thonburi, Bangkok, 10140, Thailand Emails: jaratpong.jan@kmutt.ac.th, natasha@cpe.kmutt.ac.th Abstract—In this paper, a new basis for polynomial curve modeling is presented with its linear computation. This new proposed curve can be formed by the convex combination of its blending functions and related control points. Moreover, several important geometric properties for this curve are identified, for examples, a partition of unity, convex hull property and symmetry. Later the recursive algorithm, coefficient matrix representation, the derivatives and the relationships between B´ ezier curve and this proposed curve are defined. Finally, a new proposed rectangular and triangular basis functions are also presented with their surface definitions. Keywords-B´ ezier curve, Said-Ball curve, Wang-Ball curve, DP curve, Dejdumrong curve, Linear complexity, Rectangular surfaces, Triangular surfaces. I. I NTRODUCTION In geometric modeling, there are several kinds of curves represented by different blending functions, e.g., B´ ezier - Bernstein, Ball, Volk-Schumaker (or abbrev. VS) [1], Said- Ball, Wang-Ball, Delgado-Pe˜ na (or abbrev. DP) [1], and NB1 polynomials. Some of them possess both univariate and bivariate polynomials for examples, Bernstein and Said-Ball polynomials [2,3] while the others provide only univariate functions. Particularly, Wang-Ball basis functions were gen- eralized from a cubic Ball basis by Hu et. al. [4]. It provided a recursive evaluation algorithm with linear computational complexity. It can be noticed that the Wang-Ball polynomials have coefficients with the multiplicity of two. Later in 2008, a recent basis was proposed by Dejdumrong.[5] with effi- cient algorithm and linear computational complexity. Like- wise, the coefficients of Dejdumrong’s polynomials have the multiplicity of three. However, both of them compute the curves relatively far from their control points. Although, the Wang-Ball basis has the least time computation for evaluating the curves but this curve is closed to its control points less than both B´ ezier and Dejdumrong curves. For Dejdumrong curves, it consumes a linear time complexity for generating its curves more than Wang-Ball basis while Dejdumrong curve is closer to its control point than Wang- Ball curve. In the other words, it is more shape preserving. Thus, it would be an inspiration to present a new curve modeling providing the coefficients of the multiplicity of 4. In addition, this model would offer an evaluation algorithm with linear computations. In this work, a new proposed curve representation is introduced with the most shape preserving property and linear computational complexity. The basis functions can be formed similarly to the notions of the linear interpolations of Wang-Ball and Dejdumrong basis functions. Moreover, sev- eral important geometric properties are investigated, i.e., the partition of unity, convex hull property, symmetry, recursive algorithm, coefficient matrix representation, the derivatives and the B´ ezier representation of this curve. Ultimately, a new proposed rectangular and triangular basis functions are also presented. II. NEW UNIVARIATE BASIS FUNCTIONS Given a sequence of n +1 control points, denoted by {d i } n i=0 , the proposed curve of degree n define by these points, can be expressed as D n (t)= n  i=0 D n i (t) · d i , 0 ≤ t ≤ 1. (1) where D n i (t) is a new blending function of this proposed basis function, denoted by D n i (t)= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ δ n i t i (1 − t) i+4 , for 0 ≤ i<  n 2  − 1, δ n i t i (1 − t) n−i , if i =  n 2  − 1 or i =  n 2  , D n n−1 (1 − t) , for  n+1 2  ≤ i ≤ n. (2) and δ n i is a coefficient of the polynomials of this basis function, calculated by δ n i = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 , if i = 0 or i = n, δ n−1 i−1 + δ n−1 i , for  n−1 2  ≤ i ≤  n+1 2  , δ n−1 i , if i<  n−1 2  , δ n−1 i−1 , if i>  n+1 2  . (3) For examples, the formulae for the curves of degree n = 3, 4, 5 and 6 can be shown as follows. 2009 Sixth International Conference on Computer Graphics, Imaging and Visualization 978-0-7695-3789-4/09 $25.00 © 2009 IEEE DOI 10.1109/CGIV.2009.75 222