TY - GEN

T1 - Surface representations using blossoms and buds

AU - Stefanus, Lim Yohanes

PY - 2008/9/9

Y1 - 2008/9/9

N2 - 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 (Xj(t),Yj(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.

AB - 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 (Xj(t),Yj(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.

KW - Algorithm

KW - Bivariate polynomial

KW - Blossom of a bivariate polynomial

KW - Bud of a bivariate polynomial

KW - Encoding-decoding

KW - Geometric design

KW - Sequence of representations

KW - Shape representation

KW - Surface representation

KW - Triangular Bézier patch

UR - http://www.scopus.com/inward/record.url?scp=50949086998&partnerID=8YFLogxK

U2 - 10.1109/SMI.2008.4547960

DO - 10.1109/SMI.2008.4547960

M3 - Conference contribution

AN - SCOPUS:50949086998

SN - 9781424422609

T3 - IEEE International Conference on Shape Modeling and Applications 2008, Proceedings, SMI

SP - 139

EP - 145

BT - IEEE International Conference on Shape Modeling and Applications 2008, Proceedings, SMI

T2 - IEEE International Conference on Shape Modeling and Applications 2008, SMI

Y2 - 4 June 2008 through 6 June 2008

ER -