Some remarks on Bézier curves
Parametric polynomial curves in Bézier-Bernstein representation are considered as projections of rational norm curves of degree n in n-space; from this point of view the singularities of a planar Bézier cubic are determined and expressed by its affine invariants. Secondly, for an arbitrary pair of adjacent parametric curves in homogeneous coordinates, the general conditions for geometric continuity of any order k, \(G^ k\), are established. This result generalizes the corresponding conditions in the non-homogeneous (affine) case, recently obtained by \textit{T. N. T. Goodman} [J. Approximation Theory 44, 132-153 (1984; Zbl 0569.41010)]. Some applications are given for Bézier curves. In particular, for \(\gamma\)-splines \textit{W. Boehm} [Comput. Aided Geom. Des. 4, 67-77 (1987; Zbl 0632.65005)] the existence of a global rational parameter that makes it to a \(C^ 2\) parametric curve is shown. Furthermore, for two adjacent rational Bézier curves the complete conditions for \(G^ 3\) are stated using the projective properties of the control points only.
scientific article; zbMATH DE number 4060017
scientific article