Monomial forms of two generalized ball curves and their proofs

There are several methods used to construct curves in CAGD, e.g., the de Casteljau's algorithm, Wang algorithm, Said-Ball algorithm. However, they are represented in the forms that are not suitable for performing geometric modeling. A proposed approach for evaluating Said-Ball and Wang-Ball curves by using monomial form was introduced by Aphirukmatakun and Dejdumrong in 2009. The authors did not provide any proofs for their propositions. Thus, this paper shows the proofs for the conversions from Said-Ball polynomials into its monomial form as well as that of the Wang-Ball polynomials.