Monomial Forms for Curves in CAGD with their Applications

Abstract

There are several methods used for plotting curves in CAGD, e.g., by directly computing their basis functions (polynomials) or using their recursive algorithms. For the former method, evaluating a curve using their basis functions is a tedious task because their equations need to be solved by using complicated formulae computations. Whereas for the latter method, implementing a program by using recursive algorithm is simpler than the former method but it takes more computational time. Thus, an alternative method for constructing curves by using the monomial form is introduced. Employing monomial form approach, a curve can be computed by using monomial matrix operations. Because the matrix multiplications can be done in parallel programming, the performance of generating a curve for high degree can be increased. In the mean time, there exists the monomial functions for any degree Bezier curves. However, there has been no monomial functions for any other kinds of CAGD curves. This work proposes several monomial functions for Said-Ball, Wang-Ball, DP, Dejdumrong and NB1 curves. Consequently, these monomial functions will be useful and convenient for readily computing the derivatives, degree elevations, degree reductions and conversions among these curves.