An optimization approach for biarc curve-fitting of B-spline curves

Abstract

We present an approach to the optimal fitting of a biarc-spline to a given B-spline curve. The objective is to minimize the area between the original B-spline curve and the fitted curve. Such an objective has obvious practical implications. This approach differs from conventional biarc curve-fitting techniques in two main aspects and has some desirable features. Firstly, it exploits the inherent freedom in the choice of the biarc that can be fitted to a given pair of end-points and their tangents. The conventional approach to biarc curve-fitting introduces additional constraints, such as the minimal difference in curvature or others to uniquely determine successive biarcs. In this approach, such constraints are not imposed. Instead, the freedom is exploited in the problem formulation to achieve a better fit. Secondly, the end-points do not lie on the curve so that appropriate tolerance control can be imposed through the use of additional constraints. Almost all previous biarc-fitting methods consider end-points that are on the original curve. As a result of these two aspects, the resulting biarc curve fits closely to the original curve with relatively fewer segments. This has a desirable effect on the surface finish, verification of CNC codes and memory requirement. Numerical results of the application of this approach to several examples are presented.