Splitting piecewise cubic Bézier curves and the equivalence constants for some norms
Abstract
This article is concerned with splitting piecewise cubic B\'{e}zier curves and the equivalence relations for some norms defined through control points. Piecewise cubic B\'{e}zier curves are most common to approximate curves. These curves are established by control points. The norms $\Vert \cdot \Vert^{B_{N,3}}_p$ and $\Vert \cdot \Vert^{B_{2N,3}}_p$ on the space $B_{N,3}$ are determined through control points. With the purpose of keeping the degree of the curves and offering additional flexibility for curve design, we often split $N$-piece cubic B\'{e}zier curves to become $2N-$piece cubic B\'{e}zier curves. We will concentrate on the equivalence constants for the norm $\Vert \cdot \Vert^{B_{N,3}}_p$ and the norm $\Vert \cdot \Vert^{B_{2N,3}}_p$ on the space $B_{N,3}$ of $N$-piece cubic B\'{e}zier curves. So, we can use the norm $ \Vert\cdot \Vert^{B_{N,3} }_p$ to check the convergence for sequences of piecewise cubic B\'{e}zier curves. This result is important for applying piecewise cubic B\'{e}zier curves to detect optimal orbits.