Regular algebraic curve segments (I)—Definitions and characteristics

In this paper (part one of a trilogy), we introduce the concept of a discriminating family of regular algebraic curves (real, nonsingular and connected). Several discriminating families are obtained which yield different characterizations of the Bernstein–Bézier (BB) bivariate polynomials over the plane triangle and the quadrilateral domain such that their zero contours are smooth and connected. These regular curve segments in BB basis can be smoothly joined together to form algebraic curve splines or A-splines. Algorithms for the efficient graphics display of these new A-spline families are also provided.