High-quality approximation of log-aesthetic curves based on the fourth-order derivative

We propose a new method for approximating log-aesthetic curves ${\boldsymbol C}_{\mathrm{LA}}$ using high-degree Bézier curves. By leveraging the property that higher order derivatives are more sensitive to the quality of approximation, the method minimizes an objective function based on the fourth-order derivative; consequently, ${\boldsymbol C}_{\mathrm{LA}}$ is approximated with high accuracy. In addition, the proposed method is composed of two steps to ensure stable optimization so as not to be negatively affected because of a local minimum and to evaluate the fourth-order derivative. Furthermore, we reveal the difficulty in sufficiently approximating ${\boldsymbol C}_{\mathrm{LA}}$ with Bézier curves from two aspects. One aspect entails the uncertainty of how accurately the low-degree Bézier curves can approximate ${\boldsymbol C}_{\mathrm{LA}}$. The other aspect is the existence of a subset of ${\boldsymbol C}_{\mathrm{LA}}$ that is inherently difficult to approximate with such conventional parametric curves. The experimental results and comparisons demonstrated the validity of the proposed method.