On a generalization of Bernstein polynomials and Bézier curves based on umbral calculus

• Generalization of both Bernstein and Lagrange polynomials.
• Generalization of the Stancu and Goldman polynomials.
• Generalization of the de Casteljau algorithm in a new way.
• Efficient evaluation through a linear transformation of the control polygon.
• New results are illustrated with many pictures.

In Winkel (2001) a generalization of Bernstein polynomials and Bézier curves based on umbral calculus has been introduced. In the present paper we describe new geometric and algorithmic properties of this generalization including: (1) families of polynomials introduced by Stancu (1968) and Goldman (1985), i.e., families that include both Bernstein and Lagrange polynomial, are generalized in a new way, (2) a generalized de Casteljau algorithm is discussed, (3) an efficient evaluation of generalized Bézier curves through a linear transformation of the control polygon is described, (4) a simple criterion for endpoint tangentiality is established.