A general framework for the optimal approximation of circular arcs by parametric polynomial curves
This work provides a theoretical and practical improvement for geometric modeling and CAD, where accurate circular arc approximation is needed, though it is incremental in nature.
The paper proposes a general framework for optimally approximating circular arcs with parametric polynomial curves, achieving the best known radial distance approximation for low-degree cases with closed-form solutions.
We propose a general framework for geometric approximation of circular arcs by parametric polynomial curves. The approach is based on constrained uniform approximation of an error function by scalar polynomials. The system of nonlinear equations for the unknown control points of the approximating polynomial given in the Bézier form is derived and a detailed analysis provided for some low degree cases which might be important in practice. At least for these cases the solutions can be, in principal, written in a closed form, and provide the best known approximants according to the radial distance. A general conjecture on the optimality of the solution is stated and several numerical examples conforming theoretical results are given.