Surface representations using blossoms and buds

Shape representations using polynomials in computer-aided geometric design (CAGD) and computer graphics are ubiquitous. This paper shows that any bivariate polynomial p(t,u) of total degree d ≤ n can be represented in the form of a blossom of another bivariate polynomial b(t,u) of total degree d evaluated off the diagonal at the linear function pairs (X_j(t),Y_j(u)), j = 1, . . ., n, chosen under some conditions expressed in terms of symmetric functions. The bivariate polynomial b(t,u) is called a bud of the bivariate polynomial p(t,u). An algorithm for finding a bud b(t,u) of a given bivariate polynomial p(t,u) is presented. Successively, a bud of b(t,u) can be computed and so on, to form a sequence of representations. The information represented by the original bivariate polynomial is preserved in its buds. This scheme can be used for encoding/ decoding geometric design information. The objects in the encoding/decoding sequence can be rendered graphically and manipulated geometrically like the usual parametric representations. Examples concerning triangular Bézier patches are provided as illustrations.