Equivalence constants for some norms on the space of N-piece cubic Bézier curves

Abstract

This paper is concerned with norms of N-piece cubic Bézier curves and the equivalence relations between some norms on the space BN,3 of N-piece cubic Bézier curves. We introduce a norm ∥ · ∥Bp3 on the space B3 of cubic Bézier curves and a norm ∥·∥BpN,3 on the space BN,3 of N-piece cubic Bézier curves. This article we deal with the equivalence constants for the norm ∥ · ∥BpN,3 and the Lp norm on the space BN,3 of N-piece cubic Bézier curves. An N-piece Bézier curve of degree m can be considered as an N-piece Bézier curve of degree m + 1. We also study the equivalence constants for the norm ∥ · ∥BpN,m and the norm ∥ · ∥BpN,m+1 on the space BN,m of N-piece Bézier curves of degree m.