Dynamic Bézier Curves: New Findings on Reparameterization by Arc length

The Bézier curve is a special curve developed in the 1960s, which is not to be generated "directly" with classical mathematics, therefore it is also called "free-form curve". The Bézier mathematics makes it possible to generate a lot of different curves with simple parameterization. Under dynamic aspects of the CNC process, the Bézier curve developed for static application profiles leads to Shape-dependent fluctuations of the cutting speed in orders of magnitude from 1:3 to 1:10 - and above. In Analysis, the problem of leveling different support point distances - which is causal here -, is studied under the term Parameterization by Arc length. For Bézier curves, however, a generally valid solution has not yet been published due to mathematical preconditions. Based on this background, a generally valid approximation algorithm for the Reparameterization of Bézier curves according to Arc length and their Dynamic use on CNC machines, was developed.