Null Bézier Curves in Minkowski 3-Space

Abstract

In this paper, we define and investigate the properties of null Bézier curves in Minkowski 3-space. The method applied is a theoretical literature study, applying the definitions of Bézier curves and the geometric framework of null curves in semi-Riemannian geometry. We establish several fundamental characteristics of these curves, including the causal nature of their tangent vectors at endpoints and their Frenet frame apparatus when parametrized by pseudo-arc length. Furthermore, we define the concept of a null Bertrand pair for such curves and prove that if a null Bézier curve of degree n≥3 admits a Bertrand mate, then both curves are necessarily helices. Finally, we provide a conclusive parametric representation of any null Bézier curve in terms of a single non-constant function. This representation offers a powerful tool for explicitly constructing null Bézier curves within this geometric setting.